Cubic Spline Chebyshev Polynomial Approximation for Solving Boundary Value Problems

Abstract

In this work, a Chebyshev polynomial spline function is derived and used to approximate the solution of the second order two-point boundary value problems of variable coefficients with the associated boundary conditions. In deriving the method, the cubic spline Chebyshev polynomial approximation, $S(x)$ is made to satisfy certain conditions for continuity and smoothness of functions. Numerical examples are presented to illustrate the applications of this method. The solution, $y(x)$ of these examples are obtained at some nodal points in the interval of consideration. The absolute errors in each example are estimated, and the comparison of exact values, and approximate values by the present method and other methods in literature at the nodal points are presented graphically. The comparison shows that the proposed method produces better results than Approaching Spline Techniques and collocation method.