On the analysis of mixed-index time fractional differential equation systems
This provides a theoretical foundation for solving a class of fractional differential equations, which is incremental for mathematicians working on fractional calculus.
The authors prove a theorem on the solution of linear mixed-index time fractional differential equations, generalizing the Mittag-Leffler solution and the linear sequential class, and analyze asymptotic stability using Laplace transforms, with numerical simulations.
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.