Spectral Solutions of Linear and Nonlinear BVPs Using Certain Jacobi Polynomials Generalizing Third- and Fourth-Kinds of Chebyshev Polynomials

Abstract

This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems. For this purpose, we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials. These two classes generalize the two celebrated non-symmetric classes of polynomials, namely, Chebyshev polynomials of third- and fourth-kinds. The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials. The derived formulas serve in converting the even-order linear differential equations with their boundary conditions into linear systems that can be efficiently solved. Furthermore, and based on the first-order derivatives formula of certain Jacobi polynomials, the operational matrix of derivatives is extracted and employed to present another algorithm to treat both linear and nonlinear two-point boundary value problems based on the application of the collocation method. Convergence analysis of the proposed expansions is investigated. Some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.