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Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 696179, 15 pages

http://dx.doi.org/10.1155/2012/696179

## On the Numerical Solution of Fractional Parabolic Partial Differential Equations with the Dirichlet Condition

^{1}Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey^{2}Department of Mathematics, ITTU, Ashgabat, Turkmenistan, Turkey^{3}Department of Mathematical Engineering, Gumushane University, 29100 Gumushane, Turkey

Received 13 April 2012; Revised 14 June 2012; Accepted 12 July 2012

Academic Editor: Seenith Sivasundaram

Copyright © 2012 Allaberen Ashyralyev and 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 first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are presented. Stability and almost coercive stability estimates for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations.

#### 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., [1–28] and the references therein).

The role played by stability inequalities (well posedness) in the study of boundary value problems for parabolic partial differential equations is well known (see, e.g., [29–34]). In the present paper, the mixed boundary value problem for the fractional parabolic equation 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 .

The first and second order of accuracy in and second orders of accuracy in space variables difference schemes for the approximate solution of problem (1.1) are presented. The stability and almost coercive stability estimates for the solution of these difference schemes are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations.

#### 2. 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 . It is known that is a self-adjoint positive definite operator in . Here, 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 [35]) 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 (2.5). Here

Moreover, applying the second order of approximation formula for (see [27]) 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 (2.5). Here and in the future

Theorem 2.1. *Let and be sufficiently small positive numbers. Then, the solutions of difference scheme (2.8) and (2.12) satisfy the following stability estimate:
**
where does not depend on and , . *

*Proof. *We consider the difference scheme (2.8). We have that
where
Using formula (2.15), we can write
First, we will prove that
Using formula (2.17), we get
Using formulas (2.16) and (2.19), we obtain
Now, let us estimate . Applying the triangle inequality and the estimate [34]
we get
for any . Then, using the difference analogy of integral inequality, we get (2.18).

Second, applying formula (2.17), estimates (2.18) and (2.21), we obtain
Estimate (2.14) for the solution of (2.8) is proved. The proof of estimate (2.14) for the solution of (2.12) follows the scheme of the proof of estimate (2.14) for the solution of (2.8) and rely on the estimate
Here,
Theorem 2.1 is proved.

Theorem 2.2. *Let and be sufficiently small positive numbers. Then, the solutions of difference scheme (2.8) satisfy the following almost coercive stability estimate:
**
where is independent of and , . *

*Proof. *We will prove the estimate
Using formula (2.19) and estimate (2.21), we obtain
and estimate (2.18), the triangle inequality and equation (2.8), we get (2.27). From that it follows:
Then, the proof of estimate (2.26) is based on estimates (2.27), (2.29), and the following theorem on coercivity inequality for the solution of the elliptic difference problem in .

Theorem 2.3. *For the solutions of the elliptic difference problem**the following coercivity inequality holds (see [14, 36])**where ** does not depend on ** and **. *

Theorem 2.2 is proved.

Theorem 2.4. *Let and be sufficiently small positive numbers. Then, the solutions of difference scheme (2.12) satisfy the following almost coercive stability estimate:
**
where does not depend on and , . *

The proof of Theorem 2.4 follows the proof of Theorem 2.2 and on the estimate (2.24) and the self-adjointness and positive definiteness of operator in and Theorem 2.3.

*Remark 2.5. *The stability estimates of Theorems 2.1, 2.2, and 2.4 are satisfied in the case of operator
with Dirichlet condition in . In this case, is not self-adjoint operator in . Nevertheless, and is a self-adjoint positive definite operator in and is bounded in . The proof of this statement is based on the abstract results of [14] and difference analogy of integral inequality.

The method of proofs of Theorems 2.1, 2.2, and 2.4 enables us to obtain the estimate of convergence of difference schemes of the first and second order 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 constant figuring in the stability estimates of Theorems 2.1, 2.2, and 2.4. Therefore, our interest in the present paper is studying the difference schemes (2.8) and (2.12) 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 Results

For the numerical result, the mixed problem for the one-dimensional fractional parabolic partial differential equation is considered. The exact solution of problem (3.1) is First, applying difference scheme (2.8), we obtain We can rewrite it in the system of equations with matrix coefficients Here and in the future is the zero matrix and , for and

So, we have the second-order difference equation with respect to matrix coefficients. This type system was developed by Samarskii and Nikolaev [37]. 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 (2.12), we obtain where for any , . We get the system of equations in the matrix form where , for and So, we have again the second-order difference equation with respect to matrix coefficients. Therefore, applying the same procedure of modified Gauss elimination method (3.7) and (3.8) difference equation (3.12).

Finally, we give the results of the numerical analysis. The numerical solutions are recorded for different values of and and represents the numerical solutions of these difference schemes at . The error is computed by the following formula: Table 1 is constructed for , and , respectively.

Thus, by using the Crank-Nicholson difference scheme, the accuracy of solution increases faster than the first order of accuracy difference scheme.

#### 4. Conclusion

In this study, the first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are investigated. We have obtained stability and almost coercive stability estimates for the solution of these difference schemes. The theoretical statements for the solution of these difference schemes for one-dimensional parabolic equations are supported by numerical example in computer. We showed that the second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme.

#### Acknowledgment

The authors are grateful to Professor Pavel E. Sobolevskii for his comments and suggestions to improve the quality of the paper.

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