Numerical treatment of differential equations with the τ-method

Abstract

Given an equation of the form Dy = f, where f is an algebraic polynomial and D a νth order linear ordinary differential operator with polynomial coefficients, together with ν supplementary conditions, g j ( y = σ j , j = 1,…, ν, where the g j 's are given linear functionals, the basic idea of Lanczos' τ-method is to perturb the given ODE through the addition to its r.h.s. of an algebraic polynomial H n , usually a linear combination of Chebyshev polynomials, chosen so that the perturbed problem, Dy n = f + H n , g j ( y n ) = σ j , j = 1,…, ν, has a unique polynomial solution. The choice of H n , however, is not a simple matter, as it depends essentially on structural properties of D. So, instead of starting by choosing a perturbation H n and solving the perturbed problem afterwards, we rather take an orthogonal basis for the space of algebraic polynomials of degree ⩽ n, express y n in it, and determine its coefficients by making y n satisfy the given supplementary conditions and D y n agree with D y as far as possible or desired. This approximation principle leads quite naturally to good polynomial approximants of y in the sense of the τ-method and is much more amenable to computer programming than Lanczos' original idea. Using Newton's linearization method, a given ODE with nonlinearities of polynomial form may be reduced to a sequence of linear ODEs and to each of these we may apply the approximation principle described above. Nonlinearities of nonpolynomial form require a first stage approximation to reduce them to polynomial type and this may also be carried out with that principle. Combining it with Newton's method leads to an efficient iterative scheme which has been applied with success to a number of nonlinear differential problems, for some of which a few numerical results will be given by way of illustration.