Neville J. Ford

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

Detailed Error Analysis for a Fractional Adams Method

We investigate a method for the numerical solution of the nonlinear fractional differential equation D*αy(t)=f(t,y(t)), equipped with initial conditions y(k)(0)=y0(k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardsons extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.

Numerical solution of the Bagley Torvik equation

We consider the numerical solution of the Bagley-Torvik equation Ay″(t) + BD*3/2y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.

The numerical solution of fractional differential equations: Speed versus accuracy

This paper is concerned with the development of efficient algorithms for the approximate solution of fractional differential equations of the form Dαy(t)=f(t,y(t)), α∈R+−N.(†)We briefly review standard numerical techniques for the solution of (†) and we consider how the computational cost may be reduced by taking into account the structure of the calculations to be undertaken. We analyse the fixed memory principle and present an alternative nested mesh variant that gives a good approximation to the true solution at reasonable computational cost. We conclude with some numerical examples.

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.

A finite element method for time fractional partial differential equations

In this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results.

Fractional boundary value problems: Analysis and numerical methods

In this paper we consider nonlinear boundary value problems for differential equations of fractional order α, 0 < α < 1. We study the existence and uniqueness of the solution and extend existing published results. In the last part of the paper we study a class of prototype methods to determine their numerical solution.

Numerical Hopf bifurcation for a class of delay differential equations

Abstract In this paper we consider discretization of parameter-dependent delay differential equations of the form x′(t)=f(x(t),x(t−τ),λ), λ∈ R . We show that, if the delay differential equation undergoes a Hopf bifurcation, then the discrete scheme undergoes a Hopf bifurcation of the same type. The results of this paper extend the results of our previous analysis relating to the discretization of the delay logistic equation to a wider class of problems.

Analysis and numerical methods for fractional differential equations with delay

In this paper, we consider fractional differential equations with delay. We focus on linear equations. We summarise existence and uniqueness theory based on the method of steps and we give a theorem on the propagation of derivative discontinuities. We discuss the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward difference method to the delay case.

Stability properties of a scheme for the approximate solution of a delay-integro-differential equation

Abstract We discuss some stability properties of a numerical scheme applied to a Volterra integro-differential equation with a finite memory (in which the solution is determined by an initial function). The numerical scheme is based on a strongly-stable linear multistep formula and a consistent quadrature rule, applied with a fixed step.