Coefficients perturbation methods for higher-order differential equations

Abstract

In this paper, we develop a general approach for estimating and bounding the error committed when higher-order ordinary differential equations (ODEs) are approximated by means of the coefficients perturbation methods. This class of methods was specially devised for the solution of Schrödinger equation by Ixaru in 1984. The basic principle of perturbation methods is to find the exact solution of an approximation problem obtained from the original one by perturbing the coefficients of the ODE, as well as any supplementary condition associated to it. Recently, the first author obtained practical formulae for calculating tight error bounds for the perturbation methods when this technique is applied to second-order ODEs. This paper extends those results to the case of differential equations of arbitrary order, subjected to some specified initial or boundary conditions. The results of this paper apply to any perturbation-based numerical technique such as the segmented Tau method, piecewise collocation, Constant and Linear perturbation. We will focus on the Tau method and present numerical examples that illustrate the accuracy of our results.