Preprint Article Version 1 This version not peer reviewed

On the Analysis of Mixed-Index Time Fractional Differential Equation Systems

Version 1 : Received: 13 February 2018 / Approved: 13 February 2018 / Online: 13 February 2018 (09:03:54 CET)

How to cite: Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Preprints 2018, 2018020094 (doi: 10.20944/preprints201802.0094.v1). Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Preprints 2018, 2018020094 (doi: 10.20944/preprints201802.0094.v1).

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 generalisedMittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.

Subject Areas

time fractional differential equations; mixed-index problems; analytical solutions; asymptotic stability

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