On a Functional Differential Equation

Abstract

This paper considers some analytical and numerical aspects of the problem defined by an equation or systems of equations of the type (d/dt)y(t) = ay(λt)+by(t), with a given initial condition y(0) = 1. Series, integral representations and asymptotic expansions for y are obtained and discussed for various ranges of the parameters a, b and λ(> 0), and for all positive values of the argument t. A perturbation solution is constructed for ∣1−λ∣ ≪ 1, and confirmed by direct computation. For λ > 1 the solution is not unique, and an analysis is included of the eigensolutions for which y(0) = 0. Two numerical methods are analysed and illustrated. The first, using finite differences, is applicable for λ < 1, and two techniques are demonstrated for accelerating the convergence of the finite-difference solution towards the true solution. The second, an adaptation of the Lanczos τ method, is applicable for any λ > 0, though an error analysis is available only for λ < 1. Numerical evidence suggests that for λ > 1 the method still gives good approximations to some solution of the problem.