Computational Fractional Dynamical Systems

11Boundary Characteristic Orthogonal Polynomials

11.1 Introduction

Bhat (1985, 1986) proposed boundary characteristic orthogonal polynomials (BCOPs) in 1985, which have been applied in various scientific and engineering fields. Several authors have employed BCOPs in many problems, such as Bhat and Chakraverty (2004) and Singh and Chakraverty (1994a) for two‐dimensional BCOPs first time in a systematic manner. BCOPs have been beneficial in well‐known approaches like Rayleigh–Ritz, Galerkin, and collocation. The Gram–Schmidt orthogonalization approach (Johnson 2014) can be used to generate BCOPs. The resulting BCOPs have to satisfy some of the boundary conditions of the considered models (Singh and Chakraverty 1994b; Bhat and Chakraverty 2004). Initially, the general approximation solution to the problem is assumed to be a linear combination of BCOPs. The residual can be obtained by replacing the approximate solution in the boundary value problem (Singh and Chakraverty 1994a; Chakraverty et al. 2008). A linear system of equations may be developed by employing the residual. Finally, the resultant linear system may be handled using any known analytical/numerical method. The orthogonal nature of BCOPs makes them straightforward to analyze.

Torvik and Bagley (1984) proposed a fractional model after moving a rigid plate dipped in a Newtonian fluid as follows:

(11.1)

where a, b, and ...