Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems
type: Item
instance of: scholarly article
title: Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems
zbMATH Open document ID: 1157.65047
author: S. Chandra Sekhara Rao
author: Mukesh Kumar
published in: Applied Numerical Mathematics
publication date: 2008-10-07T00:00:00Z
review text: The authors study numerical method for solving the self-adjoint boundary value problem of a singularly perturbed second order linear differential equation \[ Lu \equiv \varepsilon u''(x)+q(x)u(x)=f(x),\quad 0\leq x\leq 1, \] with homogeneous boundary conditions \(u(0)=u(1)=0\), where \(\varepsilon\) is a small parameter and \(f(x),\;q(x)\) are smooth functions and satisfy \(q(x)\geq q_*,\, \forall x\in [0,1]\) for some positive constant \(q_*\). For small \(\varepsilon\), the solution \(u(x)\) may exhibit exponential boundary layers at both ends of the interval \([0,1]\), which gives rise to difficulties in numerical solutions. Using a B-splines basis for the space of exponential spline, the authors propose a spline collocation method which leads to an easily solvable tridiagonal algebraic system. It is shown that the method can be implemented on uniform meshes, i.e. there is no need of introducing more nodal points in the boundary layers. Further, the second order uniform convergence of the numerical solution is proven. The efficiency of the method is demonstrated by several numerical experiments where the present method is shown to be superior to the scheme using cubic B-splines on fitted meshes that was suggested previously by \textit{M. K. Kadalbajoo} and \textit{V. K. Aggarwal} [Appl. Math. Comput. 161, No.~3, 973--987 (2005; Zbl 1073.65062)].
reviewed by: Vu Hoang Linh
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 5349253
zbMATH Keywords: singular perturbation
zbMATH Keywords: exponential B-spline
zbMATH Keywords: collocation method
zbMATH Keywords: uniform mesh
zbMATH Keywords: self-adjoint boundary value problem
zbMATH Keywords: second order linear differential equation
zbMATH Keywords: exponential boundary layers
zbMATH Keywords: convergence
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: j.apnum.2007.09.008
OpenAlex ID: W2080305533
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DOI: 10.1016/J.APNUM.2007.09.008
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description: scientific article; zbMATH DE number 5349253