Numerical solution of singularly perturbed initial value problems using fourth order cubic spline method

Abstract

In this paper, we present a numerical method using a fourth order cubic spline for singularly-perturbed problems: ε y′ = f(x y ε)y(a) prescribeda ≤ x ≤ b, 0 < ε ≪ 1, where f y (x y, 0) is non-singular on (a b) and has stable eigen values for all x and y. The inner and outer regions are separated by a numerical cutting point technique and using the stretching of the independent variable x, the inner region problem is solved by a fourth order cubic spline. Finally, solutions of the inner and outer region problems are combined to give a global approximation to the original singularly perturbed initial value problem. The convergence analysis of the method giving rise to fourth order approximation of the spline solution is presented. Some well known examples have been solved to demonstrate the efficiency of the method.