On a Difference Scheme for Solving Cauchy Problems wuth the Caputo Fractional Derivative in a Banach Space

TL;DR

This paper develops a time-semidiscretization scheme for solving Cauchy problems with Caputo fractional derivatives in Banach spaces, providing convergence rates, error estimates, and numerical validation.

Contribution

It introduces a novel scheme for fractional differential equations in Banach spaces with proven convergence and error bounds, utilizing advanced special functions and operator calculus.

Findings

The scheme achieves specific convergence rates.

Error estimates are established in terms of discretization step.

Numerical experiments confirm theoretical results.

Abstract

We construct and study a time--semidiscretization scheme for the Cauchy problem associated with a linear homogeneous differential equation with the Caputo fractional time derivative of order $\alpha\in(0,1)$ and a spatial sectorial operator in a Banach space. For this scheme, we obtain rate--of--convergence and error estimates in terms of the discretization step. We use properties of Mittag--Leffler functions, hypergeometric functions, and the calculus of sectorial operators in a Banach space. Results of numerical experiments are also reported.