Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Axioms2018, 7, 25.
Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Axioms 2018, 7, 25.
Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Axioms2018, 7, 25.
Burrage, K.; Burrage, P.; Turner, I.; Zeng, F. On the Analysis of Mixed-Index Time Fractional Differential Equation Systems. Axioms 2018, 7, 25.
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.
Keywords
time fractional differential equations; mixed-index problems; analytical solutions; asymptotic stability
Subject
Computer Science and Mathematics, Computational Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
