``Shooting method'' for the solution of singularly perturbed two-point boundary-value problems having less severe boundary layer.
type: Item
instance of: scholarly article
title: ``Shooting method'' for the solution of singularly perturbed two-point boundary-value problems having less severe boundary layer.
zbMATH Open document ID: 1035.65081
DOI: 10.1016/S0096-3003(01)00263-6
author: Yanyan Li
published in: Applied Mathematics and Computation
publication date: 2003-01-28T00:00:00Z
review text: This paper presents a numerical method to compute the solution of the problem \[ \begin{matrix}\epsilon u''+ a(x)u'-b(x)u= f(x), \\ a(0)u'(0)-b(0)u(0)=f(0),\,\, u(1)=B. \end{matrix}\tag{1} \] Boundary conditions are said to be ``less severe'' as there is a solution \(u_0\) of the reduced problem \[ a(x)u'-b(x)u= f(x) \] that verifies both boundary conditions in (1). Basic assumptions are: \(\epsilon >0\) is a small parameter, \(a(x)\geq \alpha>0\) and \(b(x)\geq 0\). The proposed method is to solve an inner region problem \[ \begin{matrix} \epsilon u''+ a(x)u'-b(x)u= f(x), \\ u(0)=u_0(0),\,\, u'(0)=[f(0)+b(0)u_0(0)]/a(0), \end{matrix} \] defined for \(x\in [0,k_1]\), together with an outer region problem \[ \begin{matrix}\epsilon u''+ a(x)u'-b(x)u= f(x), \\ u(k_2)=u_0(k_2),\,\, u(1)=B, \end{matrix} \] where \(k_1\leq k_2\). The approximate solution of (1) is then obtained as the juxtaposition of these inner and outer solutions. Error estimates are derived. Numerical examples are given.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 65L12
zbMATH DE Number: 1861279
zbMATH Keywords: singular perturbation
zbMATH Keywords: boundary value problem
zbMATH Keywords: shooting method
zbMATH Keywords: exponentially fitted difference scheme
zbMATH Keywords: boundary layer
zbMATH Keywords: Error estimates
zbMATH Keywords: numerical examples
reviewed by: Patrick Habets
MaRDI profile type: Publication
cites work: Q4871736
cites work: Q3909948
cites work: Q4486278
cites work: Q4865485
cites work: Asymptotic and numerical analysis of singular perturbation problems: A survey
cites work: Layer-Adapted Grids for Singular Perturbation Problems
cites work: On a Differential Equation of Boundary Layer Type
cites work: Difference approximations for singular perturbations of systems of ordinary differential equations
cites work: Analysis of Some Difference Approximations for a Singular Perturbation Problem Without Turning Points
cites work: Q3902431
cites work: A boundary value technique for singular perturbation problems
cites work: A computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers
cites work: ‘Shooting method' for singular perturbation problems arising in chemical reactor theory
cites work: ‘Shooting method’ for singularly perturbed one-dimensional reaction-diffusion neumann problems
cites work: A numerical-asymptotic solution for singular perturbation problems
cites work: Q5665539
cites work: A numerical algorithm for singular perturbation problems exhibiting weak boundary layers.
cites work: Sufficient Conditions for Uniform Convergence of a Class of Difference Schemes for a Singularly Perturbed Problem
Recommended article: Shooting method in singular perturbation problem of ordinary differential equations with boundary layers
Recommended article: Q4659509
Recommended article: Shooting Method for Nonlinear Singularly Perturbed Boundary‐Value Problems
Recommended article: On the shooting method for a class of two-point singular nonlinear boundary value problems
Recommended article: Q3339201
Recommended article: An iterative shooting method for a certain class of singular two-point boundary value problems.
Recommended article: Some modifications of the shooting method for solving non-linear two-point boundary value problems
Recommended article: Q3345712
Recommended article: A nonlinear shooting method for two-point boundary value problems
description: scientific article; zbMATH DE number 1861279