Advanced Numerical and Semi-Analytical Methods for Differential Equations

4Boundary Characteristics Orthogonal Polynomials

4.1 Introduction

Boundary characteristic orthogonal polynomials (BCOPs) proposed by Bhat [1,2] in 1985 have been used in various science and engineering problems. Further, many authors like Bhat and Chakraverty [3], Singh and Chakraverty [4] have also used BCOPs in different problems. BCOPs are found to be advantageous in well‐known methods like Rayleigh–Ritz, Galerkin, collocation, etc.

BCOPs may be generated by using the Gram–Schmidt orthogonalization procedure [5]. The generated BCOPs have to satisfy the boundary conditions of the considered problem [ 3 ,6]. Initially, the general approximated solution of the considered problem is assumed as linear combination of BCOPs. By substituting the approximated solution in the boundary value problem, one may get the residual [ 4 ,7]. Further, by using the residual and following the algorithm of a particular method, one may develop a linear system of equations. In the final step, one can handle the obtained linear system by using any known analytical/numerical procedure. The orthogonal nature of the BCOPs makes the analysis simple and straightforward.

The following section presents the Gram–Schmidt orthogonalization process for generating orthogonal polynomials.

4.2 Gram–Schmidt Orthogonalization Process

Let us suppose a set of functions (g_i(x), i = 1, 2, …) in [a, b]. From these set of functions, one can construct appropriate orthogonal functions by using the well‐known procedure known ...