Discrete Dynamics in Nature and Society

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

doi:10.1155/2012/696179

Research Article

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

Allaberen Ashyralyev1,2 and Zafer Cakir³

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

²Department of Mathematics, ITTU, Ashgabat, Turkmenistan, Turkey

³Department 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.

Table 1: Error analysis.

Method

Difference scheme (2.8)
0.0040
0.0020
0.0010

Difference scheme (2.12)
0.0006726
0.0001678
0.00004187

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.

References

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.

A. E. M. El-Mesiry, A. M. A. El-Sayed, and H. A. A. El-Saka, “Numerical methods for multi-term fractional (arbitrary) orders differential equations,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 683–699, 2005.

K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.

A. Pedas and E. Tamme, “On the convergence of spline collocation methods for solving fractional differential equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 12, pp. 3502–3514, 2011.

K. Diethelm and N. J. Ford, “Multi-order fractional differential equations and their numerical solution,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 621–640, 2004.

A. M. A. El-Sayed and F. M. Gaafar, “Fractional-order differential equations with memory and fractional-order relaxation-oscillation model,” Pure Mathematics and Applications, vol. 12, no. 3, pp. 296–310, 2001.

A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “Numerical solution for multi-term fractional (arbitrary) orders differential equations,” Computational & Applied Mathematics, vol. 23, no. 1, pp. 33–54, 2004.

D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Engineering in System Application, vol. 2, Lille, France, 1996.

A. B. Basset, “On the descent of a sphere in a viscous liquid,” Quarterly Journal of Mathematics, vol. 42, pp. 369–381, 1910.

F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., pp. 291–348, Springer, New York, NY, USA, 1997.

A. Ashyralyev, F. Dal, and Z. Pınar, “A note on the fractional hyperbolic differential and difference equations,” Applied Mathematics and Computation, vol. 217, no. 9, pp. 4654–4664, 2011.

A. Ashyralyev, “Well-posedness of the Basset problem in spaces of smooth functions,” Applied Mathematics Letters, vol. 24, no. 7, pp. 1176–1180, 2011.

M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,” Abstract and Applied Analysis, vol. 2011, Article ID 161246, 25 pages, 2011.

C. Yuan, “Two positive solutions for -type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 2, pp. 930–942, 2012.

M. De la Sen, R. P. Agarwal, A. Ibeas, and S. Alonso-Quesada, “On the existence of equilibrium points, boundedness, oscillating behavior and positivity of a SVEIRS epidemic model under constant and impulsive vaccination,” Advances in Difference Equations, vol. 2011, Article ID 748608, 32 pages, 2011.

M. De la Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2011, Article ID 867932, 19 pages, 2011.

C. Yuan, “Multiple positive solutions for semipositone -type boundary value problems of nonlinear fractional differential equations,” Analysis and Applications, vol. 9, no. 1, pp. 97–112, 2011.

C. Yuan, “Multiple positive solutions for -type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 13, pp. 1–12, 2011.

R. P. Agarwal, M. Belmekki, and M. Benchohra, “A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative,” Advances in Difference Equations, vol. 2009, Article ID 981728, 47 pages, 2009.

R. P. Agarwal, B. de Andrade, and C. Cuevas, “On type of periodicity and ergodicity to a class of fractional order differential equations,” Advances in Difference Equations, vol. 2010, Article ID 179750, 25 pages, 2010.

R. P. Agarwal, B. de Andrade, and C. Cuevas, “Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 3532–3554, 2010.

A. S. Berdyshev, A. Cabada, and E. T. Karimov, “On a non-local boundary problem for a parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator,” Nonlinear Analysis, vol. 75, no. 6, pp. 3268–3273, 2011.

A. Ashyralyev and D. Amanov, “Initial-boundary value problem for fractional partial differential equations of higher order,” Abstract and Applied Analysis, vol. 2012, Article ID 973102, 15 pages, 2012.

A. Ashyralyev and Y. A. Sharifov, “Existence and uniqueness of solutions of the system of nonlinear fractional differential equations with nonlocal and integral boundary conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 594802, 12 pages, 2012.

Z. Cakir, “Stability of difference schemes for fractional parabolic PDE with the Dirichlet-Neumann conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 463746, 17 pages, 2012.

A. Yakar and M. E. Koksal, “Existence results for solutions of nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 267108, 9 pages, 2012.

Ph. Clement and S. Guerre-Delabrire, “On the regularity of abstract Cauchy problems and bound-ary value problems,” Matematica e Applicazioni, vol. 9, no. 4, pp. 245–266, 1999.

R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in Banach-valued weighted space,” Boundary Value Problems, no. 1, pp. 9–42, 2005.

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, Boston, 1995.

P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975.

A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäauser, Basel, Switzerland, 1994.

P. E. Sobolevskii, “Some properties of the solutions of differential equations in fractional spaces,” Trudy Naucno-Issledovatel'skogi Instituta Matematiki VGU, vol. 14, pp. 68–74, 1975.

A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232–236, 2009.

P. E. Sobolevskiĭ and M. F. Tiunčik, “The difference method of approximate solution for elliptic equations,” no. 4, pp. 117–127, 1970 (Russian).

A. A. Samarskii and E. S. Nikolaev, Numerical Methods for Grid Equations. Vol. II, Birkhäuser, Basel, Switzerland, 1989.