B-spline Collocation Methods for Solving Linear Two-Point Boundary Value Problems

Abstract

Generating numerical solutions for boundary value problems (BVPs) using numerical methods can be a difficult task since the solutions may involve complex mathematical formulations. Hence, various types of B-splines collocation methods were developed to produce better numerical approximations with much simpler approaches. The present study introduced two new types of extended Cubic Hybrid B-spline Collocation Method (CHBSM) namely the Extended Cubic Hybrid B-spline Collocation Method (ECHBSM) and Extended Trigonometric Cubic Hybrid B-spline Collocation Method (ETCHBSM) for solving second order linear two-point BVPs. These methods were tested on three examples of linear two-point BVPs of order two. For comparison purposes, three established collocation methods which are the Cubic B-spline Collocation Method (CBSM), Cubic Trigonometric B-spline Collocation Method (CTBSM) and Cubic Hybrid B-spline Collocation Method (CHBSM) were also applied on these examples. The numerical results were tabulated and analysed to make comparisons with the analytical solutions and the conventional methods from past literatures. For CHBSM, ECHBSM and ETCHBSM, optimization was applied to the free parameter using a simple proposed approach. The result demonstrated that CHBSM yields the best approximation with the analytical solutions.