New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials

Abstract

A family of symmetric generalized Jacobi polynomials (GJPs) is introduced and used for solving high even-order linear elliptic differential equations in one variable by the Galerkin method. Some efficient and accurate algorithms are developed and implemented for solving such equations. The use of GJPs leads to a simplified analysis, more precise error estimates and very efficient numerical algorithms. The methods lead to linear systems with specially structured matrices that can be efficiently inverted. Two numerical examples are presented aiming to demonstrate the accuracy and the efficiency of the proposed algorithms.