Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

• Published in: Journal of optimization theory and applications Vol. 116; no. 1; pp. 167 - 182
• Main Authors: Valanarasu, T., Ramanujan, N.
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 01.01.2003; Springer Nature B.V
• Subjects: Applied mathematics; Boundary value problems; Exact sciences and technology; Mathematical analysis; Mathematics; Methods; Ordinary differential equations; Sciences and techniques of general use; ordinary differential equations; small parameters; asymptotic expansions; non- turning points; initial-value methods; Singular-perturbation problems; boundary layers; two-point boundary- value problems
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1023/A:1022118420907

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.