On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations
We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as...
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| Vydáno v: | Physica. D Ročník 462; s. 134139 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Elsevier B.V
01.06.2024
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| ISSN: | 0167-2789, 1872-8022 |
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| Abstract | We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end.
•We investigate systems of nonlinear Caputo fractional differential equations.•We prove the Cauchy-Kovalevskaya theorem on convergence of fractional power series.•We use majorants and the implicit-function theorem adequately.•Corollaries on continuity and computational examples (e.g. Gompertz) are included.•We address and discuss Caputo fractional PDEs (quasilinear, heat and Schrödinger). |
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| AbstractList | We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or compartmental. Essentially, the theorem states that if the input function has a Taylor series, then the solution can be locally expressed as a fractional power series. We use, in the real field, the method of majorants and the analytic version of the implicit-function theorem, in a way that circumvents difficulties associated to fractional calculus. Some corollaries on continuity are derived, with computational examples for illustration, and a discussion on fractional partial differential equations is included with a case study and counterexamples. Open problems are raised at the end.
•We investigate systems of nonlinear Caputo fractional differential equations.•We prove the Cauchy-Kovalevskaya theorem on convergence of fractional power series.•We use majorants and the implicit-function theorem adequately.•Corollaries on continuity and computational examples (e.g. Gompertz) are included.•We address and discuss Caputo fractional PDEs (quasilinear, heat and Schrödinger). |
| ArticleNumber | 134139 |
| Author | Jornet, Marc |
| Author_xml | – sequence: 1 givenname: Marc orcidid: 0000-0003-0748-3730 surname: Jornet fullname: Jornet, Marc email: marc.jornet@uv.es organization: Departament de Matemàtiques, Universitat de València, 46100 Burjassot, Spain |
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| Cites_doi | 10.1016/j.physa.2021.125947 10.1007/s00521-023-08268-8 10.1155/2014/238459 10.1186/s13662-017-1456-z 10.1016/j.net.2021.07.026 10.1016/j.na.2021.112340 10.3390/fractalfract4030044 10.1007/s40324-022-00314-0 10.1016/j.jmaa.2006.05.061 10.3390/e22121359 10.1016/j.jcp.2014.07.019 10.1111/j.1365-246X.1967.tb02303.x 10.3934/mbe.2021163 10.1186/s13662-021-03345-5 10.1016/j.chaos.2021.110652 10.1186/s13662-015-0613-5 10.3390/fractalfract2040023 10.1119/1.15432 10.3390/fractalfract5020057 10.1016/j.cnsns.2021.105764 |
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| Keywords | Fractional differential equation Implicit-function theorem Convergent power series Cauchy-Kovalevskaya theorem Analytical solution Vector function |
| Language | English |
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| Snippet | We aim at proving the Cauchy-Kovalevskaya theorem for systems of nonlinear fractional differential equations in the Caputo sense, not necessarily polynomial or... |
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| SubjectTerms | Analytical solution Cauchy-Kovalevskaya theorem Convergent power series Fractional differential equation Implicit-function theorem Vector function |
| Title | On the Cauchy-Kovalevskaya theorem for Caputo fractional differential equations |
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