Implicit single step methods by spline-like functions for solution of ordinary differential equations
type: Item
instance of: scholarly article
title: Implicit single step methods by spline-like functions for solution of ordinary differential equations
zbMATH Open document ID: 0668.65054
DOI: 10.1016/0898-1221(88)90005-3
author: B. Ráczkevi
author: Gy. Molnárka
publication date: 1988-01-01T00:00:00Z
review text: A family of one step implicit algorithms is given which produces continuously differentiable approximations to a first order ordinary differential equation. The approximations are not splines but are spline- like. Let \((i)\quad y(x+h)=y(x)+hg(y,h,f)\) determine an explicit one step p-th order approximation to the solution y(x) of \(y'=f(x,y)\) at nodes \(x_ n=x_ 0+nh\), \(n=0,1,..\). The new algorithm is obtained by replacing (i) by \((ii)\quad y(x+h) = y(x)+hg(y,h,f)+wh^{p+1}\) where \(w\) is chosen so that \(y'(x+h)=f(x+h,y(x+h)).\) It is proved that (ii) is a \((p+1)\)-th order implicit algorithm. When (i) is specialized to be the Euler formula (ii) turns out to be \(A\)-stable. A computer program has been written in which (i) is chosen to be a Runge-Kutta algorithm of order \(p=5\) with continuously variable weights due to Sarafyan [cf. \textit{C. Outlaw}, \textit{L. Derr} and \textit{D.Sarafyan}, ibid. 12A(6), 815-824 (1986; Zbl 0626.65069)]. Two numerical examples are provided.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 4094699
zbMATH Keywords: implicit single step methods
zbMATH Keywords: \(A\)-stability
zbMATH Keywords: \(A\)-stable methods
zbMATH Keywords: spline-like approximation
zbMATH Keywords: one step implicit algorithms
zbMATH Keywords: Runge-Kutta algorithm
zbMATH Keywords: examples
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
full work available at URL: 0898-1221(88)90005-3
OpenAlex ID: W2064144532
cites work: Spline function approximations for solutions of ordinary differential equations
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cites work: Continuous approximate solution of ordinary differential equations and their systems
cites work: A sixth-order imbedded Runge-Kutta algorithm with continuously variable weights
cites work: Q3774861
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 4094699