Fractional calculus

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class="vector-toc-numb">2</span> <span>Computing the fractional integral</span> </div> </a> <button aria-controls="toc-Computing_the_fractional_integral-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computing the fractional integral subsection</span> </button> <ul id="toc-Computing_the_fractional_integral-sublist" class="vector-toc-list"> <li id="toc-Riemann–Liouville_fractional_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemann–Liouville_fractional_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Riemann–Liouville fractional integral</span> </div> </a> <ul id="toc-Riemann–Liouville_fractional_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hadamard_fractional_integral" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hadamard_fractional_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Hadamard fractional integral</span> </div> </a> <ul id="toc-Hadamard_fractional_integral-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atangana–Baleanu_fractional_integral_(AB_fractional_integral)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atangana–Baleanu_fractional_integral_(AB_fractional_integral)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Atangana–Baleanu fractional integral (AB fractional integral)</span> </div> </a> <ul id="toc-Atangana–Baleanu_fractional_integral_(AB_fractional_integral)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fractional_derivatives" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Fractional_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Fractional derivatives</span> </div> </a> <button aria-controls="toc-Fractional_derivatives-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Fractional derivatives subsection</span> </button> <ul id="toc-Fractional_derivatives-sublist" class="vector-toc-list"> <li id="toc-Riemann–Liouville_fractional_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemann–Liouville_fractional_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Riemann–Liouville fractional derivative</span> </div> </a> <ul id="toc-Riemann–Liouville_fractional_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caputo_fractional_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caputo_fractional_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Caputo fractional derivative</span> </div> </a> <ul id="toc-Caputo_fractional_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caputo–Fabrizio_fractional_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caputo–Fabrizio_fractional_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Caputo–Fabrizio fractional derivative</span> </div> </a> <ul id="toc-Caputo–Fabrizio_fractional_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Atangana–Baleanu_fractional_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Atangana–Baleanu_fractional_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Atangana–Baleanu fractional derivative</span> </div> </a> <ul id="toc-Atangana–Baleanu_fractional_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riesz_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riesz_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Riesz derivative</span> </div> </a> <ul id="toc-Riesz_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_types" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Other types</span> </div> </a> <ul id="toc-Other_types-sublist" class="vector-toc-list"> <li id="toc-Coimbra_derivative" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Coimbra_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6.1</span> <span>Coimbra derivative</span> </div> </a> <ul id="toc-Coimbra_derivative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nature_of_the_fractional_derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nature_of_the_fractional_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Nature of the fractional derivative</span> </div> </a> <ul id="toc-Nature_of_the_fractional_derivative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Generalizations</span> </div> </a> <button aria-controls="toc-Generalizations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalizations subsection</span> </button> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> <li id="toc-Erdélyi–Kober_operator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Erdélyi–Kober_operator"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Erdélyi–Kober operator</span> </div> </a> <ul id="toc-Erdélyi–Kober_operator-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Functional_calculus" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Functional_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Functional calculus</span> </div> </a> <ul id="toc-Functional_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Fractional_conservation_of_mass" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractional_conservation_of_mass"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Fractional conservation of mass</span> </div> </a> <ul id="toc-Fractional_conservation_of_mass-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrochemical_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrochemical_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Electrochemical analysis</span> </div> </a> <ul id="toc-Electrochemical_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Groundwater_flow_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Groundwater_flow_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Groundwater flow problem</span> </div> </a> <ul id="toc-Groundwater_flow_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractional_advection_dispersion_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractional_advection_dispersion_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Fractional advection dispersion equation</span> </div> </a> <ul id="toc-Fractional_advection_dispersion_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Time-space_fractional_diffusion_equation_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Time-space_fractional_diffusion_equation_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Time-space fractional diffusion equation models</span> </div> </a> <ul id="toc-Time-space_fractional_diffusion_equation_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structural_damping_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Structural_damping_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Structural damping models</span> </div> </a> <ul id="toc-Structural_damping_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-PID_controllers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PID_controllers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>PID controllers</span> </div> </a> <ul id="toc-PID_controllers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Acoustic_wave_equations_for_complex_media" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Acoustic_wave_equations_for_complex_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.8</span> <span>Acoustic wave equations for complex media</span> </div> </a> <ul id="toc-Acoustic_wave_equations_for_complex_media-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fractional_Schrödinger_equation_in_quantum_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fractional_Schrödinger_equation_in_quantum_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.9</span> <span>Fractional Schrödinger equation in quantum theory</span> </div> </a> <ul id="toc-Fractional_Schrödinger_equation_in_quantum_theory-sublist" class="vector-toc-list"> <li id="toc-Variable-order_fractional_Schrödinger_equation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Variable-order_fractional_Schrödinger_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.9.1</span> <span>Variable-order fractional Schrödinger equation</span> </div> </a> <ul id="toc-Variable-order_fractional_Schrödinger_equation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle See also subsection</span> </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Other_fractional_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_fractional_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Other fractional theories</span> </div> </a> <ul id="toc-Other_fractional_theories-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Articles_regarding_the_history_of_fractional_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_regarding_the_history_of_fractional_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Articles regarding the history of fractional calculus</span> </div> </a> <ul id="toc-Articles_regarding_the_history_of_fractional_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.2</span> <span>Books</span> </div> </a> <ul id="toc-Books-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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Available in 16 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-16" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%81%D8%A7%D8%B6%D9%84_%D9%88%D8%AA%D9%83%D8%A7%D9%85%D9%84_%D9%83%D8%B3%D8%B1%D9%8A" title="تفاضل وتكامل كسري – Arabic" lang="ar" hreflang="ar" data-title="تفاضل وتكامل كسري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fraktionale_Infinitesimalrechnung" title="Fraktionale Infinitesimalrechnung – German" lang="de" hreflang="de" data-title="Fraktionale Infinitesimalrechnung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Murruline_diferentsiaal-_ja_integraalarvutus" title="Murruline diferentsiaal- ja integraalarvutus – Estonian" lang="et" hreflang="et" data-title="Murruline diferentsiaal- ja integraalarvutus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/C%C3%A1lculo_fraccional" title="Cálculo fraccional – Spanish" lang="es" hreflang="es" data-title="Cálculo fraccional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D8%B3%D8%A7%D8%A8_%DA%A9%D8%B3%D8%B1%DB%8C" title="حساب کسری – Persian" lang="fa" hreflang="fa" data-title="حساب کسری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_fractionnaire" title="Analyse fractionnaire – French" lang="fr" hreflang="fr" data-title="Analyse fractionnaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%EC%88%98%EA%B3%84_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="분수계 미적분학 – Korean" lang="ko" hreflang="ko" data-title="분수계 미적분학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AD%E0%A4%BF%E0%A4%A8%E0%A5%8D%E0%A4%A8%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%95%E0%A4%B2%E0%A4%A8" title="भिन्नात्मक कलन – Hindi" lang="hi" hreflang="hi" data-title="भिन्नात्मक कलन" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Calcolo_frazionario" title="Calcolo frazionario – Italian" lang="it" hreflang="it" data-title="Calcolo frazionario" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99_%D7%A9%D7%91%D7%A8%D7%99" title="חשבון אינפיניטסימלי שברי – Hebrew" lang="he" hreflang="he" data-title="חשבון אינפיניטסימלי שברי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%B0%E9%9A%8E%E5%BE%AE%E7%A9%8D%E5%88%86%E5%AD%A6" title="分数階微積分学 – Japanese" lang="ja" hreflang="ja" data-title="分数階微積分学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pochodna_u%C5%82amkowa" title="Pochodna ułamkowa – Polish" lang="pl" hreflang="pl" data-title="Pochodna ułamkowa" 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a href="/wiki/Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><br /><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a class="mw-selflink selflink">Fractional</a></li> <li><a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li> <li><a href="/wiki/Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li> <li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Miscellanea</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Precalculus" title="Precalculus">Precalculus</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">History</a></li> <li><a href="/wiki/Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li> <li><a href="/wiki/List_of_calculus_topics" title="List of calculus topics">List of topics</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li> <li><a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Calculus" title="Template:Calculus"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus" title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Fractional calculus</b> is a branch of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> that studies the several different possibilities of defining <a href="/wiki/Real_number" title="Real number">real number</a> powers or <a href="/wiki/Complex_number" title="Complex number">complex number</a> powers of the <a href="/wiki/Derivative" title="Derivative">differentiation</a> <a href="/wiki/Operator_(mathematics)" title="Operator (mathematics)">operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1c59da7454059970283a77a9a765100ce6a1f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.273ex; height:5.509ex;" alt="{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}"></span> </p><p>and of the <a href="/wiki/Integral" title="Integral">integration</a> operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>Note 1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d39490673c02687cc2d3df07b164dcca9636018" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.917ex; height:5.843ex;" alt="{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}"></span> </p><p>and developing a <a href="/wiki/Calculus" title="Calculus">calculus</a> for such operators generalizing the classical one. </p><p>In this context, the term <i>powers</i> refers to iterative application of a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> to a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>,</span> that is, repeatedly <a href="/wiki/Function_composition" title="Function composition">composing</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> with itself, as in <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}D^{n}(f)&=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)\\&=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots ))).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>D</mi> <mo>∘</mo> <mi>D</mi> <mo>∘</mo> <mi>D</mi> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <mi>D</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mo>⋯</mo> <mi>D</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}D^{n}(f)&=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)\\&=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots ))).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02e6ee53a72e5ef49af5ad57ec7ea2ad0453d184" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:34.177ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}D^{n}(f)&=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)\\&=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots ))).\end{aligned}}}"></span> </p><p>For example, one may ask for a meaningful interpretation of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {D}}=D^{\scriptstyle {\frac {1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>D</mi> </msqrt> </mrow> <mo>=</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {D}}=D^{\scriptstyle {\frac {1}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1448413c33ca0cbd5b40f11df0587cdd13b4320b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.619ex; height:3.843ex;" alt="{\displaystyle {\sqrt {D}}=D^{\scriptstyle {\frac {1}{2}}}}"></span> </p><p>as an analogue of the <a href="/wiki/Functional_square_root" title="Functional square root">functional square root</a> for the differentiation operator, that is, an expression for some linear operator that, when applied <em>twice</em> to any function, will have the same effect as <a href="/wiki/Derivative" title="Derivative">differentiation</a>. More generally, one can look at the question of defining a linear operator <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11984b2968ff1326b343a3401db1c5af97cfe92" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.026ex; height:2.343ex;" alt="{\displaystyle D^{a}}"></span> </p><p>for every real number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> in such a way that, when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> takes an <a href="/wiki/Integer" title="Integer">integer</a> value <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1cf6a513f2062531d95dbb198944936f312982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.786ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {Z} }"></span>,</span> it coincides with the usual <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-fold</span> differentiation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> if <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span>,</span> and with the <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-th</span> power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> when <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9f68fb8426c4411972241aac6c359e7812eaba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n<0}"></span>.</span> </p><p>One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is that the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> of operator powers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{D^{a}\mid a\in \mathbb {R} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>∣</mo> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{D^{a}\mid a\in \mathbb {R} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ddd07e3e3cebebecbe6fd8759a86c5b6e19573" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.037ex; height:2.843ex;" alt="{\displaystyle \{D^{a}\mid a\in \mathbb {R} \}}"></span> defined in this way are <i>continuous</i> <a href="/wiki/Semigroup" title="Semigroup">semigroups</a> with parameter <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>,</span> of which the original <i>discrete</i> semigroup of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7103633b3d6d0991222294fbdae4ab5ace36e85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.191ex; height:2.843ex;" alt="{\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}}"></span> for integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is a <a href="/wiki/Denumerable_set" class="mw-redirect" title="Denumerable set">denumerable</a> subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics. </p><p>Fractional <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>, also known as extraordinary differential equations,<sup id="cite_ref-Zwillinger2014_2-0" class="reference"><a href="#cite_note-Zwillinger2014-2"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> are a generalization of differential equations through the application of fractional calculus. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_notes">Historical notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=1" title="Edit section: Historical notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> and mathematical analysis, a <b>fractional derivative</b> is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to <a href="/wiki/Guillaume_de_l%27H%C3%B4pital" title="Guillaume de l'Hôpital">Guillaume de l'Hôpital</a> by <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a> in 1695.<sup id="cite_ref-Derivative_3-0" class="reference"><a href="#cite_note-Derivative-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>  Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2020)">citation needed</span></a></i>]</sup> </p><p>Fractional calculus was introduced in one of <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>'s early papers<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and the unified notation for differentiation and integration of arbitrary real order.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Independently, the foundations of the subject were laid by <a href="/wiki/Liouville" class="mw-redirect" title="Liouville">Liouville</a> in a paper from 1832.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> introduced the practical use of <a href="/wiki/Operational_calculus" title="Operational calculus">fractional differential operators</a> in electrical transmission line analysis circa 1890.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Computing_the_fractional_integral">Computing the fractional integral</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=2" title="Edit section: Computing the fractional integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>f</i>(<i>x</i>)</span> be a function defined for <span class="texhtml"><i>x</i> > 0</span>. Form the definite integral from 0 to <span class="texhtml mvar" style="font-style:italic;">x</span>. Call this <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Jf)(x)=\int _{0}^{x}f(t)\,dt\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>J</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Jf)(x)=\int _{0}^{x}f(t)\,dt\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a39a447696b8ce70c8cd8600133f878116656b9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.224ex; height:5.843ex;" alt="{\displaystyle (Jf)(x)=\int _{0}^{x}f(t)\,dt\,.}"></span> </p><p>Repeating this process gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(J^{2}f\right)(x)&=\int _{0}^{x}(Jf)(t)\,dt\\&=\int _{0}^{x}\left(\int _{0}^{t}f(s)\,ds\right)dt\,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>J</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(J^{2}f\right)(x)&=\int _{0}^{x}(Jf)(t)\,dt\\&=\int _{0}^{x}\left(\int _{0}^{t}f(s)\,ds\right)dt\,,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a28fbaeefa4ad370d51343035834f136c43f7ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.342ex; margin-bottom: -0.329ex; width:34.835ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\left(J^{2}f\right)(x)&=\int _{0}^{x}(Jf)(t)\,dt\\&=\int _{0}^{x}\left(\int _{0}^{t}f(s)\,ds\right)dt\,,\end{aligned}}}"></span> </p><p>and this can be extended arbitrarily. </p><p>The <a href="/wiki/Cauchy_formula_for_repeated_integration" title="Cauchy formula for repeated integration">Cauchy formula for repeated integration</a>, namely <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(J^{n}f\right)(x)={\frac {1}{(n-1)!}}\int _{0}^{x}\left(x-t\right)^{n-1}f(t)\,dt\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(J^{n}f\right)(x)={\frac {1}{(n-1)!}}\int _{0}^{x}\left(x-t\right)^{n-1}f(t)\,dt\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1fc6e89735bb0e11e01f2255bb03522bd5a5c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.099ex; height:6.176ex;" alt="{\displaystyle \left(J^{n}f\right)(x)={\frac {1}{(n-1)!}}\int _{0}^{x}\left(x-t\right)^{n-1}f(t)\,dt\,,}"></span> leads in a straightforward way to a generalization for real <span class="texhtml mvar" style="font-style:italic;">n</span>: using the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a> to remove the discrete nature of the factorial function gives us a natural candidate for applications of the fractional integral operator as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(J^{\alpha }f\right)(x)={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}\left(x-t\right)^{\alpha -1}f(t)\,dt\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(J^{\alpha }f\right)(x)={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}\left(x-t\right)^{\alpha -1}f(t)\,dt\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b50d2a48f75b14dda93d1d832b64169eedc4197" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.127ex; height:6.176ex;" alt="{\displaystyle \left(J^{\alpha }f\right)(x)={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}\left(x-t\right)^{\alpha -1}f(t)\,dt\,.}"></span> </p><p>This is in fact a well-defined operator. </p><p>It is straightforward to show that the <span class="texhtml mvar" style="font-style:italic;">J</span> operator satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&=\left(J^{\beta }\right)\left(J^{\alpha }f\right)(x)\\&=\left(J^{\alpha +\beta }f\right)(x)\\&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-t\right)^{\alpha +\beta -1}f(t)\,dt\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&=\left(J^{\beta }\right)\left(J^{\alpha }f\right)(x)\\&=\left(J^{\alpha +\beta }f\right)(x)\\&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-t\right)^{\alpha +\beta -1}f(t)\,dt\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5b3a3987bdf4997bb9ff0e5ed77be1725f0223" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:52.488ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&=\left(J^{\beta }\right)\left(J^{\alpha }f\right)(x)\\&=\left(J^{\alpha +\beta }f\right)(x)\\&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-t\right)^{\alpha +\beta -1}f(t)\,dt\,.\end{aligned}}}"></span> </p> <style data-mw-deduplicate="TemplateStyles:r1256386598">.mw-parser-output .cot-header-mainspace{background:#F0F2F5;color:inherit}.mw-parser-output .cot-header-other{background:#CCFFCC;color:inherit}@media screen{html.skin-theme-clientpref-night .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-night .mw-parser-output .cot-header-other{background:#003500;color:inherit}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cot-header-mainspace{background:#14181F;color:inherit}html.skin-theme-clientpref-os .mw-parser-output .cot-header-other{background:#003500;color:inherit}}</style> <div style="margin-left:0"> <table class="mw-collapsible mw-archivedtalk mw-collapsed" style="color:inherit; background: transparent; text-align: left; border: 1px solid Silver; margin: 0.2em auto auto; width:100%; clear: both; padding: 1px;"> <tbody><tr> <th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of this identity</div> </th></tr> <tr> <td style="color:inherit; border: solid 1px Silver; padding: 0.6em; background: var(--background-color-base, #fff);"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}(x-t)^{\alpha -1}\left(J^{\beta }f\right)(t)\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\int _{0}^{t}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}f(s)\,ds\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}f(s)\left(\int _{s}^{x}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}\,dt\right)\,ds\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}(x-t)^{\alpha -1}\left(J^{\beta }f\right)(t)\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\int _{0}^{t}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}f(s)\,ds\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}f(s)\left(\int _{s}^{x}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}\,dt\right)\,ds\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d341e8a5e8e65cc98fb84efbdb6bbeba32235fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.768ex; margin-bottom: -0.237ex; width:70.673ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}(x-t)^{\alpha -1}\left(J^{\beta }f\right)(t)\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\int _{0}^{t}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}f(s)\,ds\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}f(s)\left(\int _{s}^{x}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}\,dt\right)\,ds\end{aligned}}}"></span> </p><p>where in the last step we exchanged the order of integration and pulled out the <span class="texhtml"><i>f</i>(<i>s</i>)</span> factor from the <span class="texhtml mvar" style="font-style:italic;">t</span> integration. </p><p>Changing variables to <span class="texhtml mvar" style="font-style:italic;">r</span> defined by <span class="texhtml"><i>t</i> = <i>s</i> + (<i>x</i> − <i>s</i>)<i>r</i></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\left(\int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr\right)\,ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\left(\int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr\right)\,ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4e762156b4241d765e6042946c170fb8dc34b4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:77.198ex; height:6.509ex;" alt="{\displaystyle \left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\left(\int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr\right)\,ds}"></span> </p><p>The inner integral is the <a href="/wiki/Beta_function" title="Beta function">beta function</a> which satisfies the following property: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr=B(\alpha ,\beta )={\frac {\Gamma (\alpha )\,\Gamma (\beta )}{\Gamma (\alpha +\beta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>r</mi> <mo>=</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr=B(\alpha ,\beta )={\frac {\Gamma (\alpha )\,\Gamma (\beta )}{\Gamma (\alpha +\beta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d7ce1da3589b058ffc6876da2ef32089fdc775" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.317ex; height:6.509ex;" alt="{\displaystyle \int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr=B(\alpha ,\beta )={\frac {\Gamma (\alpha )\,\Gamma (\beta )}{\Gamma (\alpha +\beta )}}}"></span> </p><p>Substituting back into the equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\,ds\\&=\left(J^{\alpha +\beta }f\right)(x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>s</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>s</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </msup> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\,ds\\&=\left(J^{\alpha +\beta }f\right)(x)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/900cdbb1b27cddc46aefff3fe284d412fd0321ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:52.207ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\,ds\\&=\left(J^{\alpha +\beta }f\right)(x)\end{aligned}}}"></span> </p><p>Interchanging <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> shows that the order in which the <span class="texhtml mvar" style="font-style:italic;">J</span> operator is applied is irrelevant and completes the proof. </p> </td></tr></tbody></table></div> <p>This relationship is called the semigroup property of fractional <a href="/wiki/Differintegral" title="Differintegral">differintegral</a> operators. </p> <div class="mw-heading mw-heading3"><h3 id="Riemann–Liouville_fractional_integral"><span id="Riemann.E2.80.93Liouville_fractional_integral"></span>Riemann–Liouville fractional integral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=3" title="Edit section: Riemann–Liouville fractional integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The classical form of fractional calculus is given by the <a href="/wiki/Riemann%E2%80%93Liouville_integral" title="Riemann–Liouville integral">Riemann–Liouville integral</a>, which is essentially what has been described above. The theory of fractional integration for <a href="/wiki/Periodic_function" title="Periodic function">periodic functions</a> (therefore including the "boundary condition" of repeating after a period) is given by the <a href="/wiki/Weyl_integral" title="Weyl integral">Weyl integral</a>. It is defined on <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> whose integrals evaluate to zero). The Riemann–Liouville integral exists in two forms, upper and lower. Considering the interval <span class="texhtml">[<i>a</i>,<i>b</i>]</span>, the integrals are defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{-\alpha }}Df(t)&=\sideset {_{a}}{_{t}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau \\\sideset {_{t}}{_{b}^{-\alpha }}Df(t)&=\sideset {_{t}}{_{b}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{t}^{b}\left(\tau -t\right)^{\alpha -1}f(\tau )\,d\tau \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>I</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>τ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>I</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>τ</mi> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{-\alpha }}Df(t)&=\sideset {_{a}}{_{t}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau \\\sideset {_{t}}{_{b}^{-\alpha }}Df(t)&=\sideset {_{t}}{_{b}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{t}^{b}\left(\tau -t\right)^{\alpha -1}f(\tau )\,d\tau \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb153742f9f7427df87cc3696cf830833982eb2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:40.299ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{-\alpha }}Df(t)&=\sideset {_{a}}{_{t}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau \\\sideset {_{t}}{_{b}^{-\alpha }}Df(t)&=\sideset {_{t}}{_{b}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{t}^{b}\left(\tau -t\right)^{\alpha -1}f(\tau )\,d\tau \end{aligned}}}"></span> </p><p>Where the former is valid for <span class="texhtml"><i>t</i> > <i>a</i></span> and the latter is valid for <span class="texhtml"><i>t</i> < <i>b</i></span>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>It has been suggested<sup id="cite_ref-Mainardi_12-0" class="reference"><a href="#cite_note-Mainardi-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> that the integral on the positive real axis (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d476e5e765a5d77bbcff32e4584579207ec7d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=0}"></span>) would be more appropriately named the Abel–Riemann integral, on the basis of history of discovery and use, and in the same vein the integral over the entire real line be named Liouville–Weyl integral. </p><p>By contrast the <a href="/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative" title="Grünwald–Letnikov derivative">Grünwald–Letnikov derivative</a> starts with the derivative instead of the integral. </p> <div class="mw-heading mw-heading3"><h3 id="Hadamard_fractional_integral">Hadamard fractional integral</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=4" title="Edit section: Hadamard fractional integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>Hadamard fractional integral</i> was introduced by <a href="/wiki/Jacques_Hadamard" title="Jacques Hadamard">Jacques Hadamard</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and is given by the following formula, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {_{a}}{_{t}^{-\alpha }}{\mathbf {D} }f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(\log {\frac {t}{\tau }}\right)^{\alpha -1}f(\tau ){\frac {d\tau }{\tau }},\qquad t>a\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>τ</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> <mi>τ</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>t</mi> <mo>></mo> <mi>a</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sideset {_{a}}{_{t}^{-\alpha }}{\mathbf {D} }f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(\log {\frac {t}{\tau }}\right)^{\alpha -1}f(\tau ){\frac {d\tau }{\tau }},\qquad t>a\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022f1350040ca31ff704e2809eeac34a1c216ea8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.018ex; height:6.676ex;" alt="{\displaystyle \sideset {_{a}}{_{t}^{-\alpha }}{\mathbf {D} }f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(\log {\frac {t}{\tau }}\right)^{\alpha -1}f(\tau ){\frac {d\tau }{\tau }},\qquad t>a\,.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Atangana–Baleanu_fractional_integral_(AB_fractional_integral)"><span id="Atangana.E2.80.93Baleanu_fractional_integral_.28AB_fractional_integral.29"></span>Atangana–Baleanu fractional integral (AB fractional integral)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=5" title="Edit section: Atangana–Baleanu fractional integral (AB fractional integral)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Atangana–Baleanu fractional integral of a continuous function is defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {_{{\hphantom {A}}a}^{\operatorname {AB} }}{_{t}^{\alpha }}If(t)={\frac {1-\alpha }{\operatorname {AB} (\alpha )}}f(t)+{\frac {\alpha }{\operatorname {AB} (\alpha )\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>I</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mphantom> <mi>A</mi> </mphantom> </mpadded> </mrow> </mrow> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>AB</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mrow> <mi>AB</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mi>AB</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>τ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sideset {_{{\hphantom {A}}a}^{\operatorname {AB} }}{_{t}^{\alpha }}If(t)={\frac {1-\alpha }{\operatorname {AB} (\alpha )}}f(t)+{\frac {\alpha }{\operatorname {AB} (\alpha )\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27768c8874c4fc99fe58bc7d51741a556598594" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.951ex; height:6.509ex;" alt="{\displaystyle \sideset {_{{\hphantom {A}}a}^{\operatorname {AB} }}{_{t}^{\alpha }}If(t)={\frac {1-\alpha }{\operatorname {AB} (\alpha )}}f(t)+{\frac {\alpha }{\operatorname {AB} (\alpha )\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau }"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Fractional_derivatives">Fractional derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=6" title="Edit section: Fractional derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Fractal_derivative" title="Fractal derivative">Fractal derivative</a>.</div> <p>Unfortunately, the comparable process for the derivative operator <span class="texhtml mvar" style="font-style:italic;">D</span> is significantly more complex, but it can be shown that <span class="texhtml mvar" style="font-style:italic;">D</span> is neither <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> nor <a href="/wiki/Additive_map" title="Additive map">additive</a> in general.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Unlike classical Newtonian derivatives, fractional derivatives can be defined in a variety of different ways that often do not all lead to the same result even for smooth functions. Some of these are defined via a fractional integral. Because of the incompatibility of definitions, it is frequently necessary to be explicit about which definition is used. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fractionalderivative.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Fractionalderivative.gif/220px-Fractionalderivative.gif" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Fractionalderivative.gif/330px-Fractionalderivative.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/d/dd/Fractionalderivative.gif 2x" data-file-width="360" data-file-height="222" /></a><figcaption>Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Riemann–Liouville_fractional_derivative"><span id="Riemann.E2.80.93Liouville_fractional_derivative"></span>Riemann–Liouville fractional derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=7" title="Edit section: Riemann–Liouville fractional derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the <span class="texhtml mvar" style="font-style:italic;">α</span>th order derivative, the <span class="texhtml mvar" style="font-style:italic;">n</span>th order derivative of the integral of order <span class="texhtml">(<i>n</i> − <i>α</i>)</span> is computed, where <span class="texhtml mvar" style="font-style:italic;">n</span> is the smallest integer greater than <span class="texhtml mvar" style="font-style:italic;">α</span> (that is, <span class="texhtml"><i>n</i> = ⌈<i>α</i>⌉</span>). The Riemann–Liouville fractional derivative and integral has multiple applications such as in case of solutions to the equation in the case of multiple systems such as the tokamak systems, and Variable order fractional parameter.<sup id="cite_ref-Mostafanejad_15-0" class="reference"><a href="#cite_note-Mostafanejad-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Al-Raeei_16-0" class="reference"><a href="#cite_note-Al-Raeei-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{n-\alpha }}If(t)\\\sideset {_{t}}{_{b}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{n-\alpha }}If(t)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>I</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-OP"> <msub> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>I</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{n-\alpha }}If(t)\\\sideset {_{t}}{_{b}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{n-\alpha }}If(t)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ac7dd211f711572b5a519ac76a5e5ea816cef8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:29.645ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{n-\alpha }}If(t)\\\sideset {_{t}}{_{b}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{n-\alpha }}If(t)\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Caputo_fractional_derivative">Caputo fractional derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=8" title="Edit section: Caputo fractional derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Caputo_fractional_derivative" title="Caputo fractional derivative">Caputo fractional derivative</a></div> <p>Another option for computing fractional derivatives is the <a href="/wiki/Caputo_fractional_derivative" title="Caputo fractional derivative">Caputo fractional derivative</a>. It was introduced by <a href="/w/index.php?title=Michele_Caputo&action=edit&redlink=1" class="new" title="Michele Caputo (page does not exist)">Michele Caputo</a> in his 1967 paper.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In contrast to the Riemann–Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows, where again <span class="texhtml"><i>n</i> = ⌈<i>α</i>⌉</span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {^{C}}{_{t}^{\alpha }}Df(t)={\frac {1}{\Gamma (n-\alpha )}}\int _{0}^{t}{\frac {f^{(n)}(\tau )}{\left(t-\tau \right)^{\alpha +1-n}}}\,d\tau .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <msup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>τ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sideset {^{C}}{_{t}^{\alpha }}Df(t)={\frac {1}{\Gamma (n-\alpha )}}\int _{0}^{t}{\frac {f^{(n)}(\tau )}{\left(t-\tau \right)^{\alpha +1-n}}}\,d\tau .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1d0e67e4046b630424896df8d353ca8031bd1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.229ex; height:7.343ex;" alt="{\displaystyle \sideset {^{C}}{_{t}^{\alpha }}Df(t)={\frac {1}{\Gamma (n-\alpha )}}\int _{0}^{t}{\frac {f^{(n)}(\tau )}{\left(t-\tau \right)^{\alpha +1-n}}}\,d\tau .}"></span> </p><p>There is the Caputo fractional derivative defined as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{\nu }f(t)={\frac {1}{\Gamma (n-\nu )}}\int _{0}^{t}(t-u)^{(n-\nu -1)}f^{(n)}(u)\,du\qquad (n-1)<\nu <n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>u</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>ν</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo><</mo> <mi>ν</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\nu }f(t)={\frac {1}{\Gamma (n-\nu )}}\int _{0}^{t}(t-u)^{(n-\nu -1)}f^{(n)}(u)\,du\qquad (n-1)<\nu <n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b1a78fb1f0fb431ce8204decf418f3b369f595" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:67.429ex; height:6.509ex;" alt="{\displaystyle D^{\nu }f(t)={\frac {1}{\Gamma (n-\nu )}}\int _{0}^{t}(t-u)^{(n-\nu -1)}f^{(n)}(u)\,du\qquad (n-1)<\nu <n}"></span> which has the advantage that is zero when <span class="texhtml"><i>f</i>(<i>t</i>)</span> is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sideset {_{a}^{b}}{^{n}u}Df(t)&=\int _{a}^{b}\phi (\nu )\left[D^{(\nu )}f(t)\right]\,d\nu \\&=\int _{a}^{b}\left[{\frac {\phi (\nu )}{\Gamma (1-\nu )}}\int _{0}^{t}\left(t-u\right)^{-\nu }f'(u)\,du\right]\,d\nu \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-OP"> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <msup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⁡</mo> <mi>u</mi> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ν</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>u</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> </mrow> </msup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>ν</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sideset {_{a}^{b}}{^{n}u}Df(t)&=\int _{a}^{b}\phi (\nu )\left[D^{(\nu )}f(t)\right]\,d\nu \\&=\int _{a}^{b}\left[{\frac {\phi (\nu )}{\Gamma (1-\nu )}}\int _{0}^{t}\left(t-u\right)^{-\nu }f'(u)\,du\right]\,d\nu \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d42263d28b3965e20b9d49daadc8625ce5432e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:54.267ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}\sideset {_{a}^{b}}{^{n}u}Df(t)&=\int _{a}^{b}\phi (\nu )\left[D^{(\nu )}f(t)\right]\,d\nu \\&=\int _{a}^{b}\left[{\frac {\phi (\nu )}{\Gamma (1-\nu )}}\int _{0}^{t}\left(t-u\right)^{-\nu }f'(u)\,du\right]\,d\nu \end{aligned}}}"></span> </p><p>where <span class="texhtml"><i>ϕ</i>(<i>ν</i>)</span> is a weight function and which is used to represent mathematically the presence of multiple memory formalisms. </p> <div class="mw-heading mw-heading3"><h3 id="Caputo–Fabrizio_fractional_derivative"><span id="Caputo.E2.80.93Fabrizio_fractional_derivative"></span>Caputo–Fabrizio fractional derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=9" title="Edit section: Caputo–Fabrizio fractional derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}"></span> given by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {_{{\hphantom {C}}a}^{\text{CF}}}{_{t}^{\alpha }}Df(t)={\frac {1}{1-\alpha }}\int _{a}^{t}f'(\tau )\ e^{\left(-\alpha {\frac {t-\tau }{1-\alpha }}\right)}\ d\tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mphantom> <mi>C</mi> </mphantom> </mpadded> </mrow> </mrow> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>CF</mtext> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>t</mi> <mo>−</mo> <mi>τ</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mtext> </mtext> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sideset {_{{\hphantom {C}}a}^{\text{CF}}}{_{t}^{\alpha }}Df(t)={\frac {1}{1-\alpha }}\int _{a}^{t}f'(\tau )\ e^{\left(-\alpha {\frac {t-\tau }{1-\alpha }}\right)}\ d\tau ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c88582ef182d0b6f0961ce2ff39638a1537421" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.364ex; height:6.176ex;" alt="{\displaystyle \sideset {_{{\hphantom {C}}a}^{\text{CF}}}{_{t}^{\alpha }}Df(t)={\frac {1}{1-\alpha }}\int _{a}^{t}f'(\tau )\ e^{\left(-\alpha {\frac {t-\tau }{1-\alpha }}\right)}\ d\tau ,}"></span> </p><p>where <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<0,\alpha \in (0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<0,\alpha \in (0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191e796c1152367c593e26bced3b6bf24171e6b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.763ex; height:2.843ex;" alt="{\displaystyle a<0,\alpha \in (0,1]}"></span>.</span><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Atangana–Baleanu_fractional_derivative"><span id="Atangana.E2.80.93Baleanu_fractional_derivative"></span>Atangana–Baleanu fractional derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=10" title="Edit section: Atangana–Baleanu fractional derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2016, Atangana and Baleanu suggested differential operators based on the generalized <a href="/wiki/Mittag-Leffler_function" title="Mittag-Leffler function">Mittag-Leffler function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236b689dce03c17d59033e746b4e17860242cfe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle E_{\alpha }}"></span>. The aim was to introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in Riemann–Liouville sense and Caputo sense respectively. For a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd24bae0d7570018e828e19851902c09c618af91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.676ex;" alt="{\displaystyle C^{1}}"></span> given by <sup id="cite_ref-Algahtani2016_20-0" class="reference"><a href="#cite_note-Algahtani2016-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-doiserbia.nb.rs_21-0" class="reference"><a href="#cite_note-doiserbia.nb.rs-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}\int _{a}^{t}f'(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mphantom> <mi>A</mi> <mi>B</mi> </mphantom> </mpadded> </mrow> </mrow> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ABC</mtext> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>AB</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}\int _{a}^{t}f'(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2238bc8633fea409c150e04ff7c08e4e4fe815b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.019ex; height:6.343ex;" alt="{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}\int _{a}^{t}f'(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}"></span> </p><p>If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}{\frac {d}{dt}}\int _{a}^{t}f(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mrow class="MJX-TeXAtom-ORD"> <mphantom> <mi>D</mi> </mphantom> </mrow> </mpadded> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded height="0" depth="0"> <mphantom> <mi>A</mi> <mi>B</mi> </mphantom> </mpadded> </mrow> </mrow> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>ABC</mtext> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <msubsup> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mrow> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>AB</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}{\frac {d}{dt}}\int _{a}^{t}f(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/014c2d10b482e4b3265c9896dddd9b79ca20fd38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.184ex; height:6.343ex;" alt="{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(t)={\frac {\operatorname {AB} (\alpha )}{1-\alpha }}{\frac {d}{dt}}\int _{a}^{t}f(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}"></span> </p><p>The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function.  For example, for all <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in (0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in (0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea174265d33ba6eed47a00e6ebffedda57cb03e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.239ex; height:2.843ex;" alt="{\displaystyle \alpha \in (0,1]}"></span>,</span> the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236b689dce03c17d59033e746b4e17860242cfe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.999ex; height:2.509ex;" alt="{\displaystyle E_{\alpha }}"></span> is increasing on the real line, converges to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> in <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span>,</span> and <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\alpha }(0)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\alpha }(0)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292e416fd870311ff5a6c8f5ac4cc1c65ac5838a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.232ex; height:2.843ex;" alt="{\displaystyle E_{\alpha }(0)=1}"></span>.</span> Therefore, we have that, the function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76faf9a5c1222228b6b2f50367598f42711d6ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.177ex; height:2.843ex;" alt="{\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}"></span> is the cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples is called a <a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler distribution</a> of order <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>.</span> It is also very well-known that, all these probability distributions are <a href="/wiki/Absolute_continuity" title="Absolute continuity">absolutely continuous</a>. In particular, the function Mittag-Leffler has a particular case <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}"></span>,</span> which is the exponential function, the Mittag-Leffler distribution of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> is therefore an <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential distribution</a>. However, for <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \alpha \in (0,1)}"></span>,</span> the Mittag-Leffler distributions are <a href="/wiki/Heavy-tailed_distribution" title="Heavy-tailed distribution">heavy-tailed</a>. Their Laplace transform is given by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">E</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0afa259794949bf4db5ea3d9023c0250df7a359" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.515ex; height:5.509ex;" alt="{\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},}"></span> </p><p>This directly implies that, for <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df576f7940384416d1553ab063704d37bf99420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.496ex; height:2.843ex;" alt="{\displaystyle \alpha \in (0,1)}"></span>,</span> the expectation is infinite. In addition, these distributions are <a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">geometric stable distributions</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Riesz_derivative">Riesz derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=11" title="Edit section: Riesz derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Riesz derivative is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\left\{{\frac {\partial ^{\alpha }u}{\partial \left|x\right|^{\alpha }}}\right\}(k)=-\left|k\right|^{\alpha }{\mathcal {F}}\{u\}(k),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msup> <mrow> <mo>|</mo> <mi>k</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>u</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\left\{{\frac {\partial ^{\alpha }u}{\partial \left|x\right|^{\alpha }}}\right\}(k)=-\left|k\right|^{\alpha }{\mathcal {F}}\{u\}(k),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35d44d8d559713d031a60a8898412dd8cd33f10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.213ex; height:6.343ex;" alt="{\displaystyle {\mathcal {F}}\left\{{\frac {\partial ^{\alpha }u}{\partial \left|x\right|^{\alpha }}}\right\}(k)=-\left|k\right|^{\alpha }{\mathcal {F}}\{u\}(k),}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> denotes the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_types">Other types</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=12" title="Edit section: Other types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Classical fractional derivatives include: </p> <ul><li><a href="/wiki/Gr%C3%BCnwald%E2%80%93Letnikov_derivative" title="Grünwald–Letnikov derivative">Grünwald–Letnikov derivative</a><sup id="cite_ref-deOliveira2014_24-0" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Aslan2015_25-0" class="reference"><a href="#cite_note-Aslan2015-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li>Sonin–Letnikov derivative<sup id="cite_ref-Aslan2015_25-1" class="reference"><a href="#cite_note-Aslan2015-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li>Liouville derivative<sup id="cite_ref-deOliveira2014_24-1" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Differintegral" title="Differintegral">Caputo derivative</a><sup id="cite_ref-deOliveira2014_24-2" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>Hadamard derivative<sup id="cite_ref-deOliveira2014_24-3" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li> <li>Marchaud derivative<sup id="cite_ref-deOliveira2014_24-4" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>Riesz derivative<sup id="cite_ref-Aslan2015_25-2" class="reference"><a href="#cite_note-Aslan2015-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li>Miller–Ross derivative<sup id="cite_ref-deOliveira2014_24-5" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Weyl_integral" title="Weyl integral">Weyl derivative</a><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-deOliveira2014_24-6" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Erdelyi%E2%80%93Kober_operator" title="Erdelyi–Kober operator">Erdélyi–Kober derivative</a><sup id="cite_ref-deOliveira2014_24-7" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/w/index.php?title=Fractal_calculus&action=edit&redlink=1" class="new" title="Fractal calculus (page does not exist)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81fb943af52f056cca6657ad9702de8fc4603f02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:2.343ex;" alt="{\displaystyle F^{\alpha }}"></span>-derivative</a><sup id="cite_ref-Ali_29-0" class="reference"><a href="#cite_note-Ali-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li></ul> <p>New fractional derivatives include: </p> <ul><li>Coimbra derivative<sup id="cite_ref-deOliveira2014_24-8" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Katugampola_fractional_operators" title="Katugampola fractional operators">Katugampola derivative</a><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li>Hilfer derivative<sup id="cite_ref-deOliveira2014_24-9" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>Davidson derivative<sup id="cite_ref-deOliveira2014_24-10" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>Chen derivative<sup id="cite_ref-deOliveira2014_24-11" class="reference"><a href="#cite_note-deOliveira2014-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li><a href="/w/index.php?title=Caputo_Fabrizio_derivative&action=edit&redlink=1" class="new" title="Caputo Fabrizio derivative (page does not exist)">Caputo Fabrizio derivative</a><sup id="cite_ref-Algahtani2016_20-1" class="reference"><a href="#cite_note-Algahtani2016-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li> <li>Atangana–Baleanu derivative<sup id="cite_ref-Algahtani2016_20-2" class="reference"><a href="#cite_note-Algahtani2016-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-doiserbia.nb.rs_21-1" class="reference"><a href="#cite_note-doiserbia.nb.rs-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="Coimbra_derivative">Coimbra derivative</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=13" title="Edit section: Coimbra derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Coimbra derivative</b> is used for physical modeling:<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <pre>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(t)<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbf2894b0a8cbcfcb8deae1d39475604208bdec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.979ex; height:2.843ex;" alt="{\displaystyle q(t)<1}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}^{\mathbb {C} }_{a}\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [1-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{-q(t)}{\frac {d\,f(\tau )}{d\tau }}d\tau \,+\,{\frac {(f(0^{+})-f(0^{-}))\,t^{-q(t)}}{\Gamma (1-q(t))}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">[</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>τ</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}^{\mathbb {C} }_{a}\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [1-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{-q(t)}{\frac {d\,f(\tau )}{d\tau }}d\tau \,+\,{\frac {(f(0^{+})-f(0^{-}))\,t^{-q(t)}}{\Gamma (1-q(t))}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86ef386ed041f1f311995e8445ae3966aae5c81" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:75.591ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}^{\mathbb {C} }_{a}\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [1-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{-q(t)}{\frac {d\,f(\tau )}{d\tau }}d\tau \,+\,{\frac {(f(0^{+})-f(0^{-}))\,t^{-q(t)}}{\Gamma (1-q(t))}},\end{aligned}}}"></span> </pre> <p>where the lower limit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> can be taken as either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb153c3e2b753e4b1627d97a00817d6ecab67eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.673ex; height:2.509ex;" alt="{\displaystyle 0^{-}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> as long as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> is identically zero from or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb153c3e2b753e4b1627d97a00817d6ecab67eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.673ex; height:2.509ex;" alt="{\displaystyle 0^{-}}"></span>. Note that this operator returns the correct fractional derivatives for all values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> and can be applied to either the dependent function itself <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> with a variable order of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(f(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(f(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cdb625dbe9e4a39380dcb82e6860769ff44ad9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.806ex; height:2.843ex;" alt="{\displaystyle q(f(t))}"></span> or to the independent variable with a variable order of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1f8079f76d0e8a89cf19db8fc43f34ec569d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.718ex; height:2.843ex;" alt="{\displaystyle q(t)}"></span>.<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{[1]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{[1]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c97b3809485cd490e1a61475329776d6b0efee76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.969ex; height:2.676ex;" alt="{\displaystyle ^{[1]}}"></span> </p><p>The Coimbra derivative can be generalized to any order<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup>, leading to the Coimbra Generalized Order Differintegration Operator (GODO)<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <pre>For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(t)<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dac39f3efa2c564ef574faf91d7dfebbe07f7424" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.857ex; height:2.843ex;" alt="{\displaystyle q(t)<m}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}^{\mathbb {\quad C} }_{\,\,-\infty }\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [m-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{m-1-q(t)}{\frac {d^{m}f(\tau )}{d\tau ^{m}}}d\tau \,+\,\sum _{n=0}^{m-1}{\frac {({\frac {d^{n}f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}f(t)}{dt^{n}}}|_{0^{-}})\,t^{n-q(t)}}{\Gamma [n+1-q(t)]}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mi mathvariant="double-struck">C</mi> </mrow> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>τ</mi> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}^{\mathbb {\quad C} }_{\,\,-\infty }\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [m-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{m-1-q(t)}{\frac {d^{m}f(\tau )}{d\tau ^{m}}}d\tau \,+\,\sum _{n=0}^{m-1}{\frac {({\frac {d^{n}f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}f(t)}{dt^{n}}}|_{0^{-}})\,t^{n-q(t)}}{\Gamma [n+1-q(t)]}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c269b204f85974c9be81014f037ea5ef374a62c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.278ex; margin-bottom: -0.227ex; width:93.891ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}^{\mathbb {\quad C} }_{\,\,-\infty }\mathbb {D} ^{q(t)}f(t)={\frac {1}{\Gamma [m-q(t)]}}\int _{0^{+}}^{t}(t-\tau )^{m-1-q(t)}{\frac {d^{m}f(\tau )}{d\tau ^{m}}}d\tau \,+\,\sum _{n=0}^{m-1}{\frac {({\frac {d^{n}f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}f(t)}{dt^{n}}}|_{0^{-}})\,t^{n-q(t)}}{\Gamma [n+1-q(t)]}},\end{aligned}}}"></span> </pre> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is an integer larger than the larger value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b1f8079f76d0e8a89cf19db8fc43f34ec569d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.718ex; height:2.843ex;" alt="{\displaystyle q(t)}"></span> for all values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. Note that the second (summation) term on the right side of the definition above can be expressed as </p> <pre><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {1}{\Gamma [m-q(t)]}}\sum _{n=0}^{m-1}\{[{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{-}}]\,t^{n-q(t)}\prod _{j=n+1}^{m-1}[j-q(t)]\}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </msub> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">[</mo> <mi>j</mi> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{\Gamma [m-q(t)]}}\sum _{n=0}^{m-1}\{[{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{-}}]\,t^{n-q(t)}\prod _{j=n+1}^{m-1}[j-q(t)]\}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ed12d6afff071f04afe1bc677ef43225ac4b82" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.349ex; height:7.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{\Gamma [m-q(t)]}}\sum _{n=0}^{m-1}\{[{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{+}}-{\frac {d^{n}\!f(t)}{dt^{n}}}|_{0^{-}}]\,t^{n-q(t)}\prod _{j=n+1}^{m-1}[j-q(t)]\}\end{aligned}}}"></span> </pre> <p>so to keep the denominator on the positive branch of the Gamma (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span>) function and for ease of numerical calculation. </p> <div class="mw-heading mw-heading3"><h3 id="Nature_of_the_fractional_derivative">Nature of the fractional derivative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=14" title="Edit section: Nature of the fractional derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>-th</span> derivative of a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> at a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> is a <i>local property</i> only when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3642398a43ac55e2d858529b48f8e113682801d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.435ex; height:1.676ex;" alt="{\displaystyle x=c}"></span> depends on all values of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>,</span> even those far away from <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>.</span> Therefore, it is expected that the fractional derivative operation involves some sort of <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a>, involving information on the function further out.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p><p>The fractional derivative of a function of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is nowadays often defined by means of the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier</a> or <a href="/wiki/Mellin_transform" title="Mellin transform">Mellin</a> integral transforms.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Examination of recent papers does not mention this (November 2022)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=15" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Erdélyi–Kober_operator"><span id="Erd.C3.A9lyi.E2.80.93Kober_operator"></span>Erdélyi–Kober operator</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=16" title="Edit section: Erdélyi–Kober operator"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Erd%C3%A9lyi%E2%80%93Kober_operator" class="mw-redirect" title="Erdélyi–Kober operator">Erdélyi–Kober operator</a> is an integral operator introduced by <a href="/wiki/Arthur_Erd%C3%A9lyi" title="Arthur Erdélyi">Arthur Erdélyi</a> (1940).<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Hermann_Kober" title="Hermann Kober">Hermann Kober</a> (1940)<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> and is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}\left(t-x\right)^{\alpha -1}t^{-\alpha -\nu }f(t)\,dt\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> <mo>−</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mrow> <mi>t</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mi>ν</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}\left(t-x\right)^{\alpha -1}t^{-\alpha -\nu }f(t)\,dt\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672aacf66c86143469c4a65e4dad0f58810d20f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.55ex; height:6.509ex;" alt="{\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}\left(t-x\right)^{\alpha -1}t^{-\alpha -\nu }f(t)\,dt\,,}"></span> </p><p>which generalizes the <a href="#Fractional_integrals">Riemann–Liouville fractional integral</a> and the Weyl integral. </p> <div class="mw-heading mw-heading2"><h2 id="Functional_calculus">Functional calculus</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=17" title="Edit section: Functional calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, functions <span class="texhtml"><i>f</i>(<i>D</i>)</span> more general than powers are studied in the <a href="/wiki/Functional_calculus" title="Functional calculus">functional calculus</a> of <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theory</a>. The theory of <a href="/wiki/Pseudo-differential_operator" title="Pseudo-differential operator">pseudo-differential operators</a> also allows one to consider powers of <span class="texhtml mvar" style="font-style:italic;">D</span>. The operators arising are examples of <a href="/wiki/Singular_integral_operator" class="mw-redirect" title="Singular integral operator">singular integral operators</a>; and the generalisation of the classical theory to higher dimensions is called the theory of <a href="/wiki/Riesz_potential" title="Riesz potential">Riesz potentials</a>. So there are a number of contemporary theories available, within which <i>fractional calculus</i> can be discussed. See also <a href="/wiki/Erd%C3%A9lyi%E2%80%93Kober_operator" class="mw-redirect" title="Erdélyi–Kober operator">Erdélyi–Kober operator</a>, important in <a href="/wiki/Special_function" class="mw-redirect" title="Special function">special function</a> theory (<a href="#CITEREFKober1940">Kober 1940</a>), (<a href="#CITEREFErdélyi1950–1951">Erdélyi 1950–1951</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=18" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Fractional_conservation_of_mass">Fractional conservation of mass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=19" title="Edit section: Fractional conservation of mass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As described by Wheatcraft and Meerschaert (2008),<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> a fractional conservation of mass equation is needed to model fluid flow when the <a href="/wiki/Control_volume" title="Control volume">control volume</a> is not large enough compared to the scale of <a href="/wiki/Heterogeneity" class="mw-redirect" title="Heterogeneity">heterogeneity</a> and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\rho \left(\nabla ^{\alpha }\cdot {\vec {u}}\right)=\Gamma (\alpha +1)\Delta x^{1-\alpha }\rho \left(\beta _{s}+\phi \beta _{w}\right){\frac {\partial p}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msup> <mi>ρ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>+</mo> <mi>ϕ</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\rho \left(\nabla ^{\alpha }\cdot {\vec {u}}\right)=\Gamma (\alpha +1)\Delta x^{1-\alpha }\rho \left(\beta _{s}+\phi \beta _{w}\right){\frac {\partial p}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f29bebbff635e7c00e0a7ece9bdae1a487aca38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.315ex; height:5.509ex;" alt="{\displaystyle -\rho \left(\nabla ^{\alpha }\cdot {\vec {u}}\right)=\Gamma (\alpha +1)\Delta x^{1-\alpha }\rho \left(\beta _{s}+\phi \beta _{w}\right){\frac {\partial p}{\partial t}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Electrochemical_analysis">Electrochemical analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=20" title="Edit section: Electrochemical analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Neopolarogram" title="Neopolarogram">Neopolarogram</a></div> <p>When studying the redox behavior of a substrate in solution, a voltage is applied at an electrode surface to force electron transfer between electrode and substrate. The resulting electron transfer is measured as a current. The current depends upon the concentration of substrate at the electrode surface. As substrate is consumed, fresh substrate diffuses to the electrode as described by <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a>. Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{2}}{dx^{2}}}C(x,s)=sC(x,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>s</mi> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dx^{2}}}C(x,s)=sC(x,s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5d8a846d25447fdca17a284b2f545c1c667fc9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:22.684ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{2}}{dx^{2}}}C(x,s)=sC(x,s)}"></span> </p><p>whose solution <span class="texhtml"><i>C</i>(<i>x</i>,<i>s</i>)</span> contains a one-half power dependence on <span class="texhtml mvar" style="font-style:italic;">s</span>. Taking the derivative of <span class="texhtml"><i>C</i>(<i>x</i>,<i>s</i>)</span> and then the inverse Laplace transform yields the following relationship: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}C(x,t)={\frac {d^{\scriptstyle {\frac {1}{2}}}}{dt^{\scriptstyle {\frac {1}{2}}}}}C(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}C(x,t)={\frac {d^{\scriptstyle {\frac {1}{2}}}}{dt^{\scriptstyle {\frac {1}{2}}}}}C(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2940ba8a22d51bded0831c72b35278cda255395b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.665ex; height:8.176ex;" alt="{\displaystyle {\frac {d}{dx}}C(x,t)={\frac {d^{\scriptstyle {\frac {1}{2}}}}{dt^{\scriptstyle {\frac {1}{2}}}}}C(x,t)}"></span> </p><p>which relates the concentration of substrate at the electrode surface to the current.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> This relationship is applied in electrochemical kinetics to elucidate mechanistic behavior. For example, it has been used to study the rate of dimerization of substrates upon electrochemical reduction.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Groundwater_flow_problem">Groundwater flow problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=21" title="Edit section: Groundwater flow problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> In these works, the classical <a href="/wiki/Darcy_law" class="mw-redirect" title="Darcy law">Darcy law</a> is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. </p> <div class="mw-heading mw-heading3"><h3 id="Fractional_advection_dispersion_equation">Fractional advection dispersion equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=22" title="Edit section: Fractional advection dispersion equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This equation<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (January 2017)">clarification needed</span></a></i>]</sup> has been shown useful for modeling contaminant flow in heterogenous porous media.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a <a href="/w/index.php?title=Variational_order_derivative&action=edit&redlink=1" class="new" title="Variational order derivative (page does not exist)">variational order derivative</a>. The modified equation was numerically solved via the <a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a>. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives<sup id="cite_ref-Atangana2014a_57-0" class="reference"><a href="#cite_note-Atangana2014a-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Time-space_fractional_diffusion_equation_models">Time-space fractional diffusion equation models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=23" title="Edit section: Time-space fractional diffusion equation models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> The time derivative term corresponds to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mi>u</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6adbc7788741db4e7e17b847400d2adaefe6b652" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.47ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.}"></span> </p><p>A simple extension of the fractional derivative is the variable-order fractional derivative, <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml mvar" style="font-style:italic;">β</span> are changed into <span class="texhtml"><i>α</i>(<i>x</i>, <i>t</i>)</span> and <span class="texhtml"><i>β</i>(<i>x</i>, <i>t</i>)</span>. Its applications in anomalous diffusion modeling can be found in the reference.<sup id="cite_ref-Atangana2014a_57-1" class="reference"><a href="#cite_note-Atangana2014a-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Structural_damping_models">Structural damping models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=24" title="Edit section: Structural damping models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fractional derivatives are used to model <a href="/wiki/Viscoelastic" class="mw-redirect" title="Viscoelastic">viscoelastic</a> <a href="/wiki/Damping" title="Damping">damping</a> in certain types of materials like polymers.<sup id="cite_ref-Mainardi_12-1" class="reference"><a href="#cite_note-Mainardi-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="PID_controllers">PID controllers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=25" title="Edit section: PID controllers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Generalizing <a href="/wiki/PID_controller" class="mw-redirect" title="PID controller">PID controllers</a> to use fractional orders can increase their degree of freedom. The new equation relating the <i>control variable</i> <span class="texhtml"><i>u</i>(<i>t</i>)</span> in terms of a measured <i>error value</i> <span class="texhtml"><i>e</i>(<i>t</i>)</span> can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(t)=K_{\mathrm {p} }e(t)+K_{\mathrm {i} }D_{t}^{-\alpha }e(t)+K_{\mathrm {d} }D_{t}^{\beta }e(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> </mrow> </msub> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msubsup> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(t)=K_{\mathrm {p} }e(t)+K_{\mathrm {i} }D_{t}^{-\alpha }e(t)+K_{\mathrm {d} }D_{t}^{\beta }e(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd7d26c2245e2d6a8227b48ee7ffcc9e770fa90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.442ex; height:3.509ex;" alt="{\displaystyle u(t)=K_{\mathrm {p} }e(t)+K_{\mathrm {i} }D_{t}^{-\alpha }e(t)+K_{\mathrm {d} }D_{t}^{\beta }e(t)}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml">β</span> are positive fractional orders and <span class="texhtml"><i>K</i><sub>p</sub></span>, <span class="texhtml"><i>K</i><sub>i</sub></span>, and <span class="texhtml"><i>K</i><sub>d</sub></span>, all non-negative, denote the coefficients for the <a href="/wiki/Proportional_control" title="Proportional control">proportional</a>, <a href="/wiki/Integral" title="Integral">integral</a>, and <a href="/wiki/Derivative" title="Derivative">derivative</a> terms, respectively (sometimes denoted <span class="texhtml mvar" style="font-style:italic;">P</span>, <span class="texhtml mvar" style="font-style:italic;">I</span>, and <span class="texhtml mvar" style="font-style:italic;">D</span>).<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Acoustic_wave_equations_for_complex_media">Acoustic wave equations for complex media</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=26" title="Edit section: Acoustic wave equations for complex media"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msubsup> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89aba7f1c6b3d5a29461e324dd057fb57db722eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.788ex; height:7.176ex;" alt="{\displaystyle \nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0\,.}"></span> </p><p>See also Holm & Näsholm (2011)<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in Näsholm & Holm (2011b)<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> and in the survey paper,<sup id="cite_ref-Nasholm2_65-0" class="reference"><a href="#cite_note-Nasholm2-65"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> as well as the <i><a href="/wiki/Acoustic_attenuation" title="Acoustic attenuation">Acoustic attenuation</a></i> article. See Holm & Nasholm (2013)<sup id="cite_ref-HolmNasholm2014_66-0" class="reference"><a href="#cite_note-HolmNasholm2014-66"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.<sup id="cite_ref-Holm2019_67-0" class="reference"><a href="#cite_note-Holm2019-67"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p><p>Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments.<sup id="cite_ref-Pandey2016_68-0" class="reference"><a href="#cite_note-Pandey2016-68"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> Interestingly, Pandey and Holm derived <a href="/wiki/Cinna_Lomnitz" title="Cinna Lomnitz">Lomnitz's law</a> in <a href="/wiki/Seismology" title="Seismology">seismology</a> and Nutting's law in <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian rheology</a> using the framework of fractional calculus.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.<sup id="cite_ref-Pandey2016_68-1" class="reference"><a href="#cite_note-Pandey2016-68"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fractional_Schrödinger_equation_in_quantum_theory"><span id="Fractional_Schr.C3.B6dinger_equation_in_quantum_theory"></span>Fractional Schrödinger equation in quantum theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=27" title="Edit section: Fractional Schrödinger equation in quantum theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/w/index.php?title=Fractional_Schr%C3%B6dinger_equation&action=edit&redlink=1" class="new" title="Fractional Schrödinger equation (page does not exist)">fractional Schrödinger equation</a>, a fundamental equation of <a href="/w/index.php?title=Fractional_quantum_mechanics&action=edit&redlink=1" class="new" title="Fractional quantum mechanics (page does not exist)">fractional quantum mechanics</a>, has the following form:<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }\left(-\hbar ^{2}\Delta \right)^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }\left(-\hbar ^{2}\Delta \right)^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9306359d8ae6c87ae1144f721d44ea1678940f38" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.083ex; height:5.843ex;" alt="{\displaystyle i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }\left(-\hbar ^{2}\Delta \right)^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,.}"></span> </p><p>where the solution of the equation is the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a> <span class="texhtml"><i>ψ</i>(<b>r</b>, <i>t</i>)</span> – the quantum mechanical <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability amplitude</a> for the particle to have a given <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a> <span class="texhtml"><b>r</b></span> at any given time <span class="texhtml mvar" style="font-style:italic;">t</span>, and <span class="texhtml mvar" style="font-style:italic;">ħ</span> is the <a href="/wiki/Reduced_Planck_constant" class="mw-redirect" title="Reduced Planck constant">reduced Planck constant</a>. The <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> function <span class="texhtml"><i>V</i>(<b>r</b>, <i>t</i>)</span> depends on the system. </p><p>Further, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f5138716ff9f4e227d3174a3dee47a859c7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.413ex; height:4.343ex;" alt="{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}"></span> is the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a>, and <span class="texhtml mvar" style="font-style:italic;">D<sub>α</sub></span> is a scale constant with physical <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimension</a> <span class="texhtml">[<i>D<sub>α</sub></i>] = J<sup>1 − <i>α</i></sup>·m<sup><i>α</i></sup>·s<sup>−<i>α</i></sup> = kg<sup>1 − <i>α</i></sup>·m<sup>2 − <i>α</i></sup>·s<sup><i>α</i> − 2</sup></span>, (at <span class="texhtml"><i>α</i> = 2</span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle D_{2}={\frac {1}{2m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle D_{2}={\frac {1}{2m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14fe9bd3f2a4a239a67b2951311f0572bf71d47c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.178ex; height:3.509ex;" alt="{\textstyle D_{2}={\frac {1}{2m}}}"></span> for a particle of mass <span class="texhtml mvar" style="font-style:italic;">m</span>), and the operator <span class="texhtml">(−<i>ħ</i><sup>2</sup>Δ)<sup><i>α</i>/2</sup></span> is the 3-dimensional fractional quantum Riesz derivative defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\hbar ^{2}\Delta )^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{{\frac {i}{\hbar }}\mathbf {p} \cdot \mathbf {r} }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi class="MJX-variant">ℏ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>p</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-\hbar ^{2}\Delta )^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{{\frac {i}{\hbar }}\mathbf {p} \cdot \mathbf {r} }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad70321f5d5c8212f31b81e45bff4184bd76dfd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.133ex; height:6.009ex;" alt="{\displaystyle (-\hbar ^{2}\Delta )^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{{\frac {i}{\hbar }}\mathbf {p} \cdot \mathbf {r} }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t)\,.}"></span> </p><p>The index <span class="texhtml mvar" style="font-style:italic;">α</span> in the fractional Schrödinger equation is the Lévy index, <span class="texhtml">1 < <i>α</i> ≤ 2</span>. </p> <div class="mw-heading mw-heading4"><h4 id="Variable-order_fractional_Schrödinger_equation"><span id="Variable-order_fractional_Schr.C3.B6dinger_equation"></span>Variable-order fractional Schrödinger equation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=28" title="Edit section: Variable-order fractional Schrödinger equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a natural generalization of the <a href="/w/index.php?title=Fractional_Schr%C3%B6dinger_equation&action=edit&redlink=1" class="new" title="Fractional Schrödinger equation (page does not exist)">fractional Schrödinger equation</a>, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena:<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\hbar {\frac {\partial \psi ^{\alpha (\mathbf {r} )}(\mathbf {r} ,t)}{\partial t^{\alpha (\mathbf {r} )}}}=\left(-\hbar ^{2}\Delta \right)^{\frac {\beta (t)}{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Δ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial \psi ^{\alpha (\mathbf {r} )}(\mathbf {r} ,t)}{\partial t^{\alpha (\mathbf {r} )}}}=\left(-\hbar ^{2}\Delta \right)^{\frac {\beta (t)}{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f444c1da75e4b2ab19dad1ee46621eea7773a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.262ex; height:6.509ex;" alt="{\displaystyle i\hbar {\frac {\partial \psi ^{\alpha (\mathbf {r} )}(\mathbf {r} ,t)}{\partial t^{\alpha (\mathbf {r} )}}}=\left(-\hbar ^{2}\Delta \right)^{\frac {\beta (t)}{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t),}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7f5138716ff9f4e227d3174a3dee47a859c7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.413ex; height:4.343ex;" alt="{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}}"></span> is the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> and the operator <span class="texhtml">(−<i>ħ</i><sup>2</sup>Δ)<sup><i>β</i>(<i>t</i>)/2</sup></span> is the variable-order fractional quantum Riesz derivative. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=29" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Acoustic_attenuation" title="Acoustic attenuation">Acoustic attenuation</a></li> <li><a href="/wiki/Autoregressive_fractionally_integrated_moving_average" title="Autoregressive fractionally integrated moving average">Autoregressive fractionally integrated moving average</a></li> <li><a href="/wiki/Initialized_fractional_calculus" title="Initialized fractional calculus">Initialized fractional calculus</a></li> <li><a href="/wiki/Nonlocal_operator" title="Nonlocal operator">Nonlocal operator</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_fractional_theories">Other fractional theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=30" title="Edit section: Other fractional theories"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fractional-order_system" title="Fractional-order system">Fractional-order system</a></li> <li><a href="/wiki/Fractional_Fourier_transform" title="Fractional Fourier transform">Fractional Fourier transform</a></li> <li><a href="/wiki/Prabhakar_function" title="Prabhakar function">Prabhakar function</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=31" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span> is commonly used instead of the intuitive <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> in order to avoid confusion with other concepts identified by similar <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>–like</span> <a href="/wiki/Glyph" title="Glyph">glyphs</a>, such as <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=32" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Zwillinger2014-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Zwillinger2014_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDaniel_Zwillinger2014" class="citation book cs1">Daniel Zwillinger (12 May 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9QLjBQAAQBAJ"><i>Handbook of Differential Equations</i></a>. Elsevier Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4832-2096-3" title="Special:BookSources/978-1-4832-2096-3"><bdi>978-1-4832-2096-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Differential+Equations&rft.pub=Elsevier+Science&rft.date=2014-05-12&rft.isbn=978-1-4832-2096-3&rft.au=Daniel+Zwillinger&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9QLjBQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-Derivative-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Derivative_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatugampola2014" class="citation journal cs1">Katugampola, Udita N. (15 October 2014). <a rel="nofollow" class="external text" href="https://www.emis.de/journals/BMAA/repository/docs/BMAA6-4-1.pdf">"A New Approach To Generalized Fractional Derivatives"</a> <span class="cs1-format">(PDF)</span>. <i>Bulletin of Mathematical Analysis and Applications</i>. <b>6</b> (4): 1–15. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1106.0965">1106.0965</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Mathematical+Analysis+and+Applications&rft.atitle=A+New+Approach+To+Generalized+Fractional+Derivatives&rft.volume=6&rft.issue=4&rft.pages=1-15&rft.date=2014-10-15&rft_id=info%3Aarxiv%2F1106.0965&rft.aulast=Katugampola&rft.aufirst=Udita+N.&rft_id=https%3A%2F%2Fwww.emis.de%2Fjournals%2FBMAA%2Frepository%2Fdocs%2FBMAA6-4-1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNiels_Henrik_Abel1823" class="citation journal cs1">Niels Henrik Abel (1823). <a rel="nofollow" class="external text" href="https://abelprize.no/sites/default/files/2021-04/Magazin_for_Naturvidenskaberne_oplosning_av_et_par1_opt.pdf">"Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler (Solution de quelques problèmes à l'aide d'intégrales définies, Solution of a couple of problems by means of definite integrals)"</a> <span class="cs1-format">(PDF)</span>. <i>Magazin for Naturvidenskaberne</i>. Kristiania (Oslo): 55–68.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Magazin+for+Naturvidenskaberne&rft.atitle=Opl%C3%B6sning+af+et+Par+Opgaver+ved+Hjelp+af+bestemte+Integraler+%28Solution+de+quelques+probl%C3%A8mes+%C3%A0+l%27aide+d%27int%C3%A9grales+d%C3%A9finies%2C+Solution+of+a+couple+of+problems+by+means+of+definite+integrals%29&rft.pages=55-68&rft.date=1823&rft.au=Niels+Henrik+Abel&rft_id=https%3A%2F%2Fabelprize.no%2Fsites%2Fdefault%2Ffiles%2F2021-04%2FMagazin_for_Naturvidenskaberne_oplosning_av_et_par1_opt.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPodlubnyMaginTrymorush2017" class="citation journal cs1">Podlubny, Igor; Magin, Richard L.; Trymorush, Irina (2017). "Niels Henrik Abel and the birth of fractional calculus". <i>Fractional Calculus and Applied Analysis</i>. <b>20</b> (5): 1068–1075. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1802.05441">1802.05441</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Ffca-2017-0057">10.1515/fca-2017-0057</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119664694">119664694</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fractional+Calculus+and+Applied+Analysis&rft.atitle=Niels+Henrik+Abel+and+the+birth+of+fractional+calculus&rft.volume=20&rft.issue=5&rft.pages=1068-1075&rft.date=2017&rft_id=info%3Aarxiv%2F1802.05441&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119664694%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1515%2Ffca-2017-0057&rft.aulast=Podlubny&rft.aufirst=Igor&rft.au=Magin%2C+Richard+L.&rft.au=Trymorush%2C+Irina&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiouville1832" class="citation cs2"><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville, Joseph</a> (1832), <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f2.item.r=Joseph%20Liouville">"Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions"</a>, <i>Journal de l'École Polytechnique</i>, <b>13</b>, Paris: 1–69</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+l%27%C3%89cole+Polytechnique&rft.atitle=M%C3%A9moire+sur+quelques+questions+de+g%C3%A9om%C3%A9trie+et+de+m%C3%A9canique%2C+et+sur+un+nouveau+genre+de+calcul+pour+r%C3%A9soudre+ces+questions&rft.volume=13&rft.pages=1-69&rft.date=1832&rft.aulast=Liouville&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k4336778%2Ff2.item.r%3DJoseph%2520Liouville&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiouville1832" class="citation cs2"><a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Liouville, Joseph</a> (1832), <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f72.image">"Mémoire sur le calcul des différentielles à indices quelconques"</a>, <i>Journal de l'École Polytechnique</i>, <b>13</b>, Paris: 71–162</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+l%27%C3%89cole+Polytechnique&rft.atitle=M%C3%A9moire+sur+le+calcul+des+diff%C3%A9rentielles+%C3%A0+indices+quelconques&rft.volume=13&rft.pages=71-162&rft.date=1832&rft.aulast=Liouville&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k4336778%2Ff72.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">For the history of the subject, see the thesis (in French): Stéphane Dugowson, <a rel="nofollow" class="external text" href="http://s.dugowson.free.fr/recherche/dones/index.html"><i>Les différentielles métaphysiques</i></a> (<i>histoire et philosophie de la généralisation de l'ordre de dérivation</i>), Thèse, Université Paris Nord (1994)</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">For a historical review of the subject up to the beginning of the 20th century, see: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBertram_Ross1977" class="citation journal cs1">Bertram Ross (1977). "The development of fractional calculus 1695–1900". <i>Historia Mathematica</i>. <b>4</b>: 75–89. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2877%2990039-8">10.1016/0315-0860(77)90039-8</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122146887">122146887</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Historia+Mathematica&rft.atitle=The+development+of+fractional+calculus+1695%E2%80%931900&rft.volume=4&rft.pages=75-89&rft.date=1977&rft_id=info%3Adoi%2F10.1016%2F0315-0860%2877%2990039-8&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122146887%23id-name%3DS2CID&rft.au=Bertram+Ross&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFValérioMachadoKiryakova2014" class="citation journal cs1">Valério, Duarte; Machado, José; <a href="/wiki/Virginia_Kiryakova" title="Virginia Kiryakova">Kiryakova, Virginia</a> (2014-01-01). "Some pioneers of the applications of fractional calculus". <i>Fractional Calculus and Applied Analysis</i>. <b>17</b> (2): 552–578. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fs13540-014-0185-1">10.2478/s13540-014-0185-1</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10400.22%2F5491">10400.22/5491</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1314-2224">1314-2224</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121482200">121482200</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fractional+Calculus+and+Applied+Analysis&rft.atitle=Some+pioneers+of+the+applications+of+fractional+calculus&rft.volume=17&rft.issue=2&rft.pages=552-578&rft.date=2014-01-01&rft_id=info%3Ahdl%2F10400.22%2F5491&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121482200%23id-name%3DS2CID&rft.issn=1314-2224&rft_id=info%3Adoi%2F10.2478%2Fs13540-014-0185-1&rft.aulast=Val%C3%A9rio&rft.aufirst=Duarte&rft.au=Machado%2C+Jos%C3%A9&rft.au=Kiryakova%2C+Virginia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermann2014" class="citation book cs1">Hermann, Richard (2014). <i>Fractional Calculus: An Introduction for Physicists</i> (2nd ed.). New Jersey: World Scientific Publishing. p. 46. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014fcip.book.....H">2014fcip.book.....H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F8934">10.1142/8934</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4551-07-6" title="Special:BookSources/978-981-4551-07-6"><bdi>978-981-4551-07-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Calculus%3A+An+Introduction+for+Physicists&rft.place=New+Jersey&rft.pages=46&rft.edition=2nd&rft.pub=World+Scientific+Publishing&rft.date=2014&rft_id=info%3Adoi%2F10.1142%2F8934&rft_id=info%3Abibcode%2F2014fcip.book.....H&rft.isbn=978-981-4551-07-6&rft.aulast=Hermann&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-Mainardi-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Mainardi_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Mainardi_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMainardi2010" class="citation book cs1">Mainardi, Francesco (May 2010). <i>Fractional Calculus and Waves in Linear Viscoelasticity</i>. <a href="/wiki/Imperial_College_Press" title="Imperial College Press">Imperial College Press</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2Fp614">10.1142/p614</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84816-329-4" title="Special:BookSources/978-1-84816-329-4"><bdi>978-1-84816-329-4</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118719247">118719247</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Calculus+and+Waves+in+Linear+Viscoelasticity&rft.pub=Imperial+College+Press&rft.date=2010-05&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118719247%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2Fp614&rft.isbn=978-1-84816-329-4&rft.aulast=Mainardi&rft.aufirst=Francesco&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHadamard1892" class="citation journal cs1">Hadamard, J. (1892). <a rel="nofollow" class="external text" href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1892_4_8_A4_0.pdf">"Essai sur l'étude des fonctions données par leur développement de Taylor"</a> <span class="cs1-format">(PDF)</span>. <i>Journal de Mathématiques Pures et Appliquées</i>. <b>4</b> (8): 101–186.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+Math%C3%A9matiques+Pures+et+Appliqu%C3%A9es&rft.atitle=Essai+sur+l%27%C3%A9tude+des+fonctions+donn%C3%A9es+par+leur+d%C3%A9veloppement+de+Taylor&rft.volume=4&rft.issue=8&rft.pages=101-186&rft.date=1892&rft.aulast=Hadamard&rft.aufirst=J.&rft_id=http%3A%2F%2Fsites.mathdoc.fr%2FJMPA%2FPDF%2FJMPA_1892_4_8_A4_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKilbasSrivastavaTrujillo2006" class="citation book cs1">Kilbas, A. Anatolii Aleksandrovich; Srivastava, Hari Mohan; Trujillo, Juan J. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LhkO83ZioQkC"><i>Theory And Applications of Fractional Differential Equations</i></a>. 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(2002). "Fractional Schrodinger equation". <i>Phys. Rev. E</i>. <b>66</b> (5): 056108. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/quant-ph/0206098">quant-ph/0206098</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002PhRvE..66e6108L">2002PhRvE..66e6108L</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.252.6732">10.1.1.252.6732</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.66.056108">10.1103/PhysRevE.66.056108</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12513557">12513557</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7520956">7520956</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.+E&rft.atitle=Fractional+Schrodinger+equation&rft.volume=66&rft.issue=5&rft.pages=056108&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7520956%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2002PhRvE..66e6108L&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.252.6732%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F12513557&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.66.056108&rft_id=info%3Aarxiv%2Fquant-ph%2F0206098&rft.aulast=Laskin&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaskin2018" class="citation book cs1">Laskin, Nick (2018). <i>Fractional Quantum Mechanics</i>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.247.5449">10.1.1.247.5449</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F10541">10.1142/10541</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-322-379-0" title="Special:BookSources/978-981-322-379-0"><bdi>978-981-322-379-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Quantum+Mechanics&rft.date=2018&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.247.5449%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1142%2F10541&rft.isbn=978-981-322-379-0&rft.aulast=Laskin&rft.aufirst=Nick&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhrawyZaky2017" class="citation journal cs1">Bhrawy, A.H.; Zaky, M.A. (2017). "An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations". <i>Applied Numerical Mathematics</i>. <b>111</b>: 197–218. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apnum.2016.09.009">10.1016/j.apnum.2016.09.009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Numerical+Mathematics&rft.atitle=An+improved+collocation+method+for+multi-dimensional+space%E2%80%93time+variable-order+fractional+Schr%C3%B6dinger+equations&rft.volume=111&rft.pages=197-218&rft.date=2017&rft_id=info%3Adoi%2F10.1016%2Fj.apnum.2016.09.009&rft.aulast=Bhrawy&rft.aufirst=A.H.&rft.au=Zaky%2C+M.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=33" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_regarding_the_history_of_fractional_calculus">Articles regarding the history of fractional calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=34" title="Edit section: Articles regarding the history of fractional calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDebnath2004" class="citation journal cs1">Debnath, L. (2004). "A brief historical introduction to fractional calculus". <i>International Journal of Mathematical Education in Science and Technology</i>. <b>35</b> (4): 487–501. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00207390410001686571">10.1080/00207390410001686571</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122198977">122198977</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Mathematical+Education+in+Science+and+Technology&rft.atitle=A+brief+historical+introduction+to+fractional+calculus&rft.volume=35&rft.issue=4&rft.pages=487-501&rft.date=2004&rft_id=info%3Adoi%2F10.1080%2F00207390410001686571&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122198977%23id-name%3DS2CID&rft.aulast=Debnath&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Books">Books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=35" title="Edit section: Books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMillerRoss1993" class="citation book cs1">Miller, Kenneth S.; Ross, Bertram, eds. (1993). <i>An Introduction to the Fractional Calculus and Fractional Differential Equations</i>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-58884-9" title="Special:BookSources/978-0-471-58884-9"><bdi>978-0-471-58884-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Fractional+Calculus+and+Fractional+Differential+Equations&rft.pub=John+Wiley+%26+Sons&rft.date=1993&rft.isbn=978-0-471-58884-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamkoKilbasMarichev1993" class="citation book cs1">Samko, S.; Kilbas, A.A.; Marichev, O. (1993). <i>Fractional Integrals and Derivatives: Theory and Applications</i>. Taylor & Francis Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-88124-864-1" title="Special:BookSources/978-2-88124-864-1"><bdi>978-2-88124-864-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Integrals+and+Derivatives%3A+Theory+and+Applications&rft.pub=Taylor+%26+Francis+Books&rft.date=1993&rft.isbn=978-2-88124-864-1&rft.aulast=Samko&rft.aufirst=S.&rft.au=Kilbas%2C+A.A.&rft.au=Marichev%2C+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarpinteriMainardi1998" class="citation book cs1">Carpinteri, A.; Mainardi, F., eds. (1998). <i>Fractals and Fractional Calculus in Continuum Mechanics</i>. Springer-Verlag Telos. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-211-82913-4" title="Special:BookSources/978-3-211-82913-4"><bdi>978-3-211-82913-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractals+and+Fractional+Calculus+in+Continuum+Mechanics&rft.pub=Springer-Verlag+Telos&rft.date=1998&rft.isbn=978-3-211-82913-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIgor_Podlubny1998" class="citation book cs1">Igor Podlubny (27 October 1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=K5FdXohLto0C"><i>Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications</i></a>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-053198-4" title="Special:BookSources/978-0-08-053198-4"><bdi>978-0-08-053198-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Differential+Equations%3A+An+Introduction+to+Fractional+Derivatives%2C+Fractional+Differential+Equations%2C+to+Methods+of+Their+Solution+and+Some+of+Their+Applications&rft.pub=Elsevier&rft.date=1998-10-27&rft.isbn=978-0-08-053198-4&rft.au=Igor+Podlubny&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DK5FdXohLto0C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarasov2010" class="citation book cs1">Tarasov, V.E. (2010). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-642-14003-7"><i>Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media</i></a>. Nonlinear Physical Science. Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-14003-7">10.1007/978-3-642-14003-7</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-14003-7" title="Special:BookSources/978-3-642-14003-7"><bdi>978-3-642-14003-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fractional+Dynamics%3A+Applications+of+Fractional+Calculus+to+Dynamics+of+Particles%2C+Fields+and+Media&rft.series=Nonlinear+Physical+Science&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-3-642-14003-7&rft.isbn=978-3-642-14003-7&rft.aulast=Tarasov&rft.aufirst=V.E.&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-642-14003-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiCai2019" class="citation book cs1">Li, Changpin; Cai, Min (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1137/1.9781611975888"><i>Theory and Numerical Approximations of Fractional Integrals and Derivatives</i></a>. SIAM. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611975888">10.1137/1.9781611975888</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61197-587-1" title="Special:BookSources/978-1-61197-587-1"><bdi>978-1-61197-587-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+Numerical+Approximations+of+Fractional+Integrals+and+Derivatives&rft.pub=SIAM&rft.date=2019&rft_id=info%3Adoi%2F10.1137%2F1.9781611975888&rft.isbn=978-1-61197-587-1&rft.aulast=Li&rft.aufirst=Changpin&rft.au=Cai%2C+Min&rft_id=https%3A%2F%2Fdoi.org%2F10.1137%2F1.9781611975888&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fractional_calculus&action=edit&section=36" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Fractional_calculus"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/FractionalCalculus.html">"Fractional calculus"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Fractional+calculus&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FFractionalCalculus.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath616/kmath616.htm.htm">"Fractional Calculus"</a>. <i>MathPages.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathPages.com&rft.atitle=Fractional+Calculus&rft_id=http%3A%2F%2Fwww.mathpages.com%2Fhome%2Fkmath616%2Fkmath616.htm.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.degruyter.com/journal/key/fca/html">Journal of Fractional Calculus and Applied Analysis</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:1314-2224">1314-2224</a> 2015—</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLorenzoHartley2002" class="citation web cs1">Lorenzo, Carl F.; Hartley, Tom T. (2002). <a rel="nofollow" class="external text" href="https://www.techbriefs.com/component/content/article/tb/pub/briefs/information-sciences/2264">"Initialized Fractional Calculus"</a>. <i>Tech Briefs</i>. NASA John H. Glenn Research Center.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Tech+Briefs&rft.atitle=Initialized+Fractional+Calculus&rft.date=2002&rft.aulast=Lorenzo&rft.aufirst=Carl+F.&rft.au=Hartley%2C+Tom+T.&rft_id=https%3A%2F%2Fwww.techbriefs.com%2Fcomponent%2Fcontent%2Farticle%2Ftb%2Fpub%2Fbriefs%2Finformation-sciences%2F2264&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerrmann2018" class="citation web cs1">Herrmann, Richard (2018). <a rel="nofollow" class="external text" href="https://fractionalcalculus.org/">"GigaHedron"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=GigaHedron&rft.date=2018&rft.aulast=Herrmann&rft.aufirst=Richard&rft_id=https%3A%2F%2Ffractionalcalculus.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span> collection of books, articles, preprints, etc.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoverro2005" class="citation web cs1">Loverro, Adam (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20051029113800/http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf">"History, Definitions, and Applications for the Engineer"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/University_of_Notre_Dame" title="University of Notre Dame">University of Notre Dame</a>. Archived from <a rel="nofollow" class="external text" href="http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2005-10-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=History%2C+Definitions%2C+and+Applications+for+the+Engineer&rft.pub=University+of+Notre+Dame&rft.date=2005&rft.aulast=Loverro&rft.aufirst=Adam&rft_id=http%3A%2F%2Fwww.nd.edu%2F~msen%2FTeaching%2FUnderRes%2FFracCalc.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFractional+calculus" class="Z3988"></span></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox 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style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Differential_equations_topics" title="Template:Differential equations topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations_topics" title="Template talk:Differential equations topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations_topics" title="Special:EditPage/Template:Differential equations topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Differential_equations" style="font-size:114%;margin:0 4em"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classification</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Operations</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_operator" title="Differential operator">Differential operator</a></li> <li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation for differentiation</a></li> <li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a href="/wiki/Partial_differential_equation" title="Partial differential equation">Partial</a></li> <li><a href="/wiki/Differential-algebraic_system_of_equations" title="Differential-algebraic system of equations">Differential-algebraic</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential</a></li> <li><a href="/wiki/Fractional_differential_equations" class="mw-redirect" title="Fractional differential equations">Fractional</a></li> <li><a href="/wiki/Linear_differential_equation" title="Linear differential equation">Linear</a></li> <li><a href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">Non-linear</a></li> <li><a href="/wiki/Holonomic_function" title="Holonomic function">Holonomic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Attributes of variables</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a></li> <li><a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Coupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Decoupled</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Order</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Degree</a></li> <li><a href="/wiki/Autonomous_system_(mathematics)" title="Autonomous system (mathematics)">Autonomous</a></li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact differential equation</a></li> <li><a href="/wiki/Jet_bundle#Partial_differential_equations" title="Jet bundle">On jet bundles</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Relation to processes</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference</a> (discrete analogue)</li> <li><a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">Stochastic</a> <ul><li><a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial</a></li></ul></li> <li><a href="/wiki/Delay_differential_equation" title="Delay differential equation">Delay</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solutions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Existence/uniqueness</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">Picard–Lindelöf theorem</a></li> <li><a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a></li> <li><a href="/wiki/Carath%C3%A9odory%27s_existence_theorem" title="Carathéodory's existence theorem">Carathéodory's existence theorem</a></li> <li><a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li> <li><a href="/wiki/Phase_portrait" title="Phase portrait">Phase portrait</a></li> <li><a href="/wiki/Phase_space" title="Phase space">Phase space</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov stability</a></li> <li><a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic stability</a></li> <li><a href="/wiki/Exponential_stability" title="Exponential stability">Exponential stability</a></li> <li><a href="/wiki/Rate_of_convergence" title="Rate of convergence">Rate of convergence</a></li> <li><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series solutions</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> solutions</li> <li><a href="/wiki/Numerical_integration" title="Numerical integration">Numerical integration</a></li> <li><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Solution methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_jargon#Proof_techniques" class="mw-redirect" title="List of mathematical jargon">Inspection</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a></li> <li><a href="/wiki/Separation_of_variables" title="Separation of variables">Separation of variables</a></li> <li><a href="/wiki/Method_of_undetermined_coefficients" title="Method of undetermined coefficients">Method of undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li> <li><a href="/wiki/Integrating_factor" title="Integrating factor">Integrating factor</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transforms</a></li> <li><a href="/wiki/Euler_method" title="Euler method">Euler method</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></li> <li><a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson method</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta methods</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin method</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Examples</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></li> <li><a href="/wiki/List_of_linear_ordinary_differential_equations" title="List of linear ordinary differential equations">List of linear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_ordinary_differential_equations" title="List of nonlinear ordinary differential equations">List of nonlinear ordinary differential equations</a></li> <li><a href="/wiki/List_of_nonlinear_partial_differential_equations" title="List of nonlinear partial differential equations">List of nonlinear partial differential equations</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematicians</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/J%C3%B3zef_Maria_Hoene-Wro%C5%84ski" title="Józef Maria Hoene-Wroński">Józef Maria Hoene-Wroński</a></li> <li><a href="/wiki/Ernst_Leonard_Lindel%C3%B6f" title="Ernst Leonard Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/Carl_David_Tolm%C3%A9_Runge" class="mw-redirect" title="Carl David Tolmé Runge">Carl David Tolmé Runge</a></li> <li><a href="/wiki/Martin_Kutta" title="Martin Kutta">Martin Kutta</a></li> <li><a href="/wiki/Sofya_Kovalevskaya" title="Sofya Kovalevskaya">Sofya Kovalevskaya</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style 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Fractional calculus - Wikipedia