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

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.

Three-point compact finite difference scheme on non-uniform meshes for the time-fractional Black-Scholes equation

“…An important approach for analytical and numerical solution of linear and non-linear fractional differential equations is to use fractional power series. In [3] we construct finite-difference schemes for the fractional sub-diffusion equation using the L1 and the modified L1-approximations for the Caputo derivative. In all numerical experiments the difference approximations have first order accuracy in the time direction.…”

Three-Point Compact Approximation for the Caputo Fractional Derivative

“…The second-order backward difference approximation for the second derivative has first-order accuracy The fractional relaxation and subdiffusion equations are important fractional differential equations. Their analytical and numerical solutions have been studied extensively [3,9,12,13,14,17,20,31,38,42,44,54]. In this section we determine the numerical solutions of the fractional relaxation and subdiffusion equations which use the compact modified L1 approximation for the Caputo derivative…”

“…In [13,14] we derived the numerical solutions {u n } N n=0 and {v n } N n=0 of the fractional relaxation equation (30) which use the L1 approximation and the modified L1 approximation for the Caputo derivative,…”

“…In [14] we discuss a method for improving the differentiability class of the solutions of the fractional relaxation and subdiffusion equations. The method is based on a substitution using the fractional Taylor polynomials of the solution.…”

“…In [14] we discuss a method for improving the accuracy of the numerical solutions of the ordinary fractional relaxation equation…”