Numerical treatment of differential equations with the \(\tau\)-method
type: Item
instance of: scholarly article
title: Numerical treatment of differential equations with the \(\tau\)-method
zbMATH Open document ID: 0631.65079
DOI: 10.1016/0377-0427(87)90134-8
author: Manuel R. da Silva
published in: Journal of Computational and Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 4026715
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Lanczos tau-method
zbMATH Keywords: a posteriori error estimation
zbMATH Keywords: Newton's linearization
zbMATH Keywords: numerical examples
reviewed by: Taketomo Mitsui
MaRDI profile type: Publication
full work available at URL: 0377-0427(87)90134-8
OpenAlex ID: W2142279361
cites work: Q5533821
cites work: Numerical Solution of Systems of Ordinary Differential Equations with the Tau Method: An Error Analysis
cites work: Trigonometric Interpolation of Empirical and Analytical Functions
cites work: Q5816295
cites work: Q3846497
cites work: Collocation and residual correction
cites work: An operational approach to the Tau method for the numerical solution of non-linear differential equations
cites work: Q3334905
cites work: Software for a method of finite approximations for the numerical solution of differential equations
cites work: Numerical solution of high order boundary value problems for ordinary differential equations with an estimation of the error
cites work: Matrix displacement mappings in the numerical solution of functional and nonlinear differential equations with the tau method
description: scientific article; zbMATH DE number 4026715