TY  - EJOU
AU  - Abd-Elhameed, W. M. 
AU  - Alkenedri, Asmaa M. 

TI  - Spectral Solutions of Linear and Nonlinear BVPs Using Certain Jacobi Polynomials Generalizing Third- and Fourth-Kinds of Chebyshev Polynomials
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2021
VL  - 126
IS  - 3
SN  - 1526-1506

AB  - 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.
KW  - Jacobi polynomials; high-order boundary value problems; Galerkin method; collocation method; connection problem; convergence analysis

DO  - 10.32604/cmes.2021.013603