A novel approach to nonlinear fractional volterra integral equations

<!DOCTYPE html> <html lang="en" xml:lang="en"> <head> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title> A novel approach to nonlinear fractional volterra integral equations | Acta Polytechnica </title> <link rel="icon" href="https://ojs.cvut.cz/ojs/public/journals/2/favicon_en_US.ico"> <meta name="generator" content="Open Journal Systems 3.4.0.5"> <link rel="schema.DC" href="http://purl.org/dc/elements/1.1/" /> <meta name="DC.Creator.PersonalName" content="Mohammed Abdulshareef Hussein"/> <meta name="DC.Creator.PersonalName" content="Hassan Kamil Jassim"/> <meta name="DC.Date.created" scheme="ISO8601" content="2024-11-11"/> <meta name="DC.Date.dateSubmitted" scheme="ISO8601" content="2023-11-01"/> <meta name="DC.Date.issued" scheme="ISO8601" content="2024-11-11"/> <meta name="DC.Date.modified" scheme="ISO8601" content="2024-11-11"/> <meta name="DC.Description" xml:lang="en" content="Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise."/> <meta name="DC.Format" scheme="IMT" content="application/pdf"/> <meta name="DC.Identifier" content="9438"/> <meta name="DC.Identifier.pageNumber" content="414–419"/> <meta name="DC.Identifier.DOI" content="10.14311/AP.2024.64.0414"/> <meta name="DC.Identifier.URI" content="https://ojs.cvut.cz/ojs/index.php/ap/article/view/9438"/> <meta name="DC.Language" scheme="ISO639-1" content="en"/> <meta name="DC.Rights" content="Copyright (c) 2024 Mohammed Abdulshareef Hussein, Hassan Kamil Jassim"/> <meta name="DC.Rights" content="https://creativecommons.org/licenses/by/4.0"/> <meta name="DC.Source" content="Acta Polytechnica"/> <meta name="DC.Source.ISSN" content="1805-2363"/> <meta name="DC.Source.Issue" content="5"/> <meta name="DC.Source.Volume" content="64"/> <meta name="DC.Source.URI" content="https://ojs.cvut.cz/ojs/index.php/ap"/> <meta name="DC.Subject" xml:lang="en" content="integral equations"/> <meta name="DC.Subject" xml:lang="en" content="fractional calculus"/> <meta name="DC.Subject" xml:lang="en" content="Leibniz integral rule"/> <meta name="DC.Subject" xml:lang="en" content="Mitteg-Leffler function"/> <meta name="DC.Subject" xml:lang="en" content="Caputo fractional operator"/> <meta name="DC.Subject" xml:lang="en" content="Riemann-Liouville fractional operator"/> <meta name="DC.Title" content="A novel approach to nonlinear fractional volterra integral equations"/> <meta name="DC.Type" content="Text.Serial.Journal"/> <meta name="DC.Type.articleType" content="Articles"/> <meta name="gs_meta_revision" content="1.1"/> <meta name="citation_journal_title" content="Acta Polytechnica"/> <meta name="citation_journal_abbrev" content="Acta Polytech"/> <meta name="citation_issn" content="1805-2363"/> <meta name="citation_author" content="Mohammed Abdulshareef Hussein"/> <meta name="citation_author_institution" content="Al-Ayen Iraqi University, Scientific Research Center, 64001 Nasiriyah, Iraq; Ministry of Education, Education Directorate of Thi-Qar, 64001 Nasiriyah, Iraq"/> <meta name="citation_author" content="Hassan Kamil Jassim"/> <meta name="citation_author_institution" content="University of Thi-Qar, College of Education for Pure Science, Department of Mathematics, 64001 Thi-Qar, Iraq"/> <meta name="citation_title" content="A novel approach to nonlinear fractional volterra integral equations"/> <meta name="citation_language" content="en"/> <meta name="citation_date" content="2024/11/11"/> <meta name="citation_volume" content="64"/> <meta name="citation_issue" content="5"/> <meta name="citation_firstpage" content="414"/> <meta name="citation_lastpage" content="419"/> <meta name="citation_doi" content="10.14311/AP.2024.64.0414"/> <meta name="citation_abstract_html_url" content="https://ojs.cvut.cz/ojs/index.php/ap/article/view/9438"/> <meta name="citation_abstract" xml:lang="en" content="Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise."/> <meta name="citation_keywords" xml:lang="en" content="integral equations"/> <meta name="citation_keywords" xml:lang="en" content="fractional calculus"/> <meta name="citation_keywords" xml:lang="en" content="Leibniz integral rule"/> <meta name="citation_keywords" xml:lang="en" content="Mitteg-Leffler function"/> <meta name="citation_keywords" xml:lang="en" content="Caputo fractional operator"/> <meta name="citation_keywords" xml:lang="en" content="Riemann-Liouville fractional operator"/> <meta name="citation_pdf_url" content="https://ojs.cvut.cz/ojs/index.php/ap/article/download/9438/7233"/> <meta name="citation_reference" content="M. Belmekki, M. Benchohra. Existence results for fractional order semilinear functional differential equations with nondense domain. Nonlinear Analysis: Theory, Methods & Applications 72(2):925–932, 2010. https://doi.org/10.1016/j.na.2009.07.034"/> <meta name="citation_reference" content="H. K. Jassim, M. A. Hussein. A new approach for solving nonlinear fractional ordinary differential equations. Mathematics 11(7):1565, 2023. https://doi.org/10.3390/math11071565"/> <meta name="citation_reference" content="M. A. Hussein, H. K. Jassim. Analysis of fractional differential equations with Antagana-Baleanu fractional operator. Progress in Fractional Differentiation and Applications 9(4):681–686, 2023. https://doi.org/10.18576/pfda/090411"/> <meta name="citation_reference" content="M. Higazy, S. Aggarwal, T. A. Nofal. Sawi decomposition method for Volterra integral equation with application. Journal of Mathematics 2020(1):6687134, 2020. https://doi.org/10.1155/2020/6687134"/> <meta name="citation_reference" content="S. Jahanshahi, E. Babolian, D. F. M. Torres, A. Vahidi. Solving Abel integral equations of first kind via fractional calculus. Journal of King Saud University – Science 27(2):161–167, 2015. https://doi.org/10.1016/j.jksus.2014.09.004"/> <meta name="citation_reference" content="G. Pellicane, L. L. Lee, C. Caccamo. Integral-equation theories of fluid phase equilibria in simple fluids. Fluid Phase Equilibria 521:112665, 2020. https://doi.org/10.1016/j.fluid.2020.112665"/> <meta name="citation_reference" content="M. A. Hussein, H. K. Jassim, A. K. Jassim. An innovative iterative approach to solving Volterra integral equations of second kind. Acta Polytechnica 64(2):87–102, 2024. https://doi.org/10.14311/AP.2024.64.0087"/> <meta name="citation_reference" content="H. Mottaghi Golshan. Numerical solution of nonlinear m-dimensional Fredholm integral equations using iterative Newton-Cotes rules. Journal of Computational and Applied Mathematics 448:115917, 2024. https://doi.org/10.1016/j.cam.2024.115917"/> <meta name="citation_reference" content="M. A. Hussein. The approximate solutions of fractional differential equations with Antagana-Baleanu fractional operator. Mathematics and Computational Sciences 3(3):29–39, 2022. https://doi.org/10.30511/MCS.2022.560414.1077"/> <meta name="citation_reference" content="H. K. Jassim, M. A. S. Hussain. On approximate solutions for fractional system of differential equations with Caputo-Fabrizio fractional operator. Journal of Mathematics and Computer Science 23(1):58–66, 2021. https://doi.org/10.22436/jmcs.023.01.06"/> <meta name="citation_reference" content="R. Katani, S. Mckee. A hybrid Legendre block-pulse method for mixed Volterra-Fredholm integral equations. Journal of Computational and Applied Mathematics 376:112867, 2020. https://doi.org/10.1016/j.cam.2020.112867"/> <meta name="citation_reference" content="H. K. Jassim, M. A. Hussein. A novel formulation of the fractional derivative with the order α ≥ 0 and without the singular kernel. Mathematics 10(21):4123, 2022. https://doi.org/10.3390/math10214123"/> <meta name="citation_reference" content="J. Wu, Y. Zhou, C. Hang. A singularity free and derivative free approach for Abel integral equation in analyzing the laser-induced breakdown spectroscopy. Spectrochimica Acta Part B: Atomic Spectroscopy 167:105791, 2020. https://doi.org/10.1016/j.sab.2020.105791"/> <meta name="citation_reference" content="M. A. Hussein. Using the Elzaki decomposition method to solve nonlinear fractional differential equations with the Caputo-Fabrizio fractional operator. Baghdad Science Journal 21(3):1044–1054, 2024. https://doi.org/10.21123/bsj.2023.7310"/> <meta name="citation_reference" content="S. Aggarwal, N. Sharma. Laplace transform for the solution of first kind linear Volterra integral equation. Journal of Advanced Research in Applied Mathematics and Statistics 4(3&4):16–23, 2019."/> <meta name="citation_reference" content="S. Aggarwal, R. Kumar, J. Chandel. Solution of linear Volterra integral equation of second kind via Rishi transform. Journal of Scientific Research 15(1):111–119, 2023. https://doi.org/10.3329/jsr.v15i1.60337"/> <meta name="citation_reference" content="K. Sayevand, J. Tenreiro Machado, D. Baleanu. A new glance on the Leibniz rule for fractional derivatives. Communications in Nonlinear Science and Numerical Simulation 62:244–249, 2018. https://doi.org/10.1016/j.cnsns.2018.02.037"/> <meta name="citation_reference" content="S. Douglas, L. Grafakos. Norm estimates for the fractional derivative of multiple factors. 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Ministry of Education, Education Directorate of Thi-Qar, 64001 Nasiriyah, Iraq </span> </li> <li> <span class="name"> Hassan Kamil Jassim </span> <span class="affiliation"> University of Thi-Qar, College of Education for Pure Science, Department of Mathematics, 64001 Thi-Qar, Iraq </span> </li> </ul> </section> <section class="item doi"> <h2 class="label"> DOI: </h2> <span class="value"> <a href="https://doi.org/10.14311/AP.2024.64.0414"> https://doi.org/10.14311/AP.2024.64.0414 </a> </span> </section> <section class="item keywords"> <h2 class="label"> Keywords: </h2> <span class="value"> integral equations, fractional calculus, Leibniz integral rule, Mitteg-Leffler function, Caputo fractional operator, Riemann-Liouville fractional operator </span> </section> <section class="item abstract"> <h2 class="label">Abstract</h2> <p>Nonlinear Fractional Volterra integral equations (FVIEs) of the first kind present challenges due to their intricate nature, combining fractional calculus and integral equations. In this research paper, we introduce a novel method for solving such equations using Leibniz integral rules. Our study focuses on a thorough analysis and application of the proposed algorithm to solve fractional Volterra integral equations. By using Leibniz integral rules, we offer a fresh perspective on handling these equations, shedding light on their fundamental properties and behaviours. As a result of this study, we anticipate contributing distinctively to the broader development of analytical tools and techniques. By bridging the gap between fractional calculus and integral equations, our approach not only offers a valuable computational methodology but also paves the way for new insights into the application domains in which such equations arise.</p> </section> <section class="item downloads_chart"> <h2 class="label"> Downloads </h2> <div class="value"> <canvas class="usageStatsGraph" data-object-type="Submission" data-object-id="9438"></canvas> <div class="usageStatsUnavailable" data-object-type="Submission" data-object-id="9438"> Download data is not yet available. </div> </div> </section> <section class="item references"> <h2 class="label"> References </h2> <div class="value"> <p>M. Belmekki, M. Benchohra. Existence results for fractional order semilinear functional differential equations with nondense domain. Nonlinear Analysis: Theory, Methods & Applications 72(2):925–932, 2010. <a href="https://doi.org/10.1016/j.na.2009.07.034">https://doi.org/10.1016/j.na.2009.07.034</a> </p> <p>H. K. Jassim, M. A. Hussein. A new approach for solving nonlinear fractional ordinary differential equations. Mathematics 11(7):1565, 2023. <a href="https://doi.org/10.3390/math11071565">https://doi.org/10.3390/math11071565</a> </p> <p>M. A. Hussein, H. K. Jassim. Analysis of fractional differential equations with Antagana-Baleanu fractional operator. Progress in Fractional Differentiation and Applications 9(4):681–686, 2023. <a href="https://doi.org/10.18576/pfda/090411">https://doi.org/10.18576/pfda/090411</a> </p> <p>M. Higazy, S. Aggarwal, T. A. Nofal. Sawi decomposition method for Volterra integral equation with application. Journal of Mathematics 2020(1):6687134, 2020. <a href="https://doi.org/10.1155/2020/6687134">https://doi.org/10.1155/2020/6687134</a> </p> <p>S. Jahanshahi, E. Babolian, D. F. M. Torres, A. Vahidi. Solving Abel integral equations of first kind via fractional calculus. Journal of King Saud University – Science 27(2):161–167, 2015. <a href="https://doi.org/10.1016/j.jksus.2014.09.004">https://doi.org/10.1016/j.jksus.2014.09.004</a> </p> <p>G. Pellicane, L. L. Lee, C. Caccamo. Integral-equation theories of fluid phase equilibria in simple fluids. Fluid Phase Equilibria 521:112665, 2020. <a href="https://doi.org/10.1016/j.fluid.2020.112665">https://doi.org/10.1016/j.fluid.2020.112665</a> </p> <p>M. A. Hussein, H. K. Jassim, A. K. Jassim. An innovative iterative approach to solving Volterra integral equations of second kind. Acta Polytechnica 64(2):87–102, 2024. <a href="https://doi.org/10.14311/AP.2024.64.0087">https://doi.org/10.14311/AP.2024.64.0087</a> </p> <p>H. Mottaghi Golshan. Numerical solution of nonlinear m-dimensional Fredholm integral equations using iterative Newton-Cotes rules. Journal of Computational and Applied Mathematics 448:115917, 2024. <a href="https://doi.org/10.1016/j.cam.2024.115917">https://doi.org/10.1016/j.cam.2024.115917</a> </p> <p>M. A. Hussein. The approximate solutions of fractional differential equations with Antagana-Baleanu fractional operator. Mathematics and Computational Sciences 3(3):29–39, 2022. <a href="https://doi.org/10.30511/MCS.2022.560414.1077">https://doi.org/10.30511/MCS.2022.560414.1077</a> </p> <p>H. K. Jassim, M. A. S. Hussain. On approximate solutions for fractional system of differential equations with Caputo-Fabrizio fractional operator. Journal of Mathematics and Computer Science 23(1):58–66, 2021. <a href="https://doi.org/10.22436/jmcs.023.01.06">https://doi.org/10.22436/jmcs.023.01.06</a> </p> <p>R. Katani, S. Mckee. A hybrid Legendre block-pulse method for mixed Volterra-Fredholm integral equations. Journal of Computational and Applied Mathematics 376:112867, 2020. <a href="https://doi.org/10.1016/j.cam.2020.112867">https://doi.org/10.1016/j.cam.2020.112867</a> </p> <p>H. K. Jassim, M. A. Hussein. A novel formulation of the fractional derivative with the order α ≥ 0 and without the singular kernel. Mathematics 10(21):4123, 2022. <a href="https://doi.org/10.3390/math10214123">https://doi.org/10.3390/math10214123</a> </p> <p>J. Wu, Y. Zhou, C. Hang. A singularity free and derivative free approach for Abel integral equation in analyzing the laser-induced breakdown spectroscopy. Spectrochimica Acta Part B: Atomic Spectroscopy 167:105791, 2020. <a href="https://doi.org/10.1016/j.sab.2020.105791">https://doi.org/10.1016/j.sab.2020.105791</a> </p> <p>M. A. Hussein. Using the Elzaki decomposition method to solve nonlinear fractional differential equations with the Caputo-Fabrizio fractional operator. Baghdad Science Journal 21(3):1044–1054, 2024. <a href="https://doi.org/10.21123/bsj.2023.7310">https://doi.org/10.21123/bsj.2023.7310</a> </p> <p>S. Aggarwal, N. Sharma. Laplace transform for the solution of first kind linear Volterra integral equation. Journal of Advanced Research in Applied Mathematics and Statistics 4(3&4):16–23, 2019. </p> <p>S. Aggarwal, R. Kumar, J. Chandel. Solution of linear Volterra integral equation of second kind via Rishi transform. Journal of Scientific Research 15(1):111–119, 2023. <a href="https://doi.org/10.3329/jsr.v15i1.60337">https://doi.org/10.3329/jsr.v15i1.60337</a> </p> <p>K. Sayevand, J. Tenreiro Machado, D. Baleanu. A new glance on the Leibniz rule for fractional derivatives. Communications in Nonlinear Science and Numerical Simulation 62:244–249, 2018. <a href="https://doi.org/10.1016/j.cnsns.2018.02.037">https://doi.org/10.1016/j.cnsns.2018.02.037</a> </p> <p>S. Douglas, L. Grafakos. Norm estimates for the fractional derivative of multiple factors. Journal of Mathematical Analysis and Applications 537(2):128409, 2024. <a href="https://doi.org/10.1016/j.jmaa.2024.128409">https://doi.org/10.1016/j.jmaa.2024.128409</a> </p> <p>S. Aggarwal, R. Kumar, J. Chandel. Exact solution of non-linear Volterra integral equation of first kind using Rishi transform. Bulletin of Pure & Applied Sciences – Mathematics and Statistics 41(2):159–166, 2022. <a href="https://doi.org/10.5958/2320-3226.2022.00022.4">https://doi.org/10.5958/2320-3226.2022.00022.4</a>. </p> <p>E. Artin. The gamma function. Courier Dover Publications, USA, 2015. ISBN 978-0-486-80300-5. </p> <p>A. K. Shukla, J. C. Prajapati. On a generalization of Mittag-Leffler function and its properties. Journal of Mathematical Analysis and Applications 336(2):797–811, 2007. <a href="https://doi.org/10.1016/j.jmaa.2007.03.018">https://doi.org/10.1016/j.jmaa.2007.03.018</a> </p> <p>D. Pang, W. Jiang, A. U. Niazi. 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A novel approach to nonlinear fractional volterra integral equations | Acta Polytechnica