Time-domain response of nabla discrete fractional order systems

This paper investigates the time--domain response of nabla discrete fractional order systems by exploring several useful properties of the nabla discrete Laplace transform and the discrete Mittag--Leffler function. In particular, we establish two fundamental properties of a nabla discrete fractional order system with nonzero initial instant: i) the existence and uniqueness of the system time--domain response; and ii) the dynamic behavior of the zero input response. Finally, one numerical example is provided to show the validity of the theoretical results.Comment: 13 pages, 6 figure

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

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.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15 figures or sub-figures

Local discontinuous Galerkin methods for fractional ordinary differential equations

This paper discusses the upwinded local discontinuous Galerkin methods for the one-term/multi-term fractional ordinary differential equations (FODEs). The natural upwind choice of the numerical fluxes for the initial value problem for FODEs ensures stability of the methods. The solution can be computed element by element with optimal order of convergence $k+1$ in the $L^2$ norm and superconvergence of order $k+1+\min\{k,\alpha\}$ at the downwind point of each element. Here $k$ is the degree of the approximation polynomial used in an element and $\alpha$ ($\alpha\in (0,1]$) represents the order of the one-term FODEs. A generalization of this includes problems with classic $m$'th-term FODEs, yielding superconvergence order at downwind point as $k+1+\min\{k,\max\{\alpha,m\}\}$. The underlying mechanism of the superconvergence is discussed and the analysis confirmed through examples, including a discussion of how to use the scheme as an efficient way to evaluate the generalized Mittag-Leffler function and solutions to more generalized FODE's.Comment: 17 pages, 7 figure