The numerical solution of an integro-differential equation close to bifurcation points

In this paper we investigate the qualitative behaviour of numerical approximations to a Volterra integro-differential equation. We consider (as prototype) a linear problem with fading memory kernel of the form y′(t) = − ∫ t 0 e−λ(t−s)y(s)ds, y(0) = 1 and we consider the performance of simple numerical schemes applied to solve the equation. We are concerned with the preservation (or otherwise) of qualitative properties of the analytical solution in the numerical approximation. We outline the known stability behaviour and derive the values of λ at which the true solution bifurcates. We give the corresponding analysis for the discrete schemes and we show that, as the step size of the numerical scheme decreases, the bifurcation points tend towards those of the continuous scheme. We illustrate our results with some numerical examples.