A New Method for Numerical Solution of the Fractional Relaxation and Subdiffusion Equations Using Fractional Taylor Polynomials

TL;DR

This paper introduces a novel numerical method leveraging fractional Taylor polynomials to enhance the accuracy of solutions for fractional relaxation and subdiffusion equations, addressing issues caused by solution singularities at initial points.

Contribution

The paper presents a new approach using fractional Taylor polynomials to improve numerical accuracy for fractional differential equations with singular solutions.

Findings

Improved numerical accuracy demonstrated for fractional relaxation equations.

Enhanced solution precision for subdiffusion equations.

Method effectively handles initial singularities.

Abstract

The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the initial point, which may result in a lower accuracy of the numerical solutions. We propose a method for improving the accuracy of the numerical solutions of the fractional relaxation and subdiffusion equations based on the fractional Taylor polynomials of the solution at the initial point.