Single step methods for general second order singular initial value problems with spherical symmetry
type: Item
instance of: scholarly article
title: Single step methods for general second order singular initial value problems with spherical symmetry
zbMATH Open document ID: 0667.65062
DOI: 10.1007/BF01954902
published in: BIT
publication date: 1988-01-01T00:00:00Z
review text: Consider the class of singular initial value problems of the form (1) \(u''+(\alpha /r)u'=f(r,u,u')\), \(u(r=0)=A\), \(u'(r=0)=0\) with A a finite constant and suitable conditions on f as to guarantee existence and uniqueness of the solution of equation (1). The values \(\alpha =0,1,2\) correspond to plane, cylindrical and spherical geometries, respectively. In the case that f is independent of \(u'\) the application of classical explicit Runge-Kutta-schemes for solving (1) has been studied by several people in the last 25 years. However, those schemes are not self-starting and require some special procedure to obtain the solution at the first step because of the singularity at \(r=0.\) The author obtains a one parameter family of four-stage self-starting explicit Runge-Kutta-Nyström methods in the case \(\alpha =2\). This method is exact for \(u(r)=1/r,1,r,r^ 2,r^ 3,r^ 4\) and has a truncation error of \(0(h^ 5)\) in u and of \(0(h^ 4)\) in \(u'\). If in (1) f is independent of \(u'\), the method has truncation error of \(0(h^ 6)\) in both u and \(u'\).
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4092686
zbMATH Keywords: second order singular initial value problems
zbMATH Keywords: spherical symmetry
zbMATH Keywords: explicit Runge-Kutta-schemes
zbMATH Keywords: four-stage self-starting explicit Runge- Kutta-Nyström methods
author: Rakesh K. Jain
reviewed by: Hans Ade
MaRDI profile type: Publication
cites work: Q5560619
cites work: The Application of Runge-Kutta Schemes to Singular Initial Value Problems
cites work: Q3321316
cites work: Q4728633
cites work: On the Nonunique Equilibrium States and Buckling Mechanism of Spherical Shells
cites work: On the Multiplicity of Solutions Of a Differential Equation Arising in Chemical Reactor Theory
cites work: A Taylor series method for the numerical solution of two-point boundary value problems
full work available at URL: https://doi.org/10.1007/bf01954902
OpenAlex ID: W4987302
description: scientific article; zbMATH DE number 4092686