Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 434976, 15 pages

doi:10.1155/2012/434976

Research Article

Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition

Allaberen Ashyralyev1,2 and Fadime Dal³

¹Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey

²Department of Mathematics, ITTU, Ashgabad, Turkmenistan

³Department of Mathematics, Ege University, 35100 Izmir, Turkey

Received 19 December 2011; Accepted 18 April 2012

Academic Editor: Chuanxi Qian

Copyright © 2012 Allaberen Ashyralyev and Fadime Dal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The numerical and analytic solutions of the mixed problem for multidimensional fractional hyperbolic partial differential equations with the Neumann condition are presented. The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation with the Neumann condition is presented. Stability estimates for the solution of this difference scheme and for the first- and second-order difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations. He's variational iteration method is applied. The comparison of these methods is presented.

1. Introduction

It is known that various problems in fluid mechanics (dynamics, elasticity) and other areas of physics lead to fractional partial differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [1–15] and the references given therein).

The role played by stability inequalities (well posedness) in the study of boundary-value problems for hyperbolic partial differential equations is well known (see, e.g., [16–29]).

In the present paper, finite difference and He's iteration methods for the approximate solutions of the mixed boundary-value problem for the multidimensional fractional hyperbolic equation

are studied. Here is the unit open cube in the -dimensional Euclidean space: with boundary ; and are given smooth functions and .

1.1. Definition

The Caputo fractional derivative of order of a continuous function is defined by

where is the gamma function.

2. The Finite Difference Method

In this section, we consider the first order of accuracy in and the second-orders of accuracy in space variables’ stable difference scheme for the approximate solution of problem (1.1). The stability estimates for the solution of this difference scheme and its first- and second-order difference derivatives are established. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations.

2.1. The Difference Scheme: Stability Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, let us define the grid space

We introduce the Banach space of the grid functions defined on , equipped with the norm

To the differential operator generated by problem (1.1), we assign the difference operator by the formula

acting in the space of grid functions , satisfying the conditions for all . It is known that is a self-adjoint positive definite operator in . With the help of we arrive at the initial boundary value problem

for an infinite system of ordinary fractional differential equations.

In the second step, we replace problem (2.4) by the first order of accuracy difference scheme

Here .

Theorem 2.1. Let and be sufficiently small numbers. Then, the solutions of difference scheme (2.5) satisfy the following stability estimates:

Here and do not depend on , , and .

The proof of Theorem 2.1 is based on the self-adjointness and positive definitness of operator in and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 2.2. For the solutions of the elliptic difference problem

the following coercivity inequality holds [30]:

Finally, applying this difference scheme, the numerical methods are proposed in the following section for solving the one-dimensional fractional hyperbolic partial differential equation. The method is illustrated by numerical examples.

2.2. Numerical Results

For the numerical result, the mixed problem

for solving the one-dimensional fractional hyperbolic partial differential equation is considered. Applying difference scheme (2.5), we obtained

We get the system of equations in the matrix form:

where

Here

So, we have the second-order difference equation with respect to matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method for difference equation with respect to matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:

,   are square matrices, and are column matrices defined by

where

Now, we will give the results of the numerical analysis. First, we give an estimate for the constants and figuring in the stability estimates of Theorem 2.1. We have

The constants and in the case of numerical solution of initial-boundary value problem (2.9) are computed. The constants and are given in Table 1 for , and , respectively.

Table 1: Stability estimates for (2.9).

The values of
0.2096
0.2073
0.2061

The values of
0.2075
0.1223
0.0670

Second, for the accurate comparison of the difference scheme considered, the errors computed by

of the numerical solutions are recorded for higher values of , where represents the exact solution and represents the numerical solution at . The errors and results are shown in Table 2 for and , respectively.

Table 2: Comparison of the errors for the difference scheme.

Method
M = 80
M = 80
M = 60

N = 20
N = 40
N = 60

Comparison of errors () for approximate solutions
0.0071
0.0037
0.0008

Comparison of errors () for approximate solutions
0.1030
0.0521
0.0491

Comparison of errors () for approximate solutions
0.1224
0.0806
0.0882

The figure of the difference scheme solution of (2.9) is given by the Figure 2. The exact solution of (2.9) is given by as follows:

The figure of the exact solution of (2.9) is shown by the Figure 1.

Figure 1: The surface shows the exact solution for (2.9).

Figure 2: Difference scheme solution for (2.9).

3. He's Variational Iteration Method

In the present paper, the mixed boundary value problem for the multidimensional fractional hyperbolic equation (1.1) is considered. The correction functional for (1.1) can be approximately expressed as follows:

where is a general Lagrangian multiplier (see, e.g., [31]) and is considered as a restricted variation as a restricted variation (see, e.g., [32]); that is, , is its initial approximation. Using the above correction functional stationary and noticing that , we obtain

From the above relation for any , we get the Euler-Lagrange equation:

with the following natural boundary conditions:

Therefore, the Lagrange multiplier can be identified as follows:

Substituting the identified Lagrange multiplier into (3.1), the following variational iteration formula can be obtained:

In this case, let an initial approximation . Then approximate solution takes the form .

3.1. Variational Iteration Solution 1

For the numerical result, the mixed problem

for solving the one-dimensional fractional hyperbolic partial differential equation is considered.

According to formula (3.6), the iteration formula for (3.7) is given by

Now we start with an initial approximation

Using the above iteration formula (3.8), we can obtain the other components as

The figure of (3.10) is given by the Figure 3.

Figure 3: Variational iteration method for (3.10).

3.2. Variational Iteration Solution 2

For the numerical result, the mixed problem

for solving the two-dimensional fractional hyperbolic partial differential equation is considered.

According to formula (3.6), the iteration formula for (3.11) is given by

we start with an initial approximation

Using the above iteration formula (3.12), we can obtain the other components as

The exact solution of (3.11) is given by as follows:

The figure of the exact solution of (3.11) is shown by the Figure 5.

The figure of (3.14) is given by the Figure 4, and so on; in the same manner the rest of the components of the iteration formula (3.12) can be obtained using the Maple package.

Figure 4: Variational iteration method for (3.14).

Figure 5: The surface shows the exact solution for (3.11).

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