- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

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

http://dx.doi.org/10.1155/2012/434976

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

^{1}Department of Mathematics, Fatih University, Buyukcekmece 34500, Istanbul, Turkey^{2}Department of Mathematics, ITTU, Ashgabad, Turkmenistan^{3}Department 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.

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.

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.

#### 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.

##### 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.

#### 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, London, UK, 1993. View at Zentralblatt MATH - J.-L. Lavoie, T. J. Osler, and R. Tremblay, “Fractional derivatives and special functions,”
*SIAM Review*, vol. 18, no. 2, pp. 240–268, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. E. Tarasov, “Fractional derivative as fractional power of derivative,”
*International Journal of Mathematics*, vol. 18, no. 3, pp. 281–299, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, vol. 378 of*CISM Courses and Lectures*, pp. 223–276, Springer, Vienna, Austria, 1997. View at Google Scholar - D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in
*Proceedings of the Computational Engineering in System Application 2*, Lille, France, 1996. - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev, F. Dal, and Z. Pinar, “On the numerical solution of fractional hyperbolic partial differential equations,”
*Mathematical Problems in Engineering*, vol. 2009, Article ID 730465, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Dal, “Application of variational iteration method to fractional hyperbolic partial differential equations,”
*Mathematical Problems in Engineering*, vol. 2009, Article ID 824385, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Podlubny and A. M. A. El-Sayed,
*On Two Definitions of Fractional Calculus*, Solvak Academy of Science-Institute of Experimental Phys, 1996. - S. G. Kreĭn,
*Linear Differential Equations in a Banach Space,*, Nauka, Moscow, Russia, 1967. - P. E. Sobolevskii and L. M. Chebotaryeva, “Approximate solution by method of lines of the Cauchy problem for an abstract hyperbolic equations,”
*Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika*, vol. 5, pp. 103–116, 1977 (Russian). View at Google Scholar - A. Ashyralyev, M. Martinez, J. Paster, and S. Piskarev, “Weak maximal regularity for abstract hy- perbolic problems in function spaces,” in
*Proceedings of the of 6th International ISAAC Congress*, p. 90, Ankara, Turkey, 2007. - A. Ashyralyev and N. Aggez, “A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations,”
*Numerical Functional Analysis and Optimization*, vol. 25, no. 5-6, pp. 439–462, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and I. Muradov, “On difference schemes a second order of accuracy for hyperbolic equations,” in
*Modelling Processes of Explotation of Gas Places and Applied Problems of Theoretical Gasohydrodynamics*, pp. 127–138, Ashgabat, Ilim, 1998. View at Google Scholar - A. Ashyralyev and P. E. Sobolevskii,
*New Difference Schemes for Partial Differential Equations*, vol. 148 of*Operator Theory: Advances and Applications*, Birkhäuser, Boston, Mass, USA, 2004. View at Zentralblatt MATH - A. Ashyralyev and Y. Ozdemir, “On nonlocal boundary value problems for hyperbolic-parabolic equations,”
*Taiwanese Journal of Mathematics*, vol. 11, no. 4, pp. 1075–1089, 2007. View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and O. Yildirim, “On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations,”
*Taiwanese Journal of Mathematics*, vol. 14, no. 1, pp. 165–194, 2010. View at Google Scholar · View at Zentralblatt MATH - A. A. Samarskii, I. P. Gavrilyuk, and V. L. Makarov, “Stability and regularization of three-level difference schemes with unbounded operator coefficients in Banach spaces,”
*SIAM Journal on Numerical Analysis*, vol. 39, no. 2, pp. 708–723, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and P. E. Sobolevskii, “Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations,”
*Discrete Dynamics in Nature and Society*, no. 2, pp. 183–213, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and M. E. Koksal, “On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,”
*Numerical Functional Analysis and Optimization*, vol. 26, no. 7-8, pp. 739–772, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and M. E. Koksal, “On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space,”
*Numerical Functional Analysis and Optimization*, vol. 26, no. 7-8, pp. 739–772, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Ashyraliyev, “A note on the stability of the integral-differential equation of the hyperbolic type in a Hilbert space,”
*Numerical Functional Analysis and Optimization*, vol. 29, no. 7-8, pp. 750–769, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev and P. E. Sobolevskii, “A note on the difference schemes for hyperbolic equations,”
*Abstract and Applied Analysis*, vol. 6, no. 2, pp. 63–70, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. E. Sobolevskii,
*Difference Methods for the Approximate Solution of Differential Equations*, Izdatelstvo Voronezhskogo Gosud Universiteta, Voronezh, Russia, 1975. - M. Inokti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in nonlinear mathematical physic,” in
*Variational Method in the Mechanics of Solids*, S. Nemar-Nasser, Ed., Pergamon Press, Oxford, UK, 1978. View at Google Scholar - J. H. He,
*Generalized Variational Principles in Fluids*, Science & Culture Publishing House of China, Hong Kong, 2003.