Numerical treatment of differential equations with the \(\tau\)-method
Given an equation of the form \(Dy=f\), where f is a polynomial and D a \(\nu\) th order linear ordinary differential operator with polynomial coefficients, together with \(\nu\) supplementary conditions \(g_ j(y)=\sigma_ j\), \(j=1,...,\nu\), the Lanczos \(\tau\)-method is known to treat it. But the choice of the perturbation term in the \(\tau\)-method is not a simple matter. So, instead the author takes an orthogonal basis for the space of polynomials of degree \(\leq n\), expresses \(y_ n\) in it, and determines its coefficients by making \(y_ n\) satisfy the given supplementary conditions and \(Dy_ n\) agree with y as far as possible or desired. This principle enhances the original \(\tau\)-method and makes it more amenable to computer programming. Moreover, a way of a posteriori error estimation is proposed. Using Newton's linearization, a nonlinear ODE can also be solved by the new \(\tau\)-method. A few numerical examples are given by way of illustration.
scientific article; zbMATH DE number 4026715
scientific article