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class="vector-toc-link" href="#Systems_of_first-order_equations_and_characteristic_surfaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Systems of first-order equations and characteristic surfaces</span> </div> </a> <ul id="toc-Systems_of_first-order_equations_and_characteristic_surfaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analytical_solutions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Analytical_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Analytical solutions</span> </div> </a> <button aria-controls="toc-Analytical_solutions-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 Analytical solutions subsection</span> </button> <ul id="toc-Analytical_solutions-sublist" class="vector-toc-list"> <li id="toc-Separation_of_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separation_of_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Separation of variables</span> </div> </a> <ul id="toc-Separation_of_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Method_of_characteristics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Method_of_characteristics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Method of characteristics</span> </div> </a> <ul id="toc-Method_of_characteristics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_transform" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_transform"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Integral transform</span> </div> </a> <ul id="toc-Integral_transform-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Change_of_variables" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Change_of_variables"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Change of variables</span> </div> </a> <ul id="toc-Change_of_variables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fundamental_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamental_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Fundamental solution</span> </div> </a> <ul id="toc-Fundamental_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Superposition_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Superposition_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Superposition principle</span> </div> </a> <ul id="toc-Superposition_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methods_for_non-linear_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Methods_for_non-linear_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Methods for non-linear equations</span> </div> </a> <ul id="toc-Methods_for_non-linear_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_group_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_group_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Lie group method</span> </div> </a> <ul id="toc-Lie_group_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semi-analytical_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semi-analytical_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span>Semi-analytical methods</span> </div> </a> <ul id="toc-Semi-analytical_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Numerical_solutions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Numerical_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Numerical solutions</span> </div> </a> <button aria-controls="toc-Numerical_solutions-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 Numerical solutions subsection</span> </button> <ul id="toc-Numerical_solutions-sublist" class="vector-toc-list"> <li id="toc-Finite_element_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Finite element method</span> </div> </a> <ul id="toc-Finite_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_difference_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_difference_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Finite difference method</span> </div> </a> <ul id="toc-Finite_difference_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_volume_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_volume_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Finite volume method</span> </div> </a> <ul id="toc-Finite_volume_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Neural_networks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Neural_networks"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Neural networks</span> </div> </a> <ul id="toc-Neural_networks-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Weak_solutions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Weak_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Weak solutions</span> </div> </a> <ul id="toc-Weak_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Well-posedness" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Well-posedness"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Well-posedness</span> </div> </a> <button aria-controls="toc-Well-posedness-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 Well-posedness subsection</span> </button> <ul id="toc-Well-posedness-sublist" class="vector-toc-list"> <li id="toc-The_energy_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_energy_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>The energy method</span> </div> </a> <ul id="toc-The_energy_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Existence_of_local_solutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Existence_of_local_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Existence of local solutions</span> </div> </a> <ul id="toc-Existence_of_local_solutions-sublist" class="vector-toc-list"> </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">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </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">9</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">10</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">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </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">12</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 cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Partial differential equation</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 50 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-50" 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">50 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/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%AA%D9%81%D8%A7%D8%B6%D9%84%D9%8A%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9" title="معادلة تفاضلية جزئية – Arabic" lang="ar" hreflang="ar" data-title="معادلة تفاضلية جزئية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ecuaci%C3%B3n_en_derivaes_parciales" title="Ecuación en derivaes parciales – Asturian" lang="ast" hreflang="ast" data-title="Ecuación en derivaes parciales" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%82%E0%A6%B6%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%95%E0%A6%B2%E0%A6%A8%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="আংশিক ব্যবকলনীয় সমীকরণ – Bangla" lang="bn" hreflang="bn" data-title="আংশিক ব্যবকলনীয় সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B0%D1%81%D1%82%D0%BD%D0%BE_%D0%B4%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BD%D0%BE_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5" title="Частно диференциално уравнение – Bulgarian" lang="bg" hreflang="bg" data-title="Частно диференциално уравнение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_diferencial_en_derivades_parcials" title="Equació diferencial en derivades parcials – Catalan" lang="ca" hreflang="ca" data-title="Equació diferencial en derivades parcials" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B0%D1%80%D0%BF%C4%83%D1%80_%D1%82%C4%83%D1%85%C4%83%D0%BC%D1%81%D0%B5%D0%BC%D0%BB%C4%95_%D0%B4%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D0%BB%C4%83_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85" title="Харпăр тăхăмсемлĕ дифференциаллă танлăх – Chuvash" lang="cv" hreflang="cv" data-title="Харпăр тăхăмсемлĕ дифференциаллă танлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Parci%C3%A1ln%C3%AD_diferenci%C3%A1ln%C3%AD_rovnice" title="Parciální diferenciální rovnice – Czech" lang="cs" hreflang="cs" data-title="Parciální diferenciální rovnice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A9_%D8%AA%D9%81%D8%A7%D8%B6%D9%84%D9%8A%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9" title="معادلة تفاضلية جزئية – Moroccan Arabic" lang="ary" hreflang="ary" data-title="معادلة تفاضلية جزئية" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Partielle_Differentialgleichung" title="Partielle Differentialgleichung – German" lang="de" hreflang="de" data-title="Partielle Differentialgleichung" 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/Osatuletistega_diferentsiaalv%C3%B5rrand" title="Osatuletistega diferentsiaalvõrrand – Estonian" lang="et" hreflang="et" data-title="Osatuletistega diferentsiaalvõrrand" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%81%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CE%B1%CF%86%CE%BF%CF%81%CE%B9%CE%BA%CE%AE_%CE%B5%CE%BE%CE%AF%CF%83%CF%89%CF%83%CE%B7" title="Μερική διαφορική εξίσωση – Greek" lang="el" hreflang="el" data-title="Μερική διαφορική εξίσωση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_en_derivadas_parciales" title="Ecuación en derivadas parciales – Spanish" lang="es" hreflang="es" data-title="Ecuación en derivadas parciales" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Parta_diferenciala_ekvacio" title="Parta diferenciala ekvacio – Esperanto" lang="eo" hreflang="eo" data-title="Parta diferenciala ekvacio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ekuazio_diferentzial_partzial" title="Ekuazio diferentzial partzial – Basque" lang="eu" hreflang="eu" data-title="Ekuazio diferentzial partzial" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%AF%DB%8C%D9%81%D8%B1%D8%A7%D9%86%D8%B3%DB%8C%D9%84_%D8%A8%D8%A7_%D9%85%D8%B4%D8%AA%D9%82%D8%A7%D8%AA_%D8%AC%D8%B2%D8%A6%DB%8C" title="معادله دیفرانسیل با مشتقات جزئی – Persian" lang="fa" hreflang="fa" data-title="معادله دیفرانسیل با مشتقات جزئی" 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/%C3%89quation_aux_d%C3%A9riv%C3%A9es_partielles" title="Équation aux dérivées partielles – French" lang="fr" hreflang="fr" data-title="Équation aux dérivées partielles" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3n_en_derivadas_parciais" title="Ecuación en derivadas parciais – Galician" lang="gl" hreflang="gl" data-title="Ecuación en derivadas parciais" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8E%B8%EB%AF%B8%EB%B6%84_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="편미분 방정식 – Korean" lang="ko" hreflang="ko" data-title="편미분 방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%BD%D5%B6%D5%A1%D5%AF%D5%AB_%D5%A1%D5%AE%D5%A1%D5%B6%D6%81%D5%B5%D5%A1%D5%AC%D5%B6%D5%A5%D6%80%D5%B8%D5%BE_%D5%A4%D5%AB%D6%86%D5%A5%D6%80%D5%A5%D5%B6%D6%81%D5%AB%D5%A1%D5%AC_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Մասնակի ածանցյալներով դիֆերենցիալ հավասարումներ – Armenian" lang="hy" hreflang="hy" data-title="Մասնակի ածանցյալներով դիֆերենցիալ հավասարումներ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%82%E0%A4%B6%E0%A4%BF%E0%A4%95_%E0%A4%85%E0%A4%B5%E0%A4%95%E0%A4%B2_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="आंशिक अवकल समीकरण – Hindi" lang="hi" hreflang="hi" data-title="आंशिक अवकल समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_diferensial_parsial" title="Persamaan diferensial parsial – Indonesian" lang="id" hreflang="id" data-title="Persamaan diferensial parsial" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_differenziale_alle_derivate_parziali" title="Equazione differenziale alle derivate parziali – Italian" lang="it" hreflang="it" data-title="Equazione differenziale alle derivate parziali" 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%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA_%D7%97%D7%9C%D7%A7%D7%99%D7%AA" title="משוואה דיפרנציאלית חלקית – Hebrew" lang="he" hreflang="he" data-title="משוואה דיפרנציאלית חלקית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Parci%C3%A1lis_differenci%C3%A1legyenlet" title="Parciális differenciálegyenlet – Hungarian" lang="hu" hreflang="hu" data-title="Parciális differenciálegyenlet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B4%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B0" title="Парцијална диференцијална равенка – Macedonian" lang="mk" hreflang="mk" data-title="Парцијална диференцијална равенка" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Persamaan_pembezaan_separa" title="Persamaan pembezaan separa – Malay" lang="ms" hreflang="ms" data-title="Persamaan pembezaan separa" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Parti%C3%ABle_differentiaalvergelijking" title="Partiële differentiaalvergelijking – Dutch" lang="nl" hreflang="nl" data-title="Partiële differentiaalvergelijking" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E5%BC%8F" title="偏微分方程式 – Japanese" lang="ja" hreflang="ja" data-title="偏微分方程式" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Partielle_differensialligninger" title="Partielle differensialligninger – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Partielle differensialligninger" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Xususiy_hosilali_differensial_tenglama" title="Xususiy hosilali differensial tenglama – Uzbek" lang="uz" hreflang="uz" data-title="Xususiy hosilali differensial tenglama" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_r%C3%B3%C5%BCniczkowe_cz%C4%85stkowe" title="Równanie różniczkowe cząstkowe – Polish" lang="pl" hreflang="pl" data-title="Równanie różniczkowe cząstkowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%A3o_diferencial_parcial" title="Equação diferencial parcial – Portuguese" lang="pt" hreflang="pt" data-title="Equação diferencial parcial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Bie_cu_derivate_par%C8%9Biale" title="Ecuație cu derivate parțiale – Romanian" lang="ro" hreflang="ro" data-title="Ecuație cu derivate parțiale" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%83%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B2_%D1%87%D0%B0%D1%81%D1%82%D0%BD%D1%8B%D1%85_%D0%BF%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D1%8B%D1%85" title="Дифференциальное уравнение в частных производных – Russian" lang="ru" hreflang="ru" data-title="Дифференциальное уравнение в частных производных" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Pairtial_differential_equation" title="Pairtial differential equation – Scots" lang="sco" hreflang="sco" data-title="Pairtial differential equation" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ekuacionet_diferenciale_t%C3%AB_pjesshme" title="Ekuacionet diferenciale të pjesshme – Albanian" lang="sq" hreflang="sq" data-title="Ekuacionet diferenciale të pjesshme" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Partial_differential_equation" title="Partial differential equation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Partial differential equation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Parci%C3%A1lna_diferenci%C3%A1lna_rovnica" title="Parciálna diferenciálna rovnica – Slovak" lang="sk" hreflang="sk" data-title="Parciálna diferenciálna rovnica" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Parcialna_diferencialna_ena%C4%8Dba" title="Parcialna diferencialna enačba – Slovenian" lang="sl" hreflang="sl" data-title="Parcialna diferencialna enačba" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%B0%D1%80%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D0%B4%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB%D0%BD%D0%B0_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B0" title="Парцијална диференцијална једначина – Serbian" lang="sr" hreflang="sr" data-title="Парцијална диференцијална једначина" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Parcijalna_diferencijalna_jedna%C4%8Dina" title="Parcijalna diferencijalna jednačina – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Parcijalna diferencijalna jednačina" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Osittaisdifferentiaaliyht%C3%A4l%C3%B6" title="Osittaisdifferentiaaliyhtälö – Finnish" lang="fi" hreflang="fi" data-title="Osittaisdifferentiaaliyhtälö" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Partiell_differentialekvation" title="Partiell differentialekvation – Swedish" lang="sv" hreflang="sv" data-title="Partiell differentialekvation" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Ekwasyong_parsiyal_diperensiyal" title="Ekwasyong parsiyal diperensiyal – Tagalog" lang="tl" hreflang="tl" data-title="Ekwasyong parsiyal diperensiyal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%AD%E0%B8%99%E0%B8%B8%E0%B8%9E%E0%B8%B1%E0%B8%99%E0%B8%98%E0%B9%8C%E0%B8%A2%E0%B9%88%E0%B8%AD%E0%B8%A2" title="สมการเชิงอนุพันธ์ย่อย – Thai" lang="th" hreflang="th" data-title="สมการเชิงอนุพันธ์ย่อย" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C4%B1smi_diferansiyel_denklem" title="Kısmi diferansiyel denklem – Turkish" lang="tr" hreflang="tr" data-title="Kısmi diferansiyel denklem" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B5_%D1%80%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%B7_%D1%87%D0%B0%D1%81%D1%82%D0%B8%D0%BD%D0%BD%D0%B8%D0%BC%D0%B8_%D0%BF%D0%BE%D1%85%D1%96%D0%B4%D0%BD%D0%B8%D0%BC%D0%B8" title="Диференціальне рівняння з частинними похідними – Ukrainian" lang="uk" hreflang="uk" data-title="Диференціальне рівняння з частинними похідними" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_vi_ph%C3%A2n_ri%C3%AAng_ph%E1%BA%A7n" title="Phương trình vi phân riêng phần – Vietnamese" lang="vi" hreflang="vi" data-title="Phương trình vi phân riêng phần" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="偏微分方程 – Cantonese" lang="yue" hreflang="yue" data-title="偏微分方程" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%81%8F%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="偏微分方程 – Chinese" lang="zh" hreflang="zh" data-title="偏微分方程" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q271977#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit 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nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="background:#ccccff;display:block;margin-bottom:0.2em;"><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></th></tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Scope</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Fields</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="padding-bottom:0;"> <div class="hlist"><ul><li><a href="/wiki/Natural_science" title="Natural science">Natural sciences</a></li><li><a href="/wiki/Engineering" title="Engineering">Engineering</a></li></ul></div></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Astronomy" title="Astronomy">Astronomy</a></li> <li><a href="/wiki/Physics" title="Physics">Physics</a></li> <li><a href="/wiki/Chemistry" title="Chemistry">Chemistry</a></li> <li><br /><a href="/wiki/Biology" title="Biology">Biology</a></li> <li><a href="/wiki/Geology" title="Geology">Geology</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied mathematics</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;"> <ul><li><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li> <li><a href="/wiki/Chaos_theory" title="Chaos theory">Chaos theory</a></li> <li><a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">Dynamical systems</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0;"> <a href="/wiki/Social_science" title="Social science">Social sciences</a></th></tr><tr><td class="sidebar-content hlist" style="padding-bottom:0.6em;;padding-bottom:0;"> <ul><li><a href="/wiki/Economics" title="Economics">Economics</a></li> <li><a href="/wiki/Population_dynamics" title="Population dynamics">Population dynamics</a></li></ul></td> </tr></tbody></table> <hr /> <a href="/wiki/List_of_named_differential_equations" title="List of named differential equations">List of named differential equations</a></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;;display:block;margin-top:0.1em;"> Classification</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Types</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary</a></li> <li><a class="mw-selflink selflink">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></ul> </div></td> </tr><tr><th class="sidebar-heading"> By variable type</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">Dependent and independent variables</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Autonomous_differential_equation" class="mw-redirect" title="Autonomous differential equation">Autonomous</a></li> <li>Coupled / Decoupled</li> <li><a href="/wiki/Exact_differential_equation" title="Exact differential equation">Exact</a></li> <li><a href="/wiki/Homogeneous_differential_equation" title="Homogeneous differential equation">Homogeneous</a> / <a href="/wiki/Non-homogeneous_differential_equation" class="mw-redirect" title="Non-homogeneous differential equation">Nonhomogeneous</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> Features</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Ordinary_differential_equation#Definitions" title="Ordinary differential equation">Order</a></li> <li><a href="/wiki/Differential_operator" title="Differential operator">Operator</a></li></ul> </div> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Notation</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;padding-bottom:0;;color: var(--color-base)">Relation to processes</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <ul><li><a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">Difference <span style="font-size:85%;">(discrete analogue)</span></a></li></ul> <div class="hlist"> <ul><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></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> Solution</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">Existence and uniqueness</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"> <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></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;padding-bottom:0;;color: var(--color-base)">General topics</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li><a href="/wiki/Initial_condition" title="Initial condition">Initial conditions</a></li> <li><a href="/wiki/Boundary_value_problem" title="Boundary value problem">Boundary values</a> <ul><li><a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet</a></li> <li><a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann</a></li> <li><a href="/wiki/Robin_boundary_condition" title="Robin boundary condition">Robin</a></li> <li><a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul></li> <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/Lyapunov_stability" title="Lyapunov stability">Lyapunov</a> / <a href="/wiki/Asymptotic_stability" class="mw-redirect" title="Asymptotic stability">Asymptotic</a> / <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><span class="nowrap"><a href="/wiki/Power_series_solution_of_differential_equations" title="Power series solution of differential equations">Series</a> / Integral solutions</span></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></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;padding-bottom:0;;color: var(--color-base)">Solution methods</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist"> <ul><li>Inspection</li> <li><a href="/wiki/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></li> <li><br /><a href="/wiki/Euler_method" title="Euler method">Euler</a></li> <li><a href="/wiki/Exponential_response_formula" title="Exponential response formula">Exponential response formula</a></li> <li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference</a> <span style="font-size:85%;">(<a href="/wiki/Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank–Nicolson</a>)</span></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element</a> <ul><li><a href="/wiki/Infinite_element_method" title="Infinite element method">Infinite element</a></li></ul></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></li> <li><a href="/wiki/Galerkin_method" title="Galerkin method">Galerkin</a> <ul><li><a href="/wiki/Petrov%E2%80%93Galerkin_method" title="Petrov–Galerkin method">Petrov–Galerkin</a></li></ul></li> <li><a href="/wiki/Green%27s_function" title="Green's function">Green's function</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/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li> <li><a href="/wiki/Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta</a></li></ul> </div> <ul><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">Undetermined coefficients</a></li> <li><a href="/wiki/Variation_of_parameters" title="Variation of parameters">Variation of parameters</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="background:#ddddff;font-size:105%;display:block;margin-bottom:0.4em;"> People</th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;padding-bottom:0;;color: var(--color-base)">List</div><div class="sidebar-list-content mw-collapsible-content" style="padding-top:0;"><div class="hlist" style="padding-top:0.5em"> <ul><li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</a></li> <li><a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</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/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/George_Green_(mathematician)" title="George Green (mathematician)">George Green</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/Rudolf_Lipschitz" title="Rudolf Lipschitz">Rudolf Lipschitz</a></li> <li><a href="/wiki/Ernst_Lindel%C3%B6f" class="mw-redirect" title="Ernst Lindelöf">Ernst Lindelöf</a></li> <li><a href="/wiki/%C3%89mile_Picard" title="Émile Picard">Émile Picard</a></li> <li><a href="/wiki/Phyllis_Nicolson" title="Phyllis Nicolson">Phyllis Nicolson</a></li> <li><a href="/wiki/John_Crank" title="John Crank">John Crank</a></li></ul> </div></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:Differential_equations" title="Template:Differential equations"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Differential_equations" title="Template talk:Differential equations"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Differential_equations" title="Special:EditPage/Template:Differential equations"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>partial differential equation</b> (<b>PDE</b>) is an equation which computes a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> between various <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> of a <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable function</a>. </p><p>The function is often thought of as an "unknown" to be solved for, similar to how <span class="texhtml mvar" style="font-style:italic;">x</span> is thought of as an unknown number to be solved for in an algebraic equation like <span class="texhtml"><i>x</i><sup>2</sup> − 3<i>x</i> + 2 = 0</span>. However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to <a href="/wiki/Numerical_methods_for_partial_differential_equations" title="Numerical methods for partial differential equations">numerically approximate</a> solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure mathematical research</a>, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Among the many open questions are the <a href="/wiki/Navier%E2%80%93Stokes_existence_and_smoothness" title="Navier–Stokes existence and smoothness">existence and smoothness</a> of solutions to the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>, named as one of the <a href="/wiki/Millennium_Prize_Problems" title="Millennium Prize Problems">Millennium Prize Problems</a> in 2000. </p><p>Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>. For instance, they are foundational in the modern scientific understanding of <a href="/wiki/Sound" title="Sound">sound</a>, <a href="/wiki/Heat" title="Heat">heat</a>, <a href="/wiki/Diffusion" title="Diffusion">diffusion</a>, <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a>, <a href="/wiki/Electromagnetism" title="Electromagnetism">electrodynamics</a>, <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>, <a href="/wiki/General_relativity" title="General relativity">general relativity</a>, and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> (<a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, <a href="/wiki/Pauli_equation" title="Pauli equation">Pauli equation</a> etc.). They also arise from many purely mathematical considerations, such as <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>; among other notable applications, they are the fundamental tool in the proof of the <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a> from <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a>. </p><p>Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">Ordinary differential equations</a> can be viewed as a subclass of partial differential equations, corresponding to functions of a single variable. <a href="/wiki/Stochastic_partial_differential_equation" title="Stochastic partial differential equation">Stochastic partial differential equations</a> and <a href="/wiki/Fractional_calculus" title="Fractional calculus">nonlocal equations</a> are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic</a> and <a href="/wiki/Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic</a> partial differential equations, <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, <a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equations</a>, and <a href="/wiki/Dispersive_partial_differential_equation" title="Dispersive partial differential equation">dispersive partial differential equations</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A function <span class="texhtml"><i>u</i>(<i>x</i>, <i>y</i>, <i>z</i>)</span> of three variables is "<a href="/wiki/Harmonic_function" title="Harmonic function">harmonic</a>" or "a solution of the <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace equation</a>" if it satisfies the condition <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 ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}"> <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"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <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>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f510ba98fe6488d760a4aa8800eeba69042e30d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.279ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}=0.}"></span> Such functions were widely studied in the 19th century due to their relevance for <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is harmonic. For instance <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(x,y,z)={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x,y,z)={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340365dcc474271ae80e32ccd7e9b115c17d92ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.798ex; height:6.509ex;" alt="{\displaystyle u(x,y,z)={\frac {1}{\sqrt {x^{2}-2x+y^{2}+z^{2}+1}}}}"></span> and <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(x,y,z)=2x^{2}-y^{2}-z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x,y,z)=2x^{2}-y^{2}-z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c45c4a495e490ca2f135d86f8ee7232f4d73b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.465ex; height:3.176ex;" alt="{\displaystyle u(x,y,z)=2x^{2}-y^{2}-z^{2}}"></span> are both harmonic while <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(x,y,z)=\sin(xy)+z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x,y,z)=\sin(xy)+z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b22c1729223b2ad72c98189ff56e3e5a2d8acb7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.957ex; height:2.843ex;" alt="{\displaystyle u(x,y,z)=\sin(xy)+z}"></span> is not. It may be surprising that the two examples of harmonic functions are of such strikingly different form. This is a reflection of the fact that they are <i>not</i>, in any immediate way, special cases of a "general solution formula" of the Laplace equation. This is in striking contrast to the case of <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equations</a> (ODEs) <a href="/wiki/Linear_differential_equation#Homogeneous_equation_with_constant_coefficients" title="Linear differential equation">roughly similar</a> to the Laplace equation, with the aim of many introductory textbooks being to find algorithms leading to general solution formulas. For the Laplace equation, as for a large number of partial differential equations, such solution formulas fail to exist. </p><p>The nature of this failure can be seen more concretely in the case of the following PDE: for a function <span class="texhtml"><i>v</i>(<i>x</i>, <i>y</i>)</span> of two variables, consider 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 {\frac {\partial ^{2}v}{\partial x\partial y}}=0.}"> <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"> <mn>2</mn> </mrow> </msup> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}v}{\partial x\partial y}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be55e1dc6366aaa0df3f92d2eadf552d1d1c4cc3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.865ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}v}{\partial x\partial y}}=0.}"></span> It can be directly checked that any function <span class="texhtml mvar" style="font-style:italic;">v</span> of the form <span class="texhtml"><i>v</i>(<i>x</i>, <i>y</i>) = <i>f</i>(<i>x</i>) + <i>g</i>(<i>y</i>)</span>, for any single-variable functions <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions. </p><p>The nature of this choice varies from PDE to PDE. To understand it for any given equation, <i>existence and uniqueness theorems</i> are usually important organizational principles. In many introductory textbooks, the role of <a href="/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem" title="Picard–Lindelöf theorem">existence and uniqueness theorems for ODE</a> can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background in order to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate. </p><p>To discuss such existence and uniqueness theorems, it is necessary to be precise about the <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself. </p><p>The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions. </p> <ul><li>Let <span class="texhtml mvar" style="font-style:italic;">B</span> denote the unit-radius disk around the origin in the plane. For any continuous function <span class="texhtml mvar" style="font-style:italic;">U</span> on the unit circle, there is exactly one function <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml mvar" style="font-style:italic;">B</span> such that <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 ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}"> <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"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a55467a93189024042e47531367c97fa40e749b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.228ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0}"></span> and whose restriction to the unit circle is given by <span class="texhtml mvar" style="font-style:italic;">U</span>.</li> <li>For any functions <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> on the real line <span class="texhtml"><b>R</b></span>, there is exactly one function <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml"><b>R</b> × (−1, 1)</span> such that <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 ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u}{\partial y^{2}}}=0}"> <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"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <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>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u}{\partial y^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd081e05fa1c11cb033aa14d0dc4919f968243e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.228ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}u}{\partial x^{2}}}-{\frac {\partial ^{2}u}{\partial y^{2}}}=0}"></span> and with <span class="texhtml"><i>u</i>(<i>x</i>, 0) = <i>f</i>(<i>x</i>)</span> and <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">∂<i>u</i></span><span class="sr-only">/</span><span class="den">∂<i>y</i></span></span>⁠</span>(<i>x</i>, 0) = <i>g</i>(<i>x</i>)</span> for all values of <span class="texhtml mvar" style="font-style:italic;">x</span>.</li></ul> <p>Even more phenomena are possible. For instance, the <a href="/wiki/Bernstein%27s_problem" title="Bernstein's problem">following PDE</a>, arising naturally in the field of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function. </p> <ul><li>If <span class="texhtml mvar" style="font-style:italic;">u</span> is a function on <span class="texhtml"><b>R</b><sup>2</sup></span> with <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 }{\partial x}}{\frac {\frac {\partial u}{\partial x}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}+{\frac {\partial }{\partial y}}{\frac {\frac {\partial u}{\partial y}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x}}{\frac {\frac {\partial u}{\partial x}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}+{\frac {\partial }{\partial y}}{\frac {\frac {\partial u}{\partial y}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851a9f008619f05c599b66d23008be0e646d9a3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:60.7ex; height:11.176ex;" alt="{\displaystyle {\frac {\partial }{\partial x}}{\frac {\frac {\partial u}{\partial x}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}+{\frac {\partial }{\partial y}}{\frac {\frac {\partial u}{\partial y}}{\sqrt {1+\left({\frac {\partial u}{\partial x}}\right)^{2}+\left({\frac {\partial u}{\partial y}}\right)^{2}}}}=0,}"></span> then there are numbers <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span> with <span class="texhtml"><i>u</i>(<i>x</i>, <i>y</i>) = <i>ax</i> + <i>by</i> + <i>c</i></span>.</li></ul> <p>In contrast to the earlier examples, this PDE is <b>nonlinear</b>, owing to the square roots and the squares. A <b>linear</b> PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution. </p> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=2" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A partial differential equation is an equation that involves an unknown function 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 n\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bf67f9d06ca3af619657f8d20ee1322da77174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 2}"></span> variables and (some of) its partial derivatives.<sup id="cite_ref-FOOTNOTEEvans19981–2_4-0" class="reference"><a href="#cite_note-FOOTNOTEEvans19981–2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> That is, for the unknown function <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:U\rightarrow \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:U\rightarrow \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/052fdd40929ab2b19fa448a657c3b569ed2cf961" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.988ex; height:2.509ex;" alt="{\displaystyle u:U\rightarrow \mathbb {R} ,}"></span> of variables <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=(x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(x_{1},\dots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4328c0b5216791ddb0b9aea9cf7d5f460e6481c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.348ex; height:2.843ex;" alt="{\displaystyle x=(x_{1},\dots ,x_{n})}"></span> belonging to the open subset <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, the <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 k^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c348e1c6f8200f15d1d6026fc140d554b272096d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.984ex; height:2.676ex;" alt="{\displaystyle k^{th}}"></span>-order partial differential equation 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 F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>u</mi> <mo>,</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6851c09712a689ae776fbfb75da28e8f95f39557" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.921ex; height:3.176ex;" alt="{\displaystyle F[D^{k}u,D^{k-1}u,\dots ,Du,u,x]=0,}"></span> where <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 F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </msup> <mo>⋯</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>×</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ac1e7844179430b1e86fcda1e3c7ef2e61bddd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.664ex; height:3.343ex;" alt="{\displaystyle F:\mathbb {R} ^{n^{k}}\times \mathbb {R} ^{n^{k-1}}\dots \times \mathbb {R} ^{n}\times \mathbb {R} \times U\rightarrow \mathbb {R} ,}"></span> and <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 the partial derivative operator. </p> <div class="mw-heading mw-heading3"><h3 id="Notation">Notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=3" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Notation_for_differentiation#Partial_derivatives" title="Notation for differentiation">Notation for differentiation § Partial derivatives</a></div> <p>When writing PDEs, it is common to denote partial derivatives using subscripts. For example: <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_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <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> <mi>y</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d47b72d0c7076f075b47e65418747bf61cf75c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.398ex; height:6.343ex;" alt="{\displaystyle u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).}"></span> In the general situation that <span class="texhtml mvar" style="font-style:italic;">u</span> is a function of <span class="texhtml mvar" style="font-style:italic;">n</span> variables, then <span class="texhtml"><i>u</i><sub><i>i</i></sub></span> denotes the first partial derivative relative to the <span class="texhtml mvar" style="font-style:italic;">i</span>-th input, <span class="texhtml"><i>u</i><sub><i>ij</i></sub></span> denotes the second partial derivative relative to the <span class="texhtml mvar" style="font-style:italic;">i</span>-th and <span class="texhtml mvar" style="font-style:italic;">j</span>-th inputs, and so on. </p><p>The Greek letter <span class="texhtml">Δ</span> denotes the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a>; if <span class="texhtml mvar" style="font-style:italic;">u</span> is a function of <span class="texhtml mvar" style="font-style:italic;">n</span> variables, then <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 \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d640338de1d843ba0f4f2870003f43e092fb7a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.201ex; height:2.509ex;" alt="{\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.}"></span> In the physics literature, the Laplace operator is often denoted by <span class="texhtml">∇<sup>2</sup></span>; in the mathematics literature, <span class="texhtml">∇<sup>2</sup><i>u</i></span> may also denote the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a> of <span class="texhtml mvar" style="font-style:italic;">u</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Classification">Classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=4" title="Edit section: Classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Linear_and_nonlinear_equations">Linear and nonlinear equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=5" title="Edit section: Linear and nonlinear equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A PDE is called <b>linear</b> if it is linear in the unknown and its derivatives. For example, for a function <span class="texhtml mvar" style="font-style:italic;">u</span> of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>, a second order linear PDE is of the 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 a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f90217a0377da26da6f0f65113a6cb9cfa42fa1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:99.523ex; height:3.009ex;" alt="{\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+a_{5}(x,y)u_{x}+a_{6}(x,y)u_{y}+a_{7}(x,y)u=f(x,y)}"></span> where <span class="texhtml"><i>a<sub>i</sub></i></span> and <span class="texhtml mvar" style="font-style:italic;"><i>f</i></span> are functions of the independent variables <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> only. (Often the mixed-partial derivatives <span class="texhtml"><i>u<sub>xy</sub></i></span> and <span class="texhtml"><i>u<sub>yx</sub></i></span> will be equated, but this is not required for the discussion of linearity.) If the <span class="texhtml"><i>a<sub>i</sub></i></span> are constants (independent of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>) then the PDE is called <b>linear with constant coefficients</b>. If <span class="texhtml mvar" style="font-style:italic;"><i>f</i></span> is zero everywhere then the linear PDE is <b>homogeneous</b>, otherwise it is <b>inhomogeneous</b>. (This is separate from <a href="/wiki/Asymptotic_homogenization" title="Asymptotic homogenization">asymptotic homogenization</a>, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.) </p><p>Nearest to linear PDEs are <b>semi-linear</b> PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables 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 a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92bf61f31ada2c5e150a664f9f779de7d9d1bce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:75.269ex; height:3.009ex;" alt="{\displaystyle a_{1}(x,y)u_{xx}+a_{2}(x,y)u_{xy}+a_{3}(x,y)u_{yx}+a_{4}(x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}"></span> </p><p>In a <b>quasilinear</b> PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order 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 a_{1}(u_{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}(u_{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf0f117b1383ca71688fa0dee1d270f16fc14e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:112.52ex; height:3.009ex;" alt="{\displaystyle a_{1}(u_{x},u_{y},u,x,y)u_{xx}+a_{2}(u_{x},u_{y},u,x,y)u_{xy}+a_{3}(u_{x},u_{y},u,x,y)u_{yx}+a_{4}(u_{x},u_{y},u,x,y)u_{yy}+f(u_{x},u_{y},u,x,y)=0}"></span> Many of the fundamental PDEs in physics are quasilinear, such as the <a href="/wiki/Einstein_equations" class="mw-redirect" title="Einstein equations">Einstein equations</a> of <a href="/wiki/General_relativity" title="General relativity">general relativity</a> and the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> describing fluid motion. </p><p>A PDE without any linearity properties is called <b>fully <a href="/wiki/Nonlinear_partial_differential_equation" title="Nonlinear partial differential equation">nonlinear</a></b>, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the <a href="/wiki/Monge%E2%80%93Amp%C3%A8re_equation" title="Monge–Ampère equation">Monge–Ampère equation</a>, which arises in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>.<sup id="cite_ref-PrincetonCompanion_5-0" class="reference"><a href="#cite_note-PrincetonCompanion-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Second_order_equations">Second order equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=6" title="Edit section: Second order equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The elliptic/parabolic/hyperbolic classification provides a guide to appropriate <a href="/wiki/Initial_condition" title="Initial condition">initial-</a> and <a href="/wiki/Boundary_value_problem" title="Boundary value problem">boundary conditions</a> and to the <a href="/wiki/Smoothness" title="Smoothness">smoothness</a> of the solutions. Assuming <span class="texhtml"><i>u<sub>xy</sub></i> = <i>u<sub>yx</sub></i></span>, the general linear second-order PDE in two independent variables has the 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 Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>B</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>C</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>(lower order terms)</mtext> </mstyle> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef3d88027a5da9817b38af22a58b7aca87dfda3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.346ex; height:3.009ex;" alt="{\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,}"></span> where the coefficients <span class="texhtml mvar" style="font-style:italic;">A</span>, <span class="texhtml mvar" style="font-style:italic;">B</span>, <span class="texhtml mvar" style="font-style:italic;">C</span>... may depend upon <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>. If <span class="texhtml"><i>A</i><sup>2</sup> + <i>B</i><sup>2</sup> + <i>C</i><sup>2</sup> > 0</span> over a region of the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: <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 Ax^{2}+2Bxy+Cy^{2}+\cdots =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{2}+2Bxy+Cy^{2}+\cdots =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276f9f21d407268719142c0ba673232f6b055115" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.672ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+2Bxy+Cy^{2}+\cdots =0.}"></span> </p><p>More precisely, replacing <span class="texhtml">∂<sub><i>x</i></sub></span> by <span class="texhtml mvar" style="font-style:italic;">X</span>, and likewise for other variables (formally this is done by a <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomial</a>, here a <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a>) being most significant for the classification. </p><p>Just as one classifies <a href="/wiki/Conic_section" title="Conic section">conic sections</a> and quadratic forms into parabolic, hyperbolic, and elliptic based on the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> <span class="texhtml"><i>B</i><sup>2</sup> − 4<i>AC</i></span>, the same can be done for a second-order PDE at a given point. However, the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> in a PDE is given by <span class="texhtml"><i>B</i><sup>2</sup> − <i>AC</i></span> due to the convention of the <span class="texhtml mvar" style="font-style:italic;">xy</span> term being <span class="texhtml">2<i>B</i></span> rather than <span class="texhtml mvar" style="font-style:italic;">B</span>; formally, the discriminant (of the associated quadratic form) is <span class="texhtml">(2<i>B</i>)<sup>2</sup> − 4<i>AC</i> = 4(<i>B</i><sup>2</sup> − <i>AC</i>)</span>, with the factor of 4 dropped for simplicity. </p> <ol><li><span class="texhtml"><i>B</i><sup>2</sup> − <i>AC</i> < 0</span> (<i><a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic partial differential equation</a></i>): Solutions of <a href="/wiki/Elliptic_partial_differential_equation" title="Elliptic partial differential equation">elliptic PDEs</a> are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a> are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where <span class="texhtml"><i>x</i> < 0</span>. By change of variables, the equation can always be expressed in the 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 u_{xx}+u_{yy}+\cdots =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{xx}+u_{yy}+\cdots =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e26460136b65b2655845087d115fd81e5e34ed3a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.95ex; height:2.843ex;" alt="{\displaystyle u_{xx}+u_{yy}+\cdots =0,}"></span>where x and y correspond to changed variables. This justifies <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace equation</a> as an example of this type.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><span class="texhtml"><i>B</i><sup>2</sup> − <i>AC</i> = 0</span> (<i><a href="/wiki/Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic partial differential equation</a></i>): Equations that are <a href="/wiki/Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic</a> at every point can be transformed into a form analogous to the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a> by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where <span class="texhtml"><i>x</i> = 0</span>. By change of variables, the equation can always be expressed in the 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 u_{xx}+\cdots =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{xx}+\cdots =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d8841d080c242c18f879d56af3cf3012adfe2c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.914ex; height:2.509ex;" alt="{\displaystyle u_{xx}+\cdots =0,}"></span>where x correspond to changed variables. This justifies <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>, which are of 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="{\textstyle u_{t}-u_{xx}+\cdots =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle u_{t}-u_{xx}+\cdots =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c7b8c961ef2bf97b1ade977baf004be42889ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.263ex; height:2.509ex;" alt="{\textstyle u_{t}-u_{xx}+\cdots =0}"></span>, as an example of this type.<sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><span class="texhtml"><i>B</i><sup>2</sup> − <i>AC</i> > 0</span> (<i><a href="/wiki/Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">hyperbolic partial differential equation</a></i>): <a href="/wiki/Hyperbolic_partial_differential_equation" title="Hyperbolic partial differential equation">hyperbolic</a> equations retain any discontinuities of functions or derivatives in the initial data. An example is the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where <span class="texhtml"><i>x</i> > 0</span>. By change of variables, the equation can always be expressed in the 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 u_{xx}-u_{yy}+\cdots =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{xx}-u_{yy}+\cdots =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f83e680eaa960e45431d7185527d7c0ec5e38a2a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.95ex; height:2.843ex;" alt="{\displaystyle u_{xx}-u_{yy}+\cdots =0,}"></span>where x and y correspond to changed variables. This justifies <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> as an example of this type.<sup id="cite_ref-:0_6-2" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ol> <p>If there are <span class="texhtml mvar" style="font-style:italic;">n</span> independent variables <span class="texhtml"><i>x</i><sub>1</sub>, <i>x</i><sub>2 </sub>, …, <i>x</i><sub><i>n</i></sub></span>, a general linear partial differential equation of second order has the 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 Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>u</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <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> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>lower-order terms</mtext> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d62e51f94cd0adeea38ef6b5daa541be11f4ca2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.369ex; height:7.176ex;" alt="{\displaystyle Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad +{\text{lower-order terms}}=0.}"></span> </p><p>The classification depends upon the signature of the <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of the coefficient matrix <span class="texhtml"><i>a</i><sub><i>i</i>,<i>j</i></sub></span>. </p> <ol><li>Elliptic: the eigenvalues are all positive or all negative.</li> <li>Parabolic: the eigenvalues are all positive or all negative, except one that is zero.</li> <li>Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.</li> <li>Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ol> <p>The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the <a href="/wiki/Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a>, the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>, and the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>. </p><p>However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the <a href="/wiki/Euler%E2%80%93Tricomi_equation" title="Euler–Tricomi equation">Euler–Tricomi equation</a>; varying from elliptic to hyperbolic for different <a href="/wiki/Region_(mathematics)" class="mw-redirect" title="Region (mathematics)">regions</a> of the domain, as well as higher-order PDEs, but such knowledge is more specialized. </p> <div class="mw-heading mw-heading3"><h3 id="Systems_of_first-order_equations_and_characteristic_surfaces">Systems of first-order equations and characteristic surfaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=7" title="Edit section: Systems of first-order equations and characteristic surfaces"><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/First-order_partial_differential_equation" title="First-order partial differential equation">First-order partial differential equation</a></div> <p>The classification of partial differential equations can be extended to systems of first-order equations, where the unknown <span class="texhtml mvar" style="font-style:italic;">u</span> is now a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> with <span class="texhtml mvar" style="font-style:italic;">m</span> components, and the coefficient matrices <span class="texhtml mvar" style="font-style:italic;">A<sub>ν</sub></span> are <span class="texhtml mvar" style="font-style:italic;">m</span> by <span class="texhtml mvar" style="font-style:italic;">m</span> matrices for <span class="texhtml"><i>ν</i> = 1, 2, …, <i>n</i></span>. The partial differential equation takes the 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 Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>u</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>B</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8150ae0f89011c505554f92aac4b1bdf459d49e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.699ex; height:6.843ex;" alt="{\displaystyle Lu=\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial u}{\partial x_{\nu }}}+B=0,}"></span> where the coefficient matrices <span class="texhtml mvar" style="font-style:italic;">A<sub>ν</sub></span> and the vector <span class="texhtml mvar" style="font-style:italic;">B</span> may depend upon <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">u</span>. If a <a href="/wiki/Hypersurface" title="Hypersurface">hypersurface</a> <span class="texhtml mvar" style="font-style:italic;">S</span> is given in the implicit 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 \varphi (x_{1},x_{2},\ldots ,x_{n})=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x_{1},x_{2},\ldots ,x_{n})=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19381d47173180959d2881b39db35ec712eb5ea3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.765ex; height:2.843ex;" alt="{\displaystyle \varphi (x_{1},x_{2},\ldots ,x_{n})=0,}"></span> where <span class="texhtml mvar" style="font-style:italic;">φ</span> has a non-zero gradient, then <span class="texhtml mvar" style="font-style:italic;">S</span> is a <b>characteristic surface</b> for the <a href="/wiki/Differential_operator" title="Differential operator">operator</a> <span class="texhtml mvar" style="font-style:italic;">L</span> at a given point if the characteristic form vanishes: <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 Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fae8017409f092e75a9b50bf8e2febcdcdd407c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.575ex; height:7.509ex;" alt="{\displaystyle Q\left({\frac {\partial \varphi }{\partial x_{1}}},\ldots ,{\frac {\partial \varphi }{\partial x_{n}}}\right)=\det \left[\sum _{\nu =1}^{n}A_{\nu }{\frac {\partial \varphi }{\partial x_{\nu }}}\right]=0.}"></span> </p><p>The geometric interpretation of this condition is as follows: if data for <span class="texhtml mvar" style="font-style:italic;">u</span> are prescribed on the surface <span class="texhtml mvar" style="font-style:italic;">S</span>, then it may be possible to determine the normal derivative of <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml mvar" style="font-style:italic;">S</span> from the differential equation. If the data on <span class="texhtml mvar" style="font-style:italic;">S</span> and the differential equation determine the normal derivative of <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml mvar" style="font-style:italic;">S</span>, then <span class="texhtml mvar" style="font-style:italic;">S</span> is non-characteristic. If the data on <span class="texhtml mvar" style="font-style:italic;">S</span> and the differential equation <i>do not</i> determine the normal derivative of <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml mvar" style="font-style:italic;">S</span>, then the surface is <b>characteristic</b>, and the differential equation restricts the data on <span class="texhtml mvar" style="font-style:italic;">S</span>: the differential equation is <i>internal</i> to <span class="texhtml mvar" style="font-style:italic;">S</span>. </p> <ol><li>A first-order system <span class="texhtml"><i>Lu</i> = 0</span> is <i>elliptic</i> if no surface is characteristic for <span class="texhtml mvar" style="font-style:italic;">L</span>: the values of <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml mvar" style="font-style:italic;">S</span> and the differential equation always determine the normal derivative of <span class="texhtml mvar" style="font-style:italic;">u</span> on <span class="texhtml mvar" style="font-style:italic;">S</span>.</li> <li>A first-order system is <i>hyperbolic</i> at a point if there is a <b>spacelike</b> surface <span class="texhtml mvar" style="font-style:italic;">S</span> with normal <span class="texhtml mvar" style="font-style:italic;">ξ</span> at that point. This means that, given any non-trivial vector <span class="texhtml mvar" style="font-style:italic;">η</span> orthogonal to <span class="texhtml mvar" style="font-style:italic;">ξ</span>, and a scalar multiplier <span class="texhtml mvar" style="font-style:italic;">λ</span>, the equation <span class="texhtml"><i>Q</i>(<i>λξ</i> + <i>η</i>) = 0</span> has <span class="texhtml mvar" style="font-style:italic;">m</span> real roots <span class="texhtml"><i>λ</i><sub>1</sub>, <i>λ</i><sub>2</sub>, …, <i>λ</i><sub><i>m</i></sub></span>. The system is <b>strictly hyperbolic</b> if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form <span class="texhtml"><i>Q</i>(<i>ζ</i>) = 0</span> defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has <span class="texhtml mvar" style="font-style:italic;">nm</span> sheets, and the axis <span class="texhtml"><i>ζ</i> = <i>λξ</i></span> runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Analytical_solutions">Analytical solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=8" title="Edit section: Analytical solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Separation_of_variables">Separation of variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=9" title="Edit section: Separation of variables"><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/Separable_partial_differential_equation" title="Separable partial differential equation">Separable partial differential equation</a></div> <p>Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is <i>the</i> solution (this also applies to ODEs). We assume as an <a href="/wiki/Ansatz" title="Ansatz">ansatz</a> that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve. </p><p>This is possible for simple PDEs, which are called <a href="/wiki/Separable_partial_differential_equation" title="Separable partial differential equation">separable partial differential equations</a>, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to <a href="/wiki/Diagonal_matrices" class="mw-redirect" title="Diagonal matrices">diagonal matrices</a> – thinking of "the value for fixed <span class="texhtml mvar" style="font-style:italic;">x</span>" as a coordinate, each coordinate can be understood separately. </p><p>This generalizes to the <a href="/wiki/Method_of_characteristics" title="Method of characteristics">method of characteristics</a>, and is also used in <a href="/wiki/Integral_transform" title="Integral transform">integral transforms</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Method_of_characteristics">Method of characteristics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=10" title="Edit section: Method of characteristics"><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/Method_of_characteristics" title="Method of characteristics">Method of characteristics</a></div> <p>The characteristic surface in <span class="texhtml"><i>n</i> = <i>2</i>-</span>dimensional space is called a <b>characteristic curve</b>.<sup id="cite_ref-FOOTNOTEZachmanoglouThoe1986115–116_9-0" class="reference"><a href="#cite_note-FOOTNOTEZachmanoglouThoe1986115–116-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In special cases, one can find characteristic curves on which the first-order PDE reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the <a href="/wiki/Method_of_characteristics" title="Method of characteristics">method of characteristics</a>. </p><p>More generally, applying the method to first-order PDEs in higher dimensions, one may find characteristic surfaces. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_transform">Integral transform</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=11" title="Edit section: Integral transform"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <a href="/wiki/Integral_transform" title="Integral transform">integral transform</a> may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator. </p><p>An important example of this is <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, which diagonalizes the heat equation using the <a href="/wiki/Eigenbasis" class="mw-redirect" title="Eigenbasis">eigenbasis</a> of sinusoidal waves. </p><p>If the domain is finite or periodic, an infinite sum of solutions such as a <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> is appropriate, but an integral of solutions such as a <a href="/wiki/Fourier_integral" class="mw-redirect" title="Fourier integral">Fourier integral</a> is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral. </p> <div class="mw-heading mw-heading3"><h3 id="Change_of_variables">Change of variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=12" title="Edit section: Change of variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Often a PDE can be reduced to a simpler form with a known solution by a suitable <a href="/wiki/Change_of_variables_(PDE)" title="Change of variables (PDE)">change of variables</a>. For example, the <a href="/wiki/Black%E2%80%93Scholes_equation" title="Black–Scholes equation">Black–Scholes equation</a> <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 V}{\partial t}}+{\tfrac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>r</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>r</mi> <mi>V</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial V}{\partial t}}+{\tfrac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd96e615c439c1b57541069c7709060cb12242e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:37.688ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial V}{\partial t}}+{\tfrac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0}"></span> is reducible to the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a> <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 u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <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>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49feaa1e2aba0490c8eb7c2870b391704c433fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.146ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial u}{\partial \tau }}={\frac {\partial ^{2}u}{\partial x^{2}}}}"></span> by the change of variables<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<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}V(S,t)&=v(x,\tau ),\\[5px]x&=\ln \left(S\right),\\[5px]\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\[5px]v(x,\tau )&=e^{-\alpha x-\beta \tau }u(x,\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="0.8em 0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>V</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>τ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>x</mi> <mo>−</mo> <mi>β</mi> <mi>τ</mi> </mrow> </msup> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V(S,t)&=v(x,\tau ),\\[5px]x&=\ln \left(S\right),\\[5px]\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\[5px]v(x,\tau )&=e^{-\alpha x-\beta \tau }u(x,\tau ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36d50d536184b7e1291e53a595db0b79fad14d9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:25.827ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}V(S,t)&=v(x,\tau ),\\[5px]x&=\ln \left(S\right),\\[5px]\tau &={\tfrac {1}{2}}\sigma ^{2}(T-t),\\[5px]v(x,\tau )&=e^{-\alpha x-\beta \tau }u(x,\tau ).\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Fundamental_solution">Fundamental solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=13" title="Edit section: Fundamental solution"><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/Fundamental_solution" title="Fundamental solution">Fundamental solution</a></div> <p>Inhomogeneous equations<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. (July 2020)">clarification needed</span></a></i>]</sup> can often be solved (for constant coefficient PDEs, always be solved) by finding the <a href="/wiki/Fundamental_solution" title="Fundamental solution">fundamental solution</a> (the solution for a point source <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 P(D)u=\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mi>u</mi> <mo>=</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(D)u=\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e532c6de09f1aa78ff5076a4657a2c5e7f91392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.956ex; height:2.843ex;" alt="{\displaystyle P(D)u=\delta }"></span>), then taking the <a href="/wiki/Convolution" title="Convolution">convolution</a> with the boundary conditions to get the solution. </p><p>This is analogous in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> to understanding a filter by its <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Superposition_principle">Superposition principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=14" title="Edit section: Superposition principle"><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">Further information: <a href="/wiki/Superposition_principle" title="Superposition principle">Superposition principle</a></div> <p>The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example <span class="texhtml">sin <i>x</i> + sin <i>x</i> = 2 sin <i>x</i></span>. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If <span class="texhtml"><i>u</i><sub>1</sub></span> and <span class="texhtml"><i>u</i><sub>2</sub></span> are solutions of linear PDE in some function space <span class="texhtml mvar" style="font-style:italic;">R</span>, then <span class="texhtml"><i>u</i> = <i>c</i><sub>1</sub><i>u</i><sub>1</sub> + <i>c</i><sub>2</sub><i>u</i><sub>2</sub></span> with any constants <span class="texhtml"><i>c</i><sub>1</sub></span> and <span class="texhtml"><i>c</i><sub>2</sub></span> are also a solution of that PDE in the same function space. </p> <div class="mw-heading mw-heading3"><h3 id="Methods_for_non-linear_equations">Methods for non-linear equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=15" title="Edit section: Methods for non-linear equations"><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/Nonlinear_partial_differential_equation" title="Nonlinear partial differential equation">nonlinear partial differential equation</a></div> <p>There are no generally applicable analytical methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the <a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a>) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>). </p><p>Nevertheless, some techniques can be used for several types of equations. The <a href="/wiki/H-principle" class="mw-redirect" title="H-principle"><span class="texhtml mvar" style="font-style:italic;">h</span>-principle</a> is the most powerful method to solve <a href="/wiki/Underdetermined_system" title="Underdetermined system">underdetermined</a> equations. The <a href="/w/index.php?title=Riquier%E2%80%93Janet_theory&action=edit&redlink=1" class="new" title="Riquier–Janet theory (page does not exist)">Riquier–Janet theory</a> is an effective method for obtaining information about many analytic <a href="/wiki/Overdetermined_system" title="Overdetermined system">overdetermined</a> systems. </p><p>The <a href="/wiki/Method_of_characteristics" title="Method of characteristics">method of characteristics</a> can be used in some very special cases to solve nonlinear partial differential equations.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>In some cases, a PDE can be solved via <a href="/wiki/Perturbation_analysis" class="mw-redirect" title="Perturbation analysis">perturbation analysis</a> in which the solution is considered to be a correction to an equation with a known solution. Alternatives are <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> techniques from simple <a href="/wiki/Finite_difference" title="Finite difference">finite difference</a> schemes to the more mature <a href="/wiki/Multigrid" class="mw-redirect" title="Multigrid">multigrid</a> and <a href="/wiki/Finite_element_method" title="Finite element method">finite element methods</a>. Many interesting problems in science and engineering are solved in this way using <a href="/wiki/Computer" title="Computer">computers</a>, sometimes high performance <a href="/wiki/Supercomputer" title="Supercomputer">supercomputers</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Lie_group_method">Lie group method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=16" title="Edit section: Lie group method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From 1870 <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a>'s work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>, be referred, to a common source; and that ordinary differential equations which admit the same <a href="/wiki/Infinitesimal_transformation" title="Infinitesimal transformation">infinitesimal transformations</a> present comparable difficulties of integration. He also emphasized the subject of <a href="/wiki/Contact_transformation" class="mw-redirect" title="Contact transformation">transformations of contact</a>. </p><p>A general approach to solving PDEs uses the symmetry property of differential equations, the continuous <a href="/wiki/Infinitesimal_transformation" title="Infinitesimal transformation">infinitesimal transformations</a> of solutions to solutions (<a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a>). Continuous <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <a href="/wiki/Lie_algebras" class="mw-redirect" title="Lie algebras">Lie algebras</a> and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its <a href="/wiki/Lax_pair" title="Lax pair">Lax pairs</a>, recursion operators, <a href="/wiki/B%C3%A4cklund_transform" title="Bäcklund transform">Bäcklund transform</a> and finally finding exact analytic solutions to the PDE. </p><p>Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines. </p> <div class="mw-heading mw-heading3"><h3 id="Semi-analytical_methods">Semi-analytical methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=17" title="Edit section: Semi-analytical methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Adomian_decomposition_method" title="Adomian decomposition method">Adomian decomposition method</a>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> the <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Lyapunov</a> artificial small parameter method, and his <a href="/wiki/Homotopy_perturbation_method" class="mw-redirect" title="Homotopy perturbation method">homotopy perturbation method</a> are all special cases of the more general <a href="/wiki/Homotopy_analysis_method" title="Homotopy analysis method">homotopy analysis method</a>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbation theory</a>, thus giving these methods greater flexibility and solution generality. </p> <div class="mw-heading mw-heading2"><h2 id="Numerical_solutions">Numerical solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=18" title="Edit section: Numerical solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The three most widely used <a href="/wiki/Numerical_partial_differential_equations" class="mw-redirect" title="Numerical partial differential equations">numerical methods to solve PDEs</a> are the <a href="/wiki/Finite_element_analysis" class="mw-redirect" title="Finite element analysis">finite element method</a> (FEM), <a href="/wiki/Finite_volume_method" title="Finite volume method">finite volume methods</a> (FVM) and <a href="/wiki/Finite_difference_method" title="Finite difference method">finite difference methods</a> (FDM), as well other kind of methods called <a href="/wiki/Meshfree_methods" title="Meshfree methods">meshfree methods</a>, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version <a href="/wiki/Hp-FEM" title="Hp-FEM">hp-FEM</a>. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), <a href="/wiki/Extended_finite_element_method" title="Extended finite element method">extended finite element method</a> (XFEM), <a href="/wiki/Spectral_element_method" title="Spectral element method">spectral finite element method</a> (SFEM), <a href="/wiki/Meshfree_methods" title="Meshfree methods">meshfree finite element method</a>, <a href="/wiki/Discontinuous_Galerkin_method" title="Discontinuous Galerkin method">discontinuous Galerkin finite element method</a> (DGFEM), <a href="/w/index.php?title=Element-free_Galerkin_method&action=edit&redlink=1" class="new" title="Element-free Galerkin method (page does not exist)">element-free Galerkin method</a> (EFGM), <a href="/w/index.php?title=Interpolating_element-free_Galerkin_method&action=edit&redlink=1" class="new" title="Interpolating element-free Galerkin method (page does not exist)">interpolating element-free Galerkin method</a> (IEFGM), etc. </p> <div class="mw-heading mw-heading3"><h3 id="Finite_element_method">Finite element method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=19" title="Edit section: Finite element method"><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/Finite_element_method" title="Finite element method">Finite element method</a></div> <p>The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc. </p> <div class="mw-heading mw-heading3"><h3 id="Finite_difference_method">Finite difference method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=20" title="Edit section: Finite difference method"><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/Finite_difference_method" title="Finite difference method">Finite difference method</a></div> <p>Finite-difference methods are numerical methods for approximating the solutions to differential equations using <a href="/wiki/Finite_difference" title="Finite difference">finite difference</a> equations to approximate derivatives. </p> <div class="mw-heading mw-heading3"><h3 id="Finite_volume_method">Finite volume method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=21" title="Edit section: Finite volume method"><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/Finite_volume_method" title="Finite volume method">Finite volume method</a></div> <p>Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design. </p> <div class="mw-heading mw-heading3"><h3 id="Neural_networks">Neural networks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=22" title="Edit section: Neural networks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Deep_learning#Partial_differential_equations" title="Deep learning">Deep learning § Partial differential equations</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Deep_learning&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> One example is the reconstructing fluid flow governed by the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier-Stokes equations</a>. Using physics informed neural networks does not require the often expensive mesh generation that conventional <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">CFD</a> methods rely on.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></div></div> <div class="mw-heading mw-heading2"><h2 id="Weak_solutions">Weak solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=23" title="Edit section: Weak solutions"><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/Weak_solution" title="Weak solution">Weak solution</a></div> <p>Weak solutions are functions that satisfy the PDE, yet in other meanings than regular sense. The meaning for this term may differ with context, and one of the most commonly used definitions is based on the notion of <a href="/wiki/Schwartz_distribution" class="mw-redirect" title="Schwartz distribution">distributions</a>. </p><p>An example<sup id="cite_ref-FOOTNOTEEvans1998chpt._6._Second-Order_Elliptic_Equations_19-0" class="reference"><a href="#cite_note-FOOTNOTEEvans1998chpt._6._Second-Order_Elliptic_Equations-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> for the definition of a weak solution is as follows: </p><p>Consider the boundary-value problem 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 {\begin{aligned}Lu&=f\quad {\text{in }}U,\\u&=0\quad {\text{on }}\partial U,\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> <mi>L</mi> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>in </mtext> </mrow> <mi>U</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>u</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>on </mtext> </mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Lu&=f\quad {\text{in }}U,\\u&=0\quad {\text{on }}\partial U,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9493dc7a21d2c8d4c1423a97528f6686e51fd004" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.031ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}Lu&=f\quad {\text{in }}U,\\u&=0\quad {\text{on }}\partial U,\end{aligned}}}"></span> where <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 Lu=-\sum _{i,j}\partial _{j}(a^{ij}\partial _{i}u)+\sum _{i}b^{i}\partial _{i}u+cu}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>u</mi> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <mi>c</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Lu=-\sum _{i,j}\partial _{j}(a^{ij}\partial _{i}u)+\sum _{i}b^{i}\partial _{i}u+cu}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f0899e3b9930ac8421b259c04bc198749df02d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.893ex; height:5.843ex;" alt="{\displaystyle Lu=-\sum _{i,j}\partial _{j}(a^{ij}\partial _{i}u)+\sum _{i}b^{i}\partial _{i}u+cu}"></span> denotes a second-order partial differential operator in <b>divergence form</b>. </p><p>We say a <a href="/wiki/Sobolev_space" title="Sobolev space"><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 u\in H_{0}^{1}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in H_{0}^{1}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6958243dc499b25156917eda09991f6966fd122b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.92ex; height:3.176ex;" alt="{\displaystyle u\in H_{0}^{1}(U)}"></span></a> is a weak solution if <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 _{U}[\sum _{i,j}a^{ij}(\partial _{i}u)(\partial _{j}v)+\sum _{i}b^{i}(\partial _{i}u)v+cuv]dx=\int _{U}fvdx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">[</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>u</mi> <mo stretchy="false">)</mo> <mi>v</mi> <mo>+</mo> <mi>c</mi> <mi>u</mi> <mi>v</mi> <mo stretchy="false">]</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mi>f</mi> <mi>v</mi> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{U}[\sum _{i,j}a^{ij}(\partial _{i}u)(\partial _{j}v)+\sum _{i}b^{i}(\partial _{i}u)v+cuv]dx=\int _{U}fvdx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8cb5f0f1c01990c014b663f90e0f5935559169" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.535ex; height:6.676ex;" alt="{\displaystyle \int _{U}[\sum _{i,j}a^{ij}(\partial _{i}u)(\partial _{j}v)+\sum _{i}b^{i}(\partial _{i}u)v+cuv]dx=\int _{U}fvdx}"></span> for every <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 v\in H_{0}^{1}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in H_{0}^{1}(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eda4503ee7403f39fc4f9ce20131cb5e2a30de4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.718ex; height:3.176ex;" alt="{\displaystyle v\in H_{0}^{1}(U)}"></span>, which can be derived by a formal integral by parts. </p><p>An example for a weak solution is as follows: <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 \phi (x)={\frac {1}{4\pi }}{\frac {1}{|x|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (x)={\frac {1}{4\pi }}{\frac {1}{|x|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d10e34c0d707353841d99a4ee084136cdaf42b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.413ex; height:6.009ex;" alt="{\displaystyle \phi (x)={\frac {1}{4\pi }}{\frac {1}{|x|}}}"></span> is a weak solution satisfying <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}\phi =\delta {\text{ in }}R^{3}}"> <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>ϕ</mi> <mo>=</mo> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> in </mtext> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\phi =\delta {\text{ in }}R^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9a34842dbbfd6d016c08711746c5aa5110dcb3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.442ex; height:3.009ex;" alt="{\displaystyle \nabla ^{2}\phi =\delta {\text{ in }}R^{3}}"></span> in distributional sense, as formally, <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 _{R^{3}}\nabla ^{2}\phi (x)\psi (x)dx=\int _{R^{3}}\phi (x)\nabla ^{2}\psi (x)dx=\psi (0){\text{ for }}\psi \in C_{c}^{\infty }(R^{3}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msub> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>ψ</mi> <mo>∈</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{R^{3}}\nabla ^{2}\phi (x)\psi (x)dx=\int _{R^{3}}\phi (x)\nabla ^{2}\psi (x)dx=\psi (0){\text{ for }}\psi \in C_{c}^{\infty }(R^{3}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5521edc81aac94902d1f1a17612ce793d2f9d3e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:65.337ex; height:5.676ex;" alt="{\displaystyle \int _{R^{3}}\nabla ^{2}\phi (x)\psi (x)dx=\int _{R^{3}}\phi (x)\nabla ^{2}\psi (x)dx=\psi (0){\text{ for }}\psi \in C_{c}^{\infty }(R^{3}).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Well-posedness">Well-posedness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=24" title="Edit section: Well-posedness"><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/Well-posed_problem" title="Well-posed problem">Well-posed problem</a></div> <p>Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have: </p> <ul><li>an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE</li> <li>by <a href="/wiki/Continuity_(mathematics)" class="mw-redirect" title="Continuity (mathematics)">continuously</a> changing the free choices, one continuously changes the corresponding solution</li></ul> <p>This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed. </p> <div class="mw-heading mw-heading3"><h3 id="The_energy_method">The energy method</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=25" title="Edit section: The energy method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems (IBVP).<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE 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 {\partial u}{\partial t}}+\alpha {\frac {\partial u}{\partial x}}=0,\quad x\in [a,b],t>0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}+\alpha {\frac {\partial u}{\partial x}}=0,\quad x\in [a,b],t>0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cba9b9bfb3028cc78a61bac020ded390a6615292" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.42ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}+\alpha {\frac {\partial u}{\partial x}}=0,\quad x\in [a,b],t>0,}"></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 \alpha \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5770f8ec95300cbfde18eb59c49a11f12adbcd91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.749ex; height:2.676ex;" alt="{\displaystyle \alpha \neq 0}"></span> is a constant and <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 u(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</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 u(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e6a2bc1b46671c8ec2bd22f48f53e4aa6b9597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.342ex; height:2.843ex;" alt="{\displaystyle u(x,t)}"></span> is an unknown function with initial condition <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 u(x,0)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(x,0)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca093fdac61eb7caaac35db526d720decfa228c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.181ex; height:2.843ex;" alt="{\displaystyle u(x,0)=f(x)}"></span>. Multiplying with <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></span> and integrating over the domain 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 \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x+\alpha \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x=0.}"> <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> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <mi>α</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x+\alpha \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb563cc629b39a4c91a80aa771c2c6ea4ea416c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.072ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x+\alpha \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x=0.}"></span> </p><p>Using that <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 _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x={\frac {1}{2}}{\frac {\partial }{\partial t}}\|u\|^{2}\quad {\text{and}}\quad \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x={\frac {1}{2}}u(b,t)^{2}-{\frac {1}{2}}u(a,t)^{2},}"> <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> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x={\frac {1}{2}}{\frac {\partial }{\partial t}}\|u\|^{2}\quad {\text{and}}\quad \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x={\frac {1}{2}}u(b,t)^{2}-{\frac {1}{2}}u(a,t)^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72704e420504060018d0024ad78b417eac08416e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:68.973ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}u{\frac {\partial u}{\partial t}}\mathrm {d} x={\frac {1}{2}}{\frac {\partial }{\partial t}}\|u\|^{2}\quad {\text{and}}\quad \int _{a}^{b}u{\frac {\partial u}{\partial x}}\mathrm {d} x={\frac {1}{2}}u(b,t)^{2}-{\frac {1}{2}}u(a,t)^{2},}"></span> where <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a> has been used for the first relationship, we get <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 }{\partial t}}\|u\|^{2}+\alpha u(b,t)^{2}-\alpha u(a,t)^{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>α</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>α</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\|u\|^{2}+\alpha u(b,t)^{2}-\alpha u(a,t)^{2}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d3da73cb2944ebf5aa6ca92dd02256118d4e89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.627ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\|u\|^{2}+\alpha u(b,t)^{2}-\alpha u(a,t)^{2}=0.}"></span> </p><p>Here <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 \|\cdot \|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\cdot \|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.004ex; height:2.843ex;" alt="{\displaystyle \|\cdot \|}"></span> denotes the standard <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 L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"></span> <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>. For well-posedness we require that the energy of the solution is non-increasing, i.e. that <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 {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5c0d7e1d3f0a599347e08e761978304b2fc8d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.332ex; height:3.843ex;" alt="{\textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}"></span>, which is achieved by specifying <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" 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 u}"></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=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}"></span> if <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 >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> and 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=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229dd3f0f42d375fa257bdc1389f92f7b225c415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}"></span> if <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 <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9d48dc3d4d98b4c949bf36f18559a74bc3d87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <0}"></span>. This corresponds to only imposing boundary conditions at the inflow. Well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <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 {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>u</mi> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≤</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5c0d7e1d3f0a599347e08e761978304b2fc8d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.332ex; height:3.843ex;" alt="{\textstyle {\frac {\partial }{\partial t}}\|u\|^{2}\leq 0}"></span> holds when all data are set to zero. </p> <div class="mw-heading mw-heading3"><h3 id="Existence_of_local_solutions">Existence of local solutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Partial_differential_equation&action=edit&section=26" title="Edit section: Existence of local solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cauchy%E2%80%93Kowalevski_theorem" class="mw-redirect" title="Cauchy–Kowalevski theorem">Cauchy–Kowalevski theorem</a> for Cauchy initial value problems essentially states that if the terms in a partial differential equation are all made up of <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a> and a certain transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be non-characteristic with respect to the partial differential operator), then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an <a href="/wiki/Lewy%27s_example" title="Lewy's example">example</a> discovered by <a href="/wiki/Hans_Lewy" title="Hans Lewy">Hans Lewy</a> in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have derivatives of all orders) but not analytic for which no solution exists. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions. </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=Partial_differential_equation&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Some common PDEs</b> </p> <ul><li><a href="/wiki/Acoustic_wave_equation" title="Acoustic wave equation">Acoustic wave equation</a></li> <li><a href="/wiki/Burgers%27_equation" title="Burgers' equation">Burgers' equation</a></li> <li><a href="/wiki/Continuity_equation" title="Continuity equation">Continuity equation</a></li> <li><a href="/wiki/Heat_equation" title="Heat equation">Heat equation</a></li> <li><a href="/wiki/Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a></li> <li><a href="/wiki/Klein%E2%80%93Gordon_equation" title="Klein–Gordon equation">Klein–Gordon equation</a></li> <li><a href="/wiki/Jacobi_equation" class="mw-redirect" title="Jacobi equation">Jacobi equation</a></li> <li><a href="/wiki/Lagrange_equation" class="mw-redirect" title="Lagrange equation">Lagrange equation</a></li> <li><a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier-Stokes equation</a></li> <li><a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a></li> <li><a href="/wiki/Reaction%E2%80%93diffusion_system" title="Reaction–diffusion system">Reaction–diffusion system</a></li> <li><a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a></li> <li><a href="/wiki/Wave_equation" title="Wave equation">Wave equation</a></li></ul> <p><b>Types of boundary conditions</b> </p> <ul><li><a href="/wiki/Dirichlet_boundary_condition" title="Dirichlet boundary condition">Dirichlet boundary condition</a></li> <li><a href="/wiki/Neumann_boundary_condition" title="Neumann boundary condition">Neumann boundary condition</a></li> <li><a href="/wiki/Robin_boundary_condition" title="Robin boundary condition">Robin boundary condition</a></li> <li><a href="/wiki/Cauchy_problem" title="Cauchy problem">Cauchy problem</a></li></ul> <p><b>Various topics</b> </p> <ul><li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet bundle</a></li> <li><a href="/wiki/Laplace_transform_applied_to_differential_equations" title="Laplace transform applied to differential equations">Laplace transform applied to differential equations</a></li> <li><a href="/wiki/List_of_dynamical_systems_and_differential_equations_topics" title="List of dynamical systems and differential equations topics">List of dynamical systems and differential equations topics</a></li> <li><a href="/wiki/Matrix_differential_equation" title="Matrix differential equation">Matrix differential equation</a></li> <li><a href="/wiki/Numerical_partial_differential_equations" class="mw-redirect" title="Numerical partial differential equations">Numerical partial differential equations</a></li> <li><a href="/wiki/Partial_differential_algebraic_equation" title="Partial differential algebraic equation">Partial differential algebraic equation</a></li> <li><a href="/wiki/Recurrence_relation" title="Recurrence relation">Recurrence relation</a></li> <li><a href="/wiki/Stochastic_processes_and_boundary_value_problems" title="Stochastic processes and boundary value problems">Stochastic processes and boundary value problems</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=Partial_differential_equation&action=edit&section=28" 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 mw-references-columns"><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"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://cordis.europa.eu/project/id/801867">"Regularity and singularities in elliptic PDE's: beyond monotonicity formulas | EllipticPDE Project | Fact Sheet | H2020"</a>. <i>CORDIS | European Commission</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CORDIS+%7C+European+Commission&rft.atitle=Regularity+and+singularities+in+elliptic+PDE%27s%3A+beyond+monotonicity+formulas+%7C+EllipticPDE+Project+%7C+Fact+Sheet+%7C+H2020&rft_id=https%3A%2F%2Fcordis.europa.eu%2Fproject%2Fid%2F801867&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlainerman2010" class="citation book cs1"><a href="/wiki/Sergiu_Klainerman" title="Sergiu Klainerman">Klainerman, Sergiu</a> (2010). "PDE as a Unified Subject". In Alon, N.; Bourgain, J.; Connes, A.; Gromov, M.; Milman, V. (eds.). <i>Visions in Mathematics</i>. Modern Birkhäuser Classics. Basel: Birkhäuser. pp. 279–315. <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%2F978-3-0346-0422-2_10">10.1007/978-3-0346-0422-2_10</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-0346-0421-5" title="Special:BookSources/978-3-0346-0421-5"><bdi>978-3-0346-0421-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=PDE+as+a+Unified+Subject&rft.btitle=Visions+in+Mathematics&rft.place=Basel&rft.series=Modern+Birkh%C3%A4user+Classics&rft.pages=279-315&rft.pub=Birkh%C3%A4user&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-3-0346-0422-2_10&rft.isbn=978-3-0346-0421-5&rft.aulast=Klainerman&rft.aufirst=Sergiu&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErdoğanTzirakis2016" class="citation book cs1">Erdoğan, M. 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Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-14904-5" title="Special:BookSources/978-1-107-14904-5"><bdi>978-1-107-14904-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dispersive+Partial+Differential+Equations%3A+Wellposedness+and+Applications&rft.place=Cambridge&rft.series=London+Mathematical+Society+Student+Texts&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=978-1-107-14904-5&rft.aulast=Erdo%C4%9Fan&rft.aufirst=M.+Burak&rft.au=Tzirakis%2C+Nikolaos&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fdispersive-partial-differential-equations%2F2DC65286BA080B54EB659E42A553CA88&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEEvans19981–2-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEvans19981–2_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEvans1998">Evans 1998</a>, pp. 1–2.</span> </li> <li id="cite_note-PrincetonCompanion-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-PrincetonCompanion_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlainerman2008" class="citation cs2">Klainerman, Sergiu (2008), "Partial Differential Equations", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), <i>The Princeton Companion to Mathematics</i>, Princeton University Press, pp. 455–483</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Partial+Differential+Equations&rft.btitle=The+Princeton+Companion+to+Mathematics&rft.pages=455-483&rft.pub=Princeton+University+Press&rft.date=2008&rft.aulast=Klainerman&rft.aufirst=Sergiu&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></span> </li> <li id="cite_note-:0-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_6-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevandosky" class="citation web cs1">Levandosky, Julie. <a rel="nofollow" class="external text" href="https://web.stanford.edu/class/math220a/handouts/secondorder.pdf">"Classification of Second-Order Equations"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Classification+of+Second-Order+Equations&rft.aulast=Levandosky&rft.aufirst=Julie&rft_id=https%3A%2F%2Fweb.stanford.edu%2Fclass%2Fmath220a%2Fhandouts%2Fsecondorder.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" 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">Courant and Hilbert (1962), p.182.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGershenfeld2000" class="citation book cs1">Gershenfeld, Neil (2000). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/naturemathematic00gers_334"><i>The nature of mathematical modeling</i></a></span> (Reprinted (with corr.) ed.). 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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%2F978-3-540-74993-6">10.1007/978-3-540-74993-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-74992-9" title="Special:BookSources/978-3-540-74992-9"><bdi>978-3-540-74992-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=High+Order+Difference+Methods+for+Time+Dependent+PDE&rft.series=Springer+Series+in+Computational+Mathematics&rft.pub=Springer&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-3-540-74993-6&rft.isbn=978-3-540-74992-9&rft.aulast=Gustafsson&rft.aufirst=Bertil&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></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=Partial_differential_equation&action=edit&section=29" title="Edit section: References"><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 refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCourantHilbert1962" class="citation cs2"><a href="/wiki/Richard_Courant" title="Richard Courant">Courant, R.</a> & <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, D.</a> (1962), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fcZV4ohrerwC"><i>Methods of Mathematical Physics</i></a>, vol. II, New York: Wiley-Interscience, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783527617241" title="Special:BookSources/9783527617241"><bdi>9783527617241</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Mathematical+Physics&rft.place=New+York&rft.pub=Wiley-Interscience&rft.date=1962&rft.isbn=9783527617241&rft.aulast=Courant&rft.aufirst=R.&rft.au=Hilbert%2C+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfcZV4ohrerwC&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDrábekHolubová2007" class="citation book cs1">Drábek, Pavel; Holubová, Gabriela (2007). <i>Elements of partial differential equations</i> (Online ed.). Berlin: de Gruyter. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783110191240" title="Special:BookSources/9783110191240"><bdi>9783110191240</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+partial+differential+equations&rft.place=Berlin&rft.edition=Online&rft.pub=de+Gruyter&rft.date=2007&rft.isbn=9783110191240&rft.aulast=Dr%C3%A1bek&rft.aufirst=Pavel&rft.au=Holubov%C3%A1%2C+Gabriela&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEvans1998" class="citation book cs1">Evans, Lawrence C. (1998). <a rel="nofollow" class="external text" href="https://math24.wordpress.com/wp-content/uploads/2013/02/partial-differential-equations-by-evans.pdf"><i>Partial differential equations</i></a> <span class="cs1-format">(PDF)</span>. Providence (R. I.): American mathematical society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0772-2" title="Special:BookSources/0-8218-0772-2"><bdi>0-8218-0772-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+differential+equations&rft.place=Providence+%28R.+I.%29&rft.pub=American+mathematical+society&rft.date=1998&rft.isbn=0-8218-0772-2&rft.aulast=Evans&rft.aufirst=Lawrence+C.&rft_id=https%3A%2F%2Fmath24.wordpress.com%2Fwp-content%2Fuploads%2F2013%2F02%2Fpartial-differential-equations-by-evans.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIbragimov1993" class="citation cs2"><a href="/wiki/Nail_H._Ibragimov" title="Nail H. 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Olver">Olver, P.J.</a> (1995), <i>Equivalence, Invariants and Symmetry</i>, Cambridge Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Equivalence%2C+Invariants+and+Symmetry&rft.pub=Cambridge+Press&rft.date=1995&rft.aulast=Olver&rft.aufirst=P.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetrovskii1967" class="citation cs2"><a href="/wiki/Ivan_Petrovsky" title="Ivan Petrovsky">Petrovskii, I. G.</a> (1967), <i>Partial Differential Equations</i>, Philadelphia: W. B. 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(2005), <i>An Introduction to Partial Differential Equations</i>, New York: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-84886-5" title="Special:BookSources/0-521-84886-5"><bdi>0-521-84886-5</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+Partial+Differential+Equations&rft.place=New+York&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=0-521-84886-5&rft.aulast=Pinchover&rft.aufirst=Y.&rft.au=Rubinstein%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolyanin2002" class="citation cs2"><a href="/wiki/Andrei_Polyanin" title="Andrei Polyanin">Polyanin, A. 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(ed.), <i>Differential Equations: Their Solution Using Symmetries</i>, Cambridge University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Equations%3A+Their+Solution+Using+Symmetries&rft.pub=Cambridge+University+Press&rft.date=1989&rft.aulast=Stephani&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWazwaz2009" class="citation book cs1">Wazwaz, Abdul-Majid (2009). <i>Partial Differential Equations and Solitary Waves Theory</i>. Higher Education Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-00251-9" title="Special:BookSources/978-3-642-00251-9"><bdi>978-3-642-00251-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations+and+Solitary+Waves+Theory&rft.pub=Higher+Education+Press&rft.date=2009&rft.isbn=978-3-642-00251-9&rft.aulast=Wazwaz&rft.aufirst=Abdul-Majid&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWazwaz2002" class="citation book cs1">Wazwaz, Abdul-Majid (2002). <i>Partial Differential Equations Methods and Applications</i>. A.A. Balkema. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-5809-369-7" title="Special:BookSources/90-5809-369-7"><bdi>90-5809-369-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Partial+Differential+Equations+Methods+and+Applications&rft.pub=A.A.+Balkema&rft.date=2002&rft.isbn=90-5809-369-7&rft.aulast=Wazwaz&rft.aufirst=Abdul-Majid&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZwillinger1997" class="citation cs2">Zwillinger, D. (1997), <i>Handbook of Differential Equations</i> (3rd ed.), Boston: Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-784395-7" title="Special:BookSources/0-12-784395-7"><bdi>0-12-784395-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Differential+Equations&rft.place=Boston&rft.edition=3rd&rft.pub=Academic+Press&rft.date=1997&rft.isbn=0-12-784395-7&rft.aulast=Zwillinger&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGershenfeld1999" class="citation cs2"><a href="/wiki/Neil_Gershenfeld" title="Neil Gershenfeld">Gershenfeld, N.</a> (1999), <i>The Nature of Mathematical Modeling</i> (1st ed.), New York: Cambridge University Press, New York, NY, USA, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-57095-6" title="Special:BookSources/0-521-57095-6"><bdi>0-521-57095-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Nature+of+Mathematical+Modeling&rft.place=New+York&rft.edition=1st&rft.pub=Cambridge+University+Press%2C+New+York%2C+NY%2C+USA&rft.date=1999&rft.isbn=0-521-57095-6&rft.aulast=Gershenfeld&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrasil'shchikVinogradov1999" class="citation cs2">Krasil'shchik, I.S. & <a href="/wiki/Alexandre_Mikhailovich_Vinogradov" title="Alexandre Mikhailovich Vinogradov">Vinogradov, A.M., Eds.</a> (1999), <i>Symmetries and Conserwation Laws for Differential Equations of Mathematical Physics</i>, American Mathematical Society, Providence, Rhode Island, USA, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0958-X" title="Special:BookSources/0-8218-0958-X"><bdi>0-8218-0958-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetries+and+Conserwation+Laws+for+Differential+Equations+of+Mathematical+Physics&rft.pub=American+Mathematical+Society%2C+Providence%2C+Rhode+Island%2C+USA&rft.date=1999&rft.isbn=0-8218-0958-X&rft.aulast=Krasil%27shchik&rft.aufirst=I.S.&rft.au=Vinogradov%2C+A.M.%2C+Eds.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrasil'shchikLychaginVinogradov1986" class="citation cs2">Krasil'shchik, I.S.; Lychagin, V.V. & <a href="/wiki/Alexandre_Mikhailovich_Vinogradov" title="Alexandre Mikhailovich Vinogradov">Vinogradov, A.M.</a> (1986), <i>Geometry of Jet Spaces and Nonlinear Partial Differential Equations</i>, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/2-88124-051-8" title="Special:BookSources/2-88124-051-8"><bdi>2-88124-051-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+of+Jet+Spaces+and+Nonlinear+Partial+Differential+Equations&rft.pub=Gordon+and+Breach+Science+Publishers%2C+New+York%2C+London%2C+Paris%2C+Montreux%2C+Tokyo&rft.date=1986&rft.isbn=2-88124-051-8&rft.aulast=Krasil%27shchik&rft.aufirst=I.S.&rft.au=Lychagin%2C+V.V.&rft.au=Vinogradov%2C+A.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinogradov2001" class="citation cs2"><a href="/wiki/Alexandre_Mikhailovich_Vinogradov" title="Alexandre Mikhailovich Vinogradov">Vinogradov, A.M.</a> (2001), <i>Cohomological Analysis of Partial Differential Equations and Secondary Calculus</i>, American Mathematical Society, Providence, Rhode Island, USA, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-2922-X" title="Special:BookSources/0-8218-2922-X"><bdi>0-8218-2922-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cohomological+Analysis+of+Partial+Differential+Equations+and+Secondary+Calculus&rft.pub=American+Mathematical+Society%2C+Providence%2C+Rhode+Island%2C+USA&rft.date=2001&rft.isbn=0-8218-2922-X&rft.aulast=Vinogradov&rft.aufirst=A.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGustafsson2008" class="citation book cs1"><a href="/wiki/Bertil_Gustafsson" title="Bertil Gustafsson">Gustafsson, Bertil</a> (2008). <i>High Order Difference Methods for Time Dependent PDE</i>. Springer Series in Computational Mathematics. Vol. 38. 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%2F978-3-540-74993-6">10.1007/978-3-540-74993-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-74992-9" title="Special:BookSources/978-3-540-74992-9"><bdi>978-3-540-74992-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=High+Order+Difference+Methods+for+Time+Dependent+PDE&rft.series=Springer+Series+in+Computational+Mathematics&rft.pub=Springer&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-3-540-74993-6&rft.isbn=978-3-540-74992-9&rft.aulast=Gustafsson&rft.aufirst=Bertil&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZachmanoglouThoe1986" class="citation book cs1">Zachmanoglou, E. C.; Thoe, Dale W. (1986). <i>Introduction to Partial Differential Equations with Applications</i>. New York: Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65251-3" title="Special:BookSources/0-486-65251-3"><bdi>0-486-65251-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Partial+Differential+Equations+with+Applications&rft.place=New+York&rft.pub=Courier+Corporation&rft.date=1986&rft.isbn=0-486-65251-3&rft.aulast=Zachmanoglou&rft.aufirst=E.+C.&rft.au=Thoe%2C+Dale+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li></ul> </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=Partial_differential_equation&action=edit&section=30" title="Edit section: Further reading"><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="CITEREFCajori1928" class="citation journal cs1"><a href="/wiki/Florian_Cajori" title="Florian Cajori">Cajori, Florian</a> (1928). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181123102253/http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf">"The Early History of Partial Differential Equations and of Partial Differentiation and Integration"</a> <span class="cs1-format">(PDF)</span>. <i>The American Mathematical Monthly</i>. <b>35</b> (9): 459–467. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2298771">10.2307/2298771</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2298771">2298771</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.harvard.edu/archive/21a_fall_14/exhibits/cajori/cajori.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2018-11-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-05-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=The+Early+History+of+Partial+Differential+Equations+and+of+Partial+Differentiation+and+Integration&rft.volume=35&rft.issue=9&rft.pages=459-467&rft.date=1928&rft_id=info%3Adoi%2F10.2307%2F2298771&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2298771%23id-name%3DJSTOR&rft.aulast=Cajori&rft.aufirst=Florian&rft_id=http%3A%2F%2Fwww.math.harvard.edu%2Farchive%2F21a_fall_14%2Fexhibits%2Fcajori%2Fcajori.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><a href="/wiki/Louis_Nirenberg" title="Louis Nirenberg">Nirenberg, Louis</a> (1994). "Partial differential equations in the first half of the century." Development of mathematics 1900–1950 (Luxembourg, 1992), 479–515, Birkhäuser, Basel.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrezisBrowder1998" class="citation journal cs1"><a href="/wiki/Ha%C3%AFm_Brezis" title="Haïm Brezis">Brezis, Haïm</a>; <a href="/wiki/Felix_Browder" title="Felix Browder">Browder, Felix</a> (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Faima.1997.1713">"Partial Differential Equations in the 20th Century"</a>. <i><a href="/wiki/Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a></i>. <b>135</b> (1): 76–144. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Faima.1997.1713">10.1006/aima.1997.1713</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Mathematics&rft.atitle=Partial+Differential+Equations+in+the+20th+Century&rft.volume=135&rft.issue=1&rft.pages=76-144&rft.date=1998&rft_id=info%3Adoi%2F10.1006%2Faima.1997.1713&rft.aulast=Brezis&rft.aufirst=Ha%C3%AFm&rft.au=Browder%2C+Felix&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Faima.1997.1713&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" 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=Partial_differential_equation&action=edit&section=31" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 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class="side-box-abovebelow"> <b>Partial differential equation</b> at Wikipedia's <a href="/wiki/Wikipedia:Wikimedia_sister_projects" title="Wikipedia:Wikimedia sister projects"><span id="sister-projects">sister projects</span></a></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span><span class="sister-link"><a href="https://commons.wikimedia.org/wiki/Category:Solutions_of_PDE" class="extiw" title="c:Category:Solutions of PDE">Media</a> from Commons</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" decoding="async" width="23" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/35px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/46px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></span><span class="sister-link"><a href="https://en.wikiquote.org/wiki/Partial_differential_equation" class="extiw" title="q:Partial differential equation">Quotations</a> from Wikiquote</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/27px-Wikibooks-logo.svg.png" decoding="async" width="27" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/41px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/54px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></span><span class="sister-link"><a href="https://en.wikibooks.org/wiki/Partial_Differential_Equations" class="extiw" title="b:Partial Differential Equations">Textbooks</a> from Wikibooks</span></li></ul></div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Differential_equation,_partial">"Differential equation, partial"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Differential+equation%2C+partial&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDifferential_equation%2C_partial&rfr_id=info%3Asid%2Fen.wikipedia.org%3APartial+differential+equation" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/pde-en.htm">Partial Differential Equations: Exact Solutions</a> at EqWorld: The World of Mathematical Equations.</li> <li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-pde.htm">Partial Differential Equations: Index</a> at EqWorld: The World of Mathematical Equations.</li> <li><a rel="nofollow" class="external text" href="http://eqworld.ipmnet.ru/en/methods/meth-pde.htm">Partial Differential Equations: Methods</a> at EqWorld: The World of Mathematical Equations.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170701144823/http://www.exampleproblems.com/wiki/index.php?title=Partial_Differential_Equations">Example problems with solutions</a> at exampleproblems.com</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PartialDifferentialEquation.html">Partial Differential Equations</a> at mathworld.wolfram.com</li> <li><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/FEMDocumentation/tutorial/SolvingPDEwithFEM.html">Partial Differential Equations</a> with Mathematica</li> <li><a rel="nofollow" class="external text" href="http://www.mathworks.com/moler/pdes.pdf">Partial Differential Equations</a> in Cleve Moler: Numerical Computing with MATLAB</li> <li><a rel="nofollow" class="external text" href="http://www.nag.com/numeric/fl/nagdoc_fl24/html/D03/d03intro.html">Partial Differential Equations</a> at nag.com</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanderson2019" class="citation web cs1">Sanderson, Grant (April 21, 2019). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ly4S0oi3Yz8&list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6">"But what is a partial differential equation?"</a>. <i>3Blue1Brown</i>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211102/ly4S0oi3Yz8">Archived</a> from the original on 2021-11-02 – via <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</cite><span 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.navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Differential_equations" 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 class="mw-selflink selflink">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"></div><div role="navigation" class="navbox" aria-labelledby="Major_topics_in_mathematical_analysis" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th 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