Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems
type: Item
instance of: scholarly article
title: Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems
zbMATH Open document ID: 0929.65046
DOI: 10.1016/S0168-9274(98)00087-7
author: K. Appert
published in: Applied Numerical Mathematics
publication date: 2000-01-11T00:00:00Z
review text: This paper is concerned with the numerical solution of boundary-value problems for the generalized Emden-Fowler equation \(y''(x) + c x^p y(x)^q = 0, x \in[0,1]\) with \(y(0)=1, y(1)=0\). Here \(c,p,q\) are real constants, \( p \in (-2,-1), q>1,\) and \(c<0\). The approach proposed by the authors combines a Picard iteration in which the step \(y_{\nu -1}(x) \to y_{\nu}(x)\) satisfies the linear equation \( y_{\nu}'' + c q x^q y_{\nu} = - c x^p ( y_{\nu-1}^q - q y_{\nu-1})\) together with finite differences and extrapolation. In order to justify the extrapolation, asymptotic expansions for the Picard iterates around the two end points have been obtained. It is shown that these expansions depend crucially on the fact that \( (p+2)^{-1}\) is an integer or not. Taking into account these results, extrapolation was used to approximate the solution in the cases \((p,q,c)=(-1.5, 2.5, -1), (-1.7, 2.7, -1), (-1.4, 1, -0.6)\). Furthermore a change of the independent variable has been considered to improve the performance of the algorithm. Finally some numerical results are presented to show the improvement of the new method over the standard finite-difference technique for several grids.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 65L12
zbMATH DE Number: 1311312
zbMATH Keywords: singular boundary-value problems
zbMATH Keywords: generalized Emden-Fowler equations
zbMATH Keywords: extrapolation methods
zbMATH Keywords: Picard iteration
zbMATH Keywords: finite differences
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: algorithm
zbMATH Keywords: numerical results
reviewed by: Manuel Calvo
MaRDI profile type: Publication
full work available at URL: s0168-9274(98)00087-7
OpenAlex ID: W2061039074
description: scientific article; zbMATH DE number 1311312