An Extension of the Legendre‐Galerkin Method for Solving Sixth‐Order Differential Equations with Variable Polynomial Coefficients

A H Bhrawy,S I El-Soubhy,A S Alofi

doi:10.1155/2012/896575

Abstract

We extend the application of Legendre‐Galerkin algorithms for sixth‐order elliptic problems with constant coefficients to sixth‐order elliptic equations with variable polynomial coefficients. The complexities of the algorithm are O(N) operations for a one‐dimensional domain with (N − 5) unknowns. An efficient and accurate direct solution for algorithms based on the Legendre‐Galerkin approximations developed for the two‐dimensional sixth‐order elliptic equations with variable coefficients relies upon a tensor product process. The proposed Legendre‐Galerkin method for solving variable coefficients problem is more efficient than pseudospectral method. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques.

Highlights

Spectral methods are preferable in numerical solutions of ordinary and partial differential equations due to its high-order accuracy whenever it works 1, 2
We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomial coefficients
We introduce a generalization of Shen’s basis to numerically solve the sixth-order differential equations with variable polynomial coefficients

Summary

Introduction

Spectral methods are preferable in numerical solutions of ordinary and partial differential equations due to its high-order accuracy whenever it works 1, 2. Renewed interest in the Galerkin technique has been prompted by the decisive work of Shen 3 , where new Legendre polynomial bases for which the matrices systems are sparse are introduced. We introduce a generalization of Shen’s basis to numerically solve the sixth-order differential equations with variable polynomial coefficients. Sixth-order boundary-value problems arise in astrophysics; the narrow convecting layers bounded by stable layers, which are believed to surround A-type stars, may be modeled by sixth-order boundary-value problems 4, 5. Further discussion of the sixth-order boundary-value problems is given in 6. The literature of numerical analysis contains little work on the solution of the sixth-order boundary-value problems 4, 5, 7, 8.

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