Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials

Abstract

Two new families of general parameters generalized Jacobi polynomials are introduced. Some efficient and accurate algorithms based on these families are developed and implemented for solving third- and fifth-order differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a dual Petrov-Galerkin method. The use of general parameters generalized Jacobi polynomials leads to simplified analysis, more precise error estimates and well conditioned algorithms. The method leads to linear systems with specially structured matrices that can be efficiently inverted. Numerical results indicating the high accuracy and effectiveness of the proposed algorithms are presented.

Keywords: Dual-Petrov-Galerkin method, general parameters generalized Jacobi polynomials, nonhomogeneous Dirichlet conditions

Quaestiones Mathematicae 36(2013), 15–38