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
Hlavní autor: Jornet, Marc
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.06.2024
Témata:
Analytical solution
Cauchy-Kovalevskaya theorem
Convergent power series
Fractional differential equation
Implicit-function theorem
Vector function
Fractional differential equation
Implicit-function theorem
Convergent power series
Cauchy-Kovalevskaya theorem
Analytical solution
Vector function
ISSN:0167-2789, 1872-8022
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Popis
Shrnutí: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).
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2024.134139