Quadratic Spline and Two-Point Boundary Value Problem

Abstract

where f ( t , x, y) is defined and twice continuously differentiable in a region D of the (t, x} :y)-space intercepted by two hyperplanes £=0 and t=l. The analytical solution of (1.1) for arbitrary choices of the function / cannot be found in general. We usually resort to some numerical method for obtaining an approximate solution of the problem (1.1). The standard numerical methods for the numerical treatment of (1.1) consist of finite difference methods, shooting methods, RayleighRitz and Galerkin's procedure and collocation methods. A long list of references of all of these methods is given by Keller [4] in [2]. The subject of obtaining spline solutions for the initial as well as boundary value problems is briefly discussed in [1]. Since then many papers have appeared dealing with the continuous approximation of x(t) satisfying (1.1) via cubic and quintic splines mainly (see [3, 9]). The collocation methods using spline functions have been developed and analysed by Sakai (see [5], [6], [7]) again employing cubic and quintic splines at equi-distant knots. Recently Usmani and Sakai [11] have also used quadratic spline function for solving a two-point boundary value problem involving a fourth order differential equation. In this brief report, we propose a second order collocation method using quadratic spline employing 5-splines. Ill the sequel, it will be shown that our method is an 0(/i)-convergent procedure. In the last section some numerical evidence is included to show the practical applicability of our method by solving