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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

Spectral methods are preferable in numerical solutions of ordinary and partial differential equations due to its high-order accuracy whenever it works [

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 [

From the numerical point of view, Shen [

For sixth-order differential equations, Twizell and Boutayeb [

The main aim of this paper is to extend the application of Legendre-Galerkin method (LGM) to solve sixth-order elliptic differential equations with variable coefficients by using the expansion coefficients of the moments of the Legendre polynomials and their high-order derivatives. We present appropriate basis functions for the Legendre-Galerkin method applied to these equations. This leads to discrete systems with sparse matrices that can be efficiently inverted. The complexities of the algorithm is

This paper is organized as follows. In the next section, we discuss an algorithm for solving the one-dimensional sixth-order elliptic equations with variable polynomial coefficients. In Section

We first introduce some basic notation which will be used in the sequel. We denote by

We recall that the

Some other useful relations are

In this section, we are interested in using the Legendre-Galerkin method to solve the variable polynomial coefficients sixth-order differential equation in the form:

The problem of approximating solutions of ordinary or partial differential equations by Galerkin approximation involves the projection onto the span of some appropriate set of basis functions, typically arising as the eigenfunctions of a singular Sturm-Liouville problem. The members of the basis may satisfy automatically the boundary conditions imposed on the problem. As suggested in [

A more general situation which often arises in the numerical solution of differential equations with polynomial coefficients by using the Legendre Galerkin method is the evaluation of the expansion coefficients of the moments of high-order derivatives of infinitely differentiable functions. The formula of Legendre coefficients of the moments of one single Legendre polynomials of any degree is

We have, for arbitrary constants

Immediately obtained from relations (

Hence, by setting

If we take

The proof of this theorem is rather lengthy, but it is not difficult once Lemma

From Theorem

In general, the expense of calculating an

In the special case, (

If

Note that the results of Corollary

It is worthy to note here that if

Obviously

If the boundary conditions are nonhomogeneous, one can split the solution

In this section, we extend the results of Section

The Legendre-Galerkin approximation to (

We denote

The direct solution algorithm here developed for the sixth-order elliptic differential equation in two dimensions relies upon a tensor product process, which is defined as follows. Let P and R be two matrices of size

We can also rewrite (

Compute the matrices

from Theorem

Find the tensor product of the matrices in the previous step.

Compute

Obtain a column vector

We report on two numerical examples by using the algorithms presented in the previous sections. It is worthy to mention that the pure spectral-Galerkin technique is rarely used in practice, since for a general right-hand side function

Here we present some numerical results for a one-dimensional sixth-order equation with polynomial coefficients. We only consider the case

For

Maximum pointwise error of

20 | 30 | 1 | 1 | 1 | 3 | 5 | ||

30 | ||||||||

40 |

In order to examine the algorithm proposed in Section

Consider the two-dimensional sixth-order equation

In Table

Maximum pointwise error of

20 | 3 | 1 | 1 | 0 | 0 | 0 | ||

30 | ||||||||

40 |

We have presented stable and efficient spectral Galerkin method using Legendre polynomials as basis functions for sixth-order elliptic differential equations in one and two dimensions. We concentrated on applying our algorithms to solve variable polynomials coefficients differential equations by using the expansion coefficients of the moments of the Legendre polynomials and their high-order derivatives. Numerical results are presented which exhibit the high accuracy of the proposed algorithms.

The authors are very grateful to the referees for carefully reading the paper and for their comments and suggestions which have improved the paper. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.