On the analysis of mixed-index time fractional differential equation systems

TL;DR

This paper investigates mixed-index time fractional differential equations, providing solutions, stability analysis, and numerical simulations, thereby generalizing existing models with different fractional derivatives in system components.

Contribution

It introduces a theorem for solving mixed-index fractional systems, extending classical Mittag-Leffler solutions and analyzing their stability using Laplace transforms.

Findings

Derived a solution formula for mixed-index fractional systems.

Established stability criteria using Laplace transform techniques.

Validated results through numerical simulations.

Abstract

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.