On Approximate Solutions of Second-Order Linear Partial Differential Equations

Abstract

In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate value of the unknown Chebyshev coefficients can be found.

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Hanna, Y. (2012) On Approximate Solutions of Second-Order Linear Partial Differential Equations. Applied Mathematics, 3, 1001-1007. doi: 10.4236/am.2012.39148.

1. Introduction

Let the second-order partial differential equation be in the form [1,2]

We assume that it has a Chebyshev series solution in the form

(1.2)

where denotes a sum whose first term is halved. The unknown coefficients can be determined by using so called Chebyshev matrix method.

2. Calculation of Chebyshev Coefficients

Let we have a function, and its nth derivatives with respect to x can be expanded in Chebyshev series

and

Respectively, where and are Chebyshev coefficients; clearly, and . Then the recurrence relation between the coefficients of and is obtained as

(2.1)

From Equation (2.1), we can deduce the relations

And adding these side by side, we get

or

(2.2)

Specially, we can express the coefficients and in terms of the by means of Equation (2.2), in the forms

(2.3)

and

or

(2.4)

Now, let us take and assume for; then the system (2.2) can be transformed into the matrix form,

(2.5)

where M is given in [3].

For it follows from Equation (2.5) that

(2.6)

where clearly.

Let us assume, in the range, that the nth derivatives of with respect to y can be expanded in Chebyshev series

Respectively, where and are Chebyshev coefficients; clearly and . Then the recurrence relation between the coefficients of and is obtained as

(2.7)

From Equation (2.7), we can deduce the relations

and adding these side by side, we get

or

(2.8)

Specially, we can express the coefficients and in terms of the, by means of Equation (2.8), in the forms

(2.9)

and

or

(2.10)

Now, let us take and assume for; then the system (2.8) can be transformed into the matrix form,

(2.11)

where

For it follows from Equation (2.11) that

(2.12)

where clearly. Furthermore, can be expressed as follows:

(2.13)

3. Fundamental Relations

Now consider Equation (1.1), where A, B, C, D, E, F and G are functions of x and y, or constant, defined in the range. Our purpose is to investigate the truncated Chebyshev series solution of Equation (1.1), under the given conditions, in the series form

or in the matrix form

(3.1)

where, are the Chebyshev coefficients to be determined are the bivariate Chebyshev polynomials defined in [4], and matrices, and A are defined by

To obtain the solution of Equation (1.1) in the form of Equation (3.1), first we must reduce Equation (1.1) to a differential Equation whose coefficients are polynomials [5]. For this purpose, we assume that the functions, , , , , , and can be expressed in the form

(3.2)

Which are Taylor polynomials at. By using the expressions (3.2) in Equation (1.1), we get

(3.3)

The Chebyshev expansions of terms

in Equation (3.3) are obtained by means of the formulae

(3.4)

4. Matrix Forms of Terms in the Equation

The matrix representation of Equation (3.4) can be given by

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

where and, .

And for;, (and) is a matrix of size. The elements of Mp are given in [6].

Substituting the expressions (4.1)-(4.7) into Equation (3.3), and simplifying the result, we have the matrix equation

(4.8)

Which corresponds to a system algebraic equations for the unknown Chebyshev coefficients. Briefly, we can assume that Equation (4.8) is given in the form

(4.9)

where

Matrix Equation (4.9) can be reduce to new matrix equation by making use of

Then the new matrix Equation (4.7) becomes

(4.10)

where

and

5. Matrix Forms of Conditions

Let the conditions of Equation (1.1) be given by

(5.1)

(5.2)

(5.3)

where f is a function of x, g is a function of y and is constant.

Then, there are the following matrix forms at x = −1, 0, 1 and similar way for y = −1, 0, 1;

Derivative of Tx at x = −1, 0, 1 and similar way for y = −1, 0, 1;

We assume that the functions and can be expanded as

and

or in the matrix form

and

where

In addition, at x = −1, 0, 1 and y = −1, 0, 1, we obtain the matrix forms

Substituting theses matrices forms into conditions (5.1)-(5.3), and then simplifying, we get the fundamental matrix equations of conditions as follows:

(5.4)

where

6. Former Method for the Solution

We can assume that Equation (6.1) is in the form

(6.1)

where.

Then the augmented matrix of Equation (6.1) becomes or

(6.2)

If we take the new matrix forms of the conditions as, and, respectively, the augmented matrices of them become, and or more clearly

(6.3)

(6.4)

and

(6.5)

Consequently, by replacing Equations (6.3)-(6.5) by the last 2N + 1 rows of Equation (6.2), we have the new augment matrix

From the solution of this system we can find matrix C or matrix A.

7. Applications

The Chebyshev matrix method applied in this study is useful in finding approximate solutions of second-order linear partial differential equations in both homogeneous and non-homogeneous cases, in terms of Chebyshev polynomials. We illustrate it by the following examples.

Example 1. We now consider the problem [7]:

(1)

And seek the solution in the form

(2)

Then we obtain the matrix equation

(3)

where

And the condition matrices are

(4)

(5)

By replacing the new matrix form of Equations (4) and (5) in the new matrix form of Equation (3), we have the matrix equation under given conditions as follows:

Hence, we obtain the augmented matrix

The solution of this system is

and thereby the solution of the problem (1) becomes

or

This is exact solution [7].

Example 2

Let us now study the equation

with conditions which are

The first four terms of the series expansions:

Chebyshev matrix forms of the conditions,

Matrix form of the equation is

From the solution of this matrix equation under the given conditions, we get the Chebyshev coefficients matrix as

The solution of problem is obtained as

Which is the first four terms of.

8. Conclusions

Analytic solutions of the second order linear partial differential equations with variable coefficients are usually difficult. In many cases, it is required to approximate solutions. For this purpose, the Chebyshev matrix method can be proposed.

In this study, the usefulness of the Chebyshev matrix method presented for the approximate solution of the second order linear partial differential equations is discussed. Also, the method can be applied to both the nonhomogeneous and homogeneous cases.

A considerable advantage of the method is that the solution is expressed as a truncated Chebyshev series and thereby a Taylor polynomial. Furthermore, after calculation of the series coefficients, the solution of the equations can be easily evaluated for arbitrary values of at low computation effort.

An interesting feature of the Chebyshev matrix method is that the method can be used in finding exact solutions in much cases. The method can be also extending to the solution of the higher order linear partial differential equations.

Conflicts of Interest

The authors declare no conflicts of interest.

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