John T. Edwards

The numerical solution of linear multi-term fractional differential equations: systems of equations

In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.

Boundedness and stability of solutions to difference equations

This paper is concerned with the qualitative behaviour of solutions to difference equations. We focus on boundedness and stability of solutions and we present a unified theory that applies both to autonomous and nonautonomous equations and to nonlinear equations as well as linear equations. Our presentation brings together new, established, and hard-to-find results from the literature and provides a theory that is both memorable and easy to apply. We show how the theoretical results given here relate to some of those in the established literature and by means of simple examples we indicate how the use of Lipschitz constants in this way can provide useful insights into the qualitative behaviour of solutions to some nonlinear problems including those arising in numerical analysis.

Bifurcations in numerical methods for volterra integro-differential equations

We are interested in finding approximate solutions to parameter-dependent Volterra integro-differential equations over long time intervals using numerical schemes. This paper concentrates on changes in qualitative behavior (bifurcations) in the solutions and extends the work of Brunner and Lambert and Matthys (who considered only changes in stability behavior) to consider other bifurcations. We begin by considering a one-parameter equation with fading memory separable convolution kernel: we give an analytical discussion of bifurcations in this case and provide details of the behavior of numerical schemes. We extend our analysis to consider an equation with two-parameter fading memory convolution kernel and show the relationship to the classical test equation studied by the earlier authors. We draw attention to the fact that known stability results may not provide a reliable framework for choice of numerical scheme when other changes in qualitative behavior are also of interest. We give bifurcation plots for a variety of methods and show how, for known values of the parameters, stepsizes h>0 may be chosen to preserve the correct qualitative behavior in the numerical solution of the Volterra integro-differential equation.