Research Article  Open Access
Stability of Difference Schemes for Fractional Parabolic PDE with the DirichletNeumann Conditions
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 onedimensional 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 [1â€“18]. In this study, initialboundaryvalue problem for the fractional parabolic equation with Dirichlet and Neumann conditions is considered. Here is the standard RiemannLiouvilleâ€™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 selfadjoint positive definite operator in . With the help of we arrive at the initialboundaryvalue 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 CrankNicholson 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 2.1â€“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 initialboundaryvalue 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 onedimensional 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 initialboundaryvalue problem
for the onedimensional 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 secondorder 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 initialboundaryvalue problem
for the onedimensional 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 secondorder 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.


The results in Tables 1 and 2 show that, by using the CrankNicholson 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.
References
 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at: Zentralblatt MATH
 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at: Zentralblatt MATH
 A. A. Kilbas, H. M. Sristava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, 2006.
 J.L. Lavoie, T. J. Osler, and R. Tremblay, â€śFractional derivatives and special functions,â€ť SIAM Review, vol. 18, no. 2, pp. 240â€“268, 1976. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 V. E. Tarasov, â€śFractional derivative as fractional power of derivative,â€ť International Journal of Mathematics, vol. 18, no. 3, pp. 281â€“299, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 R. Gorenflo and F. Mainardi, â€śFractional calculus: integral and differential equations of fractional order,â€ť in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, pp. 223â€“276, Springer, Vienna, Austria, 1997. View at: Google Scholar
 D. Matignon, â€śStability results for fractional differential equations with applications to control processing,â€ť in Computational Engineeringin System Application 2, Lille, France, 1996. View at: