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

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