A numerical verification method for solutions of singularly perturbed problems with nonlinearity
type: Item
instance of: scholarly article
title: A numerical verification method for solutions of singularly perturbed problems with nonlinearity
zbMATH Open document ID: 1071.65110
DOI: 10.1007/BF03167479
author: Kouji Hashimoto
author: Ryohei Abe
author: Mitsuhiro T. Nakao
author: Yoshitaka Watanabe
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 2005-04-29T00:00:00Z
review text: This paper is concerned with the computer assisted proof to verify solutions of two-point singularly perturbed boundary value problems of type: \( L u \equiv - \epsilon u'' -b(x) u' + c(x) u = f(u),\) \( x \in (0,1)\), \( u(0)=u(1)=0\), where \(\epsilon\) is a small positive parameter, \(f\) is a bounded continuous non linear map, \( b(x), c(x) \in W_{\infty}^1(0,1)\) with \( c(x) \geq \gamma >0\). The proposed technique is based in previous researches of one of the authors [\textit{M. T. Nakao}, Numer. Funct. Anal. Optimization 22, No. 3--4, 321--356 (2001; Zbl 1106.65315)] and requires some a priori estimates of related linear problems. However for the above nonlinear singularly perturbed problems, standard estimates contain negative powers of the small parameter and therefore cannot be applied. In the paper under consideration, the authors derive a priori estimates for the L-spline method based on an exponential fitting with the Green's function, that are not badly affected by the small parameter. Such estimates allow accurate verification of solutions of the above problems. Finally, the results of two numerical examples are presented to show the effectiveness of the proposed verification results.
reviewed by: Manuel Calvo
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 65G20
Mathematics Subject Classification ID: 68T15
zbMATH DE Number: 2162990
zbMATH Keywords: nonlinear two point boundary value problems
zbMATH Keywords: computer assisted proof
zbMATH Keywords: singular perturbations
zbMATH Keywords: verification of finite element solutions
zbMATH Keywords: L-spline method
zbMATH Keywords: exponential fitting
zbMATH Keywords: numerical examples
describes a project that uses: INTLAB
MaRDI profile type: Publication
cites work: NUMERICAL VERIFICATION METHODS FOR SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS
cites work: Q2765620
cites work: An Analysis of a Superconvergence Result for a Singularly Perturbed Boundary Value Problem
cites work: Q4871736
cites work: A finite element method for a singularly perturbed boundary value problem
cites work: An Analysis of a Singularly Perturbed Two-Point Boundary Value Problem Using Only Finite Element Techniques
cites work: Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element
full work available at URL: https://doi.org/10.1007/bf03167479
OpenAlex ID: W1966877291
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description: scientific article; zbMATH DE number 2162990