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Axioms 2018, 7(2), 25; https://doi.org/10.3390/axioms7020025

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

1,2,†
,
1,2,†,* , 1,2,†
and
2,†
1
ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology (QUT), Brisbane 4001, Australia
2
School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane 4001, Australia
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Received: 13 February 2018 / Revised: 11 April 2018 / Accepted: 11 April 2018 / Published: 17 April 2018
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
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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 that 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. View Full-Text
Keywords: time fractional differential equations; mixed-index problems; analytical solution; asymptotic stability time fractional differential equations; mixed-index problems; analytical solution; asymptotic stability
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Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Axioms 2018, 7, 25.

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