Numerical solutions of ordinary fractional differential equations with singularities

TL;DR

This paper introduces a method to improve the numerical accuracy of solving ordinary linear fractional differential equations with initial singularities by employing fractional Taylor polynomials.

Contribution

It proposes a novel approach using fractional Taylor polynomials to enhance numerical solutions of FDEs with singularities, addressing accuracy issues at the initial point.

Findings

Improved numerical accuracy for FDEs with singularities.

Effective application to two-term and three-term FDEs.

Demonstrated benefits over traditional methods.

Abstract

The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solutions of ordinary linear FDEs with constant coefficients which uses the fractional Taylor polynomials of the solutions. The numerical solutions of the two-term and three-term FDEs are studied in the paper.