Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

• Published in: International journal of computer mathematics Vol. 79; no. 6; pp. 747 - 763
• Main Authors: Valarmathi, S., Ramanujam, N.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis Group 2002; Taylor and Francis
• Subjects: Asymptotic Approximation; Boundary Layer; Classical Finite Difference Scheme; Exact sciences and technology; Exponentially Fitted Finite Difference Scheme; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Sciences and techniques of general use; Self Adjoint Boundary Value Problem; Singular Perturbation; Third Order Differential Equation; Third order equation; Boundary value problem; Asymptotic expansion; Differential equation; Two point boundary value problem; Singular perturbation; Maximum principle; Asymptotic approximation; Boundary layer; Finite difference method
• DOI: 10.1080/00207160211284
• ISSN: 0020-7160

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.