Matrix approach to discrete fractional calculus II: partial fractional differential equations

TL;DR

This paper introduces a matrix-based numerical method for discretizing and solving partial differential equations with fractional derivatives and delays, demonstrated through various fractional diffusion equation examples.

Contribution

It extends Podlubny's matrix approach to efficiently handle partial fractional derivatives and delays in PDEs, providing MATLAB routines and sample implementations.

Findings

Successfully solved fractional diffusion equations with various derivative combinations.

Demonstrated the method's simplicity and generality across multiple examples.

Validated the approach for equations with delays.

Abstract

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, integer/fractional, and fractional/fractional) with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach. The fifth example illustrates that the method can be equally simply used for fractional differential equations with delays. A set of MATLAB routines for the implementation of the method as well as sample code used to solve the examples have been developed.