Higher order graded mesh scheme for time fractional differential equations

TL;DR

This paper introduces a higher order graded mesh scheme for time fractional differential equations, improving accuracy and effectively handling singularities at the origin through a novel approximation of the Caputo derivative.

Contribution

It presents a new higher order approximation for the Caputo fractional derivative using graded meshes, with stability analysis and numerical validation for solving time fractional PDEs.

Findings

Effective handling of singularity at the origin

Higher order accuracy demonstrated through numerical examples

Stable and reliable scheme for time fractional PDEs

Abstract

In this article, we propose a higher order approximation to Caputo fractional (C-F) derivative using graded mesh and standard central difference approximation for space derivatives, in order to obtain the approximate solution of time fractional partial differential equations (TFPDE). The proposed approximation for C-F derivative tackles the singularity at origin effectively and is easily applicable to diverse problems. The stability analysis and truncation error bounds of the proposed scheme are discussed, along with this, analyzed the required regularity of the solution. Few numerical examples are presented to support the theory.