Initial Value Method for Solving Second Order Singularly Perturbed Two Point Boundary Value Problems

K. Selvakumar

Department of   Mathematics , Anna University of Technology  Tirunelveli, Tirunelveli—627 007,  Tamil Nadu, India.

Abstract:  Initial value  method is presented for solving  singularly perturbed two point boundary value problems . The  presence of first derivative term leads to  a  boundary layer region nearer the  left  end point of the interval .  In this method, the approximate solution is obtained by solving the reduced problem and  an initial value problem associated with the given singularly perturbed problem  numerically. The reduced problem  is solved by  Runge Kutta method of order four and the other initial value problem is solved  by a stiff method(exponentially fitted method of  Doolan et al.,) of order one. The method do not require the matrix  inversion for the numerical convergence. The method presented in this paper is a modified form of  the method of Gasparo and Macconi. The error estimates for the numerical convergence of the method of Gasparo and Macconi and the method presented in this paper are derived.  Numerical results are given in this paper to demonstrate the applicability of the initial value  method.

Keywords:   singular perturbation problems,   exponentially fitted, uniformly convergent, asymptotic  expansion,  finite difference schemes, initial value method.




International eJournal of Mathematics and Engineering

Volume 2, Issue 2, Pages:  920 - 931