Abstract and Applied Analysis

Volume 2012 (2012), Article ID 463746, 17 pages

doi:10.1155/2012/463746

Research Article

Stability of Difference Schemes for Fractional Parabolic PDE with the Dirichlet-Neumann Conditions

Zafer Cakir

Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey

Received 25 March 2012; Accepted 3 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 Zafer Cakir. 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 stable difference schemes for the fractional parabolic equation with Dirichlet and Neumann boundary conditions are presented. Stability estimates and almost coercive stability estimates with ln for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes of one-dimensional fractional parabolic partial differential equations.

1. Introduction

Theory and applications, methods of solutions of problems for fractional differential equations have been studied extensively by many researchers [1–18]. In this study, initial-boundary-value problem for the fractional parabolic equation

with Dirichlet and Neumann conditions is considered. Here is the standard Riemann-Liouville’s derivative of order and is the open cube in the -dimensional Euclidean space

with boundary and are given smooth functions, , , and is the normal vector to .

The first and second orders of accuracy stable difference schemes for the numerical solution of problem (1.1) are presented. Stability estimates and almost coercive stability estimates with for the solution of these difference schemes are obtained. The method is illustrated by numerical examples.

2. The First and Second Orders of Accuracy Stable Difference Schemes and 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 Hilbert space of the grid function 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 and for all . Here is the approximation of . 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 a finite system of ordinary fractional differential equations.

In the second step, applying the first order of approximation formula

for

(see [19]) and using the first order of accuracy stable difference scheme for parabolic equations, one can present the first order of accuracy difference scheme with respect to

for the approximate solution of problem (2.4). Here

Moreover, applying the second order of approximation formula

for

and the Crank-Nicholson difference scheme for parabolic equations, one can present the second order of accuracy difference scheme with respect to and to and

for the approximate solution of problem (2.4). Here

Theorem 2.1. Let and be sufficiently small numbers. Then, the solutions of difference scheme (2.7) and (2.11) satisfy the following stability estimate:

Here does not depend on , and , .

Proof. For the solution of difference scheme (2.7), we have the following formulas:

where

The proof of (2.13) for (2.7) is based on (2.14) and estimate

and the triangle inequality.

In the same manner, we can obtain (2.13) for (2.11) using the inequality

Theorem 2.2. Let and be sufficiently small numbers. Then, the solutions of difference scheme (2.7) satisfy the following almost coercive stability estimate:

Here is independent of , , and , .

Proof. The proof of (2.18) for (2.7) is based on (2.14) and estimate (2.16) and the triangle inequality.

Theorem 2.3. Let and be sufficiently small numbers. Then, the solutions of difference scheme (2.11) satisfy the following almost coercive stability estimate:

Here does not depend on , and , .

Proof. The proof of (2.19) for (2.11) is based on (2.14) and estimate (2.17) and the triangle inequality.

Remark 2.4. The method of proofs of Theorems 2.1–2.3 enables us to obtain the estimate of convergence of difference schemes of the first and second orders of accuracy for approximate solutions of the initial-boundary-value problem

for semilinear fractional parabolic partial differential equations.

Note that one has not been able to obtain a sharp estimate for the constants figuring in the stability estimates of Theorems 2.1, 2.2, and 2.3. Therefore, our interest in the present paper is studying the difference schemes (2.7) and (2.11) by numerical experiments. Applying these difference schemes, the numerical methods are proposed in the following section for solving the one-dimensional fractional parabolic partial differential equation. The method is illustrated by numerical experiments.

3. Numerical Applications

For numerical results we consider two examples.

Example 3.1. We consider the initial-boundary-value problem

for the one-dimensional fractional parabolic partial differential equation. The exact solution of problem (3.1) is

First, applying difference scheme (2.7), we obtain

We get the system of equations in the matrix form

where

for and

So, we have the second-order difference equation with respect to matrix coefficients. This type system was developed by Samarskii and Nikolaev [20]. To solve this difference equation we have applied a procedure for difference equation with respect to matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:

where are square matrices and are column matrices defined by

where is the zero matrix and is the zero matrix.

Second, applying difference scheme (2.11), we obtain the second order of accuracy difference scheme in and in

Here is defined same as in (2.9). We get the system of equations in the matrix form

where

for and

For the solution of the matrix equation (3.15), we use the same algorithm as in the first order of accuracy difference scheme, where .

Example 3.2. We consider the initial-boundary-value problem

for the one-dimensional fractional parabolic partial differential equation. The exact solution of problem (3.23) is

First, applying difference scheme (2.7), we obtain

We get the system of equations in the matrix form

where matrices are defined same as in (3.5), (3.6), (3.7), (3.8), (3.9), respectively.

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

where are square matrices and are column matrices defined by

where is the identity matrix and is the zero matrix.

Second, applying the formulas

and applying difference scheme (2.11), we obtain the second order of accuracy difference scheme in and in

Here is defined similar to (2.9). We get the system of equations in the matrix form

where matrices are defined same as in (3.16), (3.17), (3.18), (3.19), (3.20), respectively.

For the solution of the matrix equation (3.31), we use the same algorithm as in the first order of accuracy difference scheme, where

for and

3.1. Error Analysis

Finally, we give the results of the numerical analysis. The error is computed by the following formula:

where represents the exact solution and represents the numerical solutions of these difference schemes at . The numerical solutions are recorded for different values of and . Tables 1 and 2 are constructed for , and 100, respectively.

Table 1: Error analysis of first and second order of accuracy difference schemes for Example 3.1.

Method

1st order of accuracy
0.2553
0.1256
0.0622

2nd order of accuracy
0.0062

Table 2: Error analysis of first and second order of accuracy difference schemes for Example 3.2.

Method

1st order of accuracy
0.1653
0.0807
0.0399

2nd order of accuracy
0.0025

The results in Tables 1 and 2 show that, by using the Crank-Nicholson difference scheme, more accurate approximate results can be obtained rather than the first order of accuracy difference scheme.

Acknowledgment

The author is grateful to Professor Allaberen Ashyralyev (Fatih University, Turkey) for his comments and suggestions to improve the quality of the paper.

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