High-order nonlinear initial-value problems countably determined
type: Item
instance of: scholarly article
title: High-order nonlinear initial-value problems countably determined
zbMATH Open document ID: 1169.65071
author: J. Martínez
published in: Journal of Computational and Applied Mathematics
publication date: 2009-05-28T00:00:00Z
review text: An iterative approach to approximate global solutions of high order initial value problems (IVPs): \[ x^{(m)}(t)= f(t, x(t), x'(t), \dots x^{(m-1)}(t) ), \quad t \in [t_0, t_0+T]\tag{1} \] with \( x^{(j)}(t_0)= x_0^j,\) \(j=0, \dots ,m-1, (1)\) is proposed. The approach extends previous results of the same authors for first order non linear IVPs [Nonlinear Anal., Theory Methods Appl. 63, No.~1 (A), 97--105 (2005; Zbl 1097.34005)]. Under continuity and uniform Lipschitz conditions on the nonlinear function \(f\), the IVP (1) is equivalent to an integral equation \( x(t) = T x(t)\) with some integral operator \(T\) and the problem reduces to the computation of a fixed point of \(T\) in a Banach space. For the approximation of such a fixed point a suitable Faber-Schauder basis \( ( \Gamma_j )_{j \geq 0}\) of piecewise linear functions associated to a sequence of nodes \( (t_j)_{j \geq 0} \) with \( t_1 = t_0+T\) dense in \([t_0, t_0+T]\) is proposed such that \( ( \varphi_k = \sum_{i \geq 0} \lambda_k^i \; \Gamma_i )\) with \( \lambda_k^i\) appropriately chosen tends to the exact solution. Several convergence results for an iterative to the solution as well as error estimates are proposed and finally the technique is applied to an IVP second order test problem to show the convergence of the method depending on the number of the nodes \(t_j\) and the number of iterations.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L70
zbMATH DE Number: 5559031
zbMATH Keywords: nonlinear high order initial value problems
zbMATH Keywords: Global approximation of solutions
zbMATH Keywords: Schauder basis
zbMATH Keywords: numerical examples
zbMATH Keywords: integral equation
zbMATH Keywords: fixed point
zbMATH Keywords: convergence
zbMATH Keywords: error estimates
MaRDI profile type: Publication
full work available at URL: j.cam.2008.08.039
OpenAlex ID: W1991609695
cites work: Linear Volterra integro-differential equation and Schauder bases
cites work: High order linear initial-value problems and Schauder bases
cites work: Nonlinear initial-value problems and Schauder bases
cites work: Q4772243
cites work: An Introduction to Banach Space Theory
cites work: ISOMORPHISMS, SCHAUDER BASES IN BANACH SPACES, AND NUMERICAL SOLUTION OF INTEGRAL AND DIFFERENTIAL EQUATIONS
cites work: Schauder bases in Banach spaces of continuous functions
DOI: 10.1016/J.CAM.2008.08.039
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description: scientific article; zbMATH DE number 5559031