- 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

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 965367, 8 pages

http://dx.doi.org/10.1155/2012/965367

## Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method

^{1}Department of Mathematics Engineering, Gümüşhane University, 29100 Gümüşhane, Turkey^{2}Department of Mathematics, Ege University, 35000 İzmir, Turkey^{3}Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA^{4}HITEC University, Taxila, Wah Cantt, Pakistan

Received 4 August 2011; Accepted 28 September 2011

Academic Editor: Shaher M. Momani

Copyright © 2012 Mehmet Merdan et al. 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

An approximate analytical solution of fractional Fornberg-Whitham equation was obtained with the help of the two-dimensional differential transformation method (DTM). It is indicated that the solutions obtained by the two-dimensional DTM are reliable and present an effective method for strongly nonlinear partial equations. Exact solutions can also be obtained from the known forms of the series solutions.

#### 1. Introduction

A homogeneous nonlinear fractional Fornberg-Whitham equation [1] is considered as in the following form: with boundary conditions and with initial conditions where is the fluid velocity, is constant and lies in the interval , is the time and is the spatial coordinate.

Subscripts denote the partial differentiation unless stated otherwise. Fornberg and Whitham obtained a peaked solution of the form , where is an arbitrary constant. In recent years, considerable interest in fractional calculus used in many fields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, accustics, material science, and signal processing can be successfully modelled by linear or nonlinear fractional order differential equations [2–8].

See fractional diffusion equation with absorbent term and external force by Das and Gupta [9], fractional convection-diffusion equation with nonlinear source term by Momani and Yıldırım [10], space-time fractional advection-dispersion equation by Yıldırım and Koçak [11], fractional Zakharov-Kuznetsov equations by Yıldırım and Gülkanat [12], boundary value problems by He [13], integro-differential equation by El-Shahed [14], non-Newtonian flow by Siddiqui et al. [15], fractional PDEs in fluid mechanics by Yıldırım [16], fractional Schrödinger equation [17, 18] and nonlinear fractional predator-prey model [19] by HPM, linear PDEs of fractional order by He [20], Momani, and Odibat [21], and so forth. In 2009, Tian and Gao [22] studied the proof of the existence of the attractor for the one-dimensional viscous Fornberg-Whitham equation. Abidi and Omrani [23] have solved the Fornberg-Whitham equation by the homotopy analysis method. Recently, Gupta and Singh [24] have used homotopy perturbation method to numerical solution of fractional Fornberg-Whitham Equation.

The goal of this paper is to extend the two-dimensional differential transform method to solve fractional Fornberg-Whitham equation.

This paper is organized as follows.

In Section 2, we are giving definitions related to the fractional calculus theory briefly. To show in efficiency of this method, we give the implementation of the DTM for the Fornberg-Whitham equation and numerical results in Sections 3 and 4. The conclusions are then given in the final Section 5.

#### 2. Basic Definitions

Here are some basic definitions and properties of the fractional calculus theory which can be found in [5, 6, 25, 26].

*Definition 2.1. *A real function , in the space , if there exists a real number , such that , where and it is said to be in the space if , .

*Definition 2.2. *The left-sided Riemann-Liouville fractional integral operator of order , of a function is defined as
The properties of the operator can be found in Jang et al. [25].

*Definition 2.3. *The fractional derivative of in the Caputo [6] sense is defined as
The unknown function is assumed to be a casual function of fractional derivatives (i.e., vanishing for ) taken in Caputo sense as follows.

*Definition 2.4. *For as the smallest integer that exceeds , the Caputo time-fractional derivative operator of order is defined as

#### 3. Two-Dimensional Differential Transformation Method

DTM is an analytic method based on the Taylor series expansion which constructs an analytical solution in the form of a polynomial. The traditional high order Taylor series method requires symbolic computation. However, the DTM obtains a polynomial series solution by means of an iterative procedure. The method is well addressed by Odibat and Momani [26]. The proposed method is based on the combination of the classical two-dimensional DTM and generalized Taylor’s Table 1 formula. Consider a function of two variables and suppose that it can be represented as a product of two single-variable functions, that is, . The basic definitions and fundamental operations of the two-dimensional differential transform of the function are expressed as follows [25–38]. Two-dimensional differential transform of can be represented as: where , is called the spectrum of . The generalized two-dimensional differential transform of the function is given by where .

In case of , and , the generalized two-dimensional differential transform (3.2) reduces to the classical two-dimensional differential transform.

From the above definitions, it can be found that the concept of two-dimensional differential transform is derived from two-dimensional differential transform which is obtained from two-dimensional Taylor series expansion.

#### 4. The DTM Applied to Fractional Fornberg-Whitham Equation

In this section, we will research the solution of fractional Fornberg-Whitham equation, which has been widely examined in the literature. We described the implementation of the DTM for the fractional Fornberg-Whitham equation in detail. To solve (1.1)–(1.3), according to DTM, (1.2)-(1.3) with initial condition become with boundary conditions Applying the differential transform of (1.1), (4.1), and (4.2), then Substituting (4.3) into (4.4), we obtain the closed form solution as As , this series has the closed form , which is an exact solution of the classical gas dynamics equation.

The graphs of exact and DTM solutions belonging to examples examined above are shown in Figure 1. It can be deduced that DTM solution corresponds to the exact solutions.

Both the exact results and the approximate solutions obtained for the DTM approximations are plotted in Figure 1. There are no visible differences in the two solutions of each pair of diagrams.

#### 5. Conclusions

In this paper, the applicability of the fractional differential transformation method to the solution of fractional Fornberg-Whitham equation with a number of initial and boundary values has been proved. DTM can be applied to many complicated linear and strongly nonlinear partial differential equations and does not require linearization, discretization, or perturbation. The obtained results indicate that this method is powerful and meaningful for solving the nonlinear fractional Fornberg-Whitham type differential equations.

#### References

- J. Zhou and L. Tian, “A type of bounded traveling wave solutions for the Fornberg-Whitham equation,”
*Journal of Mathematical Analysis and Applications*, vol. 346, no. 1, pp. 255–261, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. B. Oldham and J. Spanier,
*The Fractional Calculus*, Academic Press, London, UK, 1974. - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, San Diego, Calif, USA, 1999. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. - I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, San Diego, Calif, USA, 1999. - M. Caputo, “Linear models of dissipation whose Q is almost frequency independent—part II,”
*Geophysical Journal of the Royal Astronomical Society*, vol. 13, p. 529, 1967. View at Google Scholar - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. - S. Das and P. K. Gupta, “An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy perturbation method,”
*Zeitschrift fur Naturforschung*, vol. 65, no. 3, pp. 182–190, 2010. View at Google Scholar - S. Momani and A. Yıldırım, “Analytical approximate solutions of the fractional convection-diffusion equation with nonlinear source term by He's homotopy perturbation method,”
*International Journal of Computer Mathematics*, vol. 87, no. 5, pp. 1057–1065, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Yıldırım and H. Koçak, “Homotopy perturbation method for solving the space-time fractional advection-dispersion equation,”
*Advances in Water Resources*, vol. 32, no. 12, pp. 1711–1716, 2009. View at Publisher · View at Google Scholar - A. Yıldırım and Y. Gülkanat, “Analytical approach to fractional Zakharov-Kuznetsov equations by He's homotopy perturbation method,”
*Communications in Theoretical Physics*, vol. 53, no. 6, pp. 1005–1010, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-H. He, “Homotopy perturbation method for solving boundary value problems,”
*Physics Letters A*, vol. 350, no. 1-2, pp. 87–88, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. El-Shahed, “Application of He's homotopy perturbation method to Volterra's integro-differential equation,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 2, pp. 163–168, 2005. View at Google Scholar - A. M. Siddiqui, R. Mahmood, and Q. K. Ghori, “Thin film flow of a third grade fluid on a moving belt by he's homotopy perturbation method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 7, no. 1, pp. 7–14, 2006. View at Google Scholar - A. Yıldırım, “Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method,”
*International Journal of Numerical Methods for Heat & Fluid Flow*, vol. 20, no. 2, pp. 186–200, 2010. View at Publisher · View at Google Scholar - S. Das, P. K. Gupta, and S. Barat, “A note on fractional Schrödinger equation,”
*Nonlinear Science Letters A*, vol. 1, no. 1, pp. 91–94, 2010. View at Google Scholar - A. Yıldırım, “An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 4, pp. 445–450, 2009. View at Google Scholar · View at Scopus - S. Das, P. K. Gupta, and Rajeev, “A fractional predator-prey model and its solution,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 7, pp. 873–876, 2009. View at Google Scholar · View at Scopus - J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 167, no. 1-2, pp. 57–68, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 910–919, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Tian and Y. Gao, “The global attractor of the viscous Fornberg-Whitham equation,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 11, pp. 5176–5186, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Abidi and K. Omrani, “The homotopy analysis method for solving the Fornberg-Whitham equation and comparison with Adomian's decomposition method,”
*Computers & Mathematics with Applications*, vol. 59, no. 8, pp. 2743–2750, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. K. Gupta and M. Singh, “Homotopy perturbation method for fractional Fornberg-Whitham equation,”
*Computers & Mathematics with Applications*, vol. 61, no. 2, pp. 250–254, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M.-J. Jang, C.-L. Chen, and Y.-C. Liu, “Two-dimensional differential transform for partial differential equations,”
*Applied Mathematics and Computation*, vol. 121, no. 2-3, pp. 261–270, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. M. Odibat and S. Momani, “Approximate solutions for boundary value problems of time-fractional wave equation,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 767–774, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transform method,”
*Chaos, Solitons and Fractals*, vol. 34, no. 5, pp. 1473–1481, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Adomian,
*Solving Frontier Problems of Physics: The Decomposition Method*, vol. 60, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. - D. J. Evans and H. Bulut, “A new approach to the gas dynamics equation: an application of the decomposition method,”
*International Journal of Computer Mathematics*, vol. 79, no. 7, pp. 817–822, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Kangalgil and F. Ayaz, “Solitary wave solutions for the KdV and mKdV equations by differential transform method,”
*Chaos, Solitons and Fractals*, vol. 41, no. 1, pp. 464–472, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Y. Luchko and R. Groreflo, “The initial value problem for some fractional differential equations with the Caputo derivative,” preprint series A08–98, Fachbreich Mathematik und Informatik, Freic Universitat, Berlin, Germany, 1998.
- S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,”
*Physics Letters A*, vol. 370, no. 5-6, pp. 379–387, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. S. V. Ravi Kanth and K. Aruna, “Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations,”
*Chaos, Solitons and Fractals*, vol. 41, no. 5, pp. 2277–2281, 2009. View at Publisher · View at Google Scholar - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Springer, New York, NY, USA, 2003. - J. K. Zhou,
*Differential Transform and Its Applications for Electrical Circuits*, Huazhong University Press, Wuhan, China, 1986. - M. Merdan and A. Gökdoğan, “Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method,”
*Mathematical & Computational Applications*, vol. 16, no. 3, pp. 761–772, 2011. View at Google Scholar - A. Kurnaz and G. Oturanç, “The differential transform approximation for the system of ordinary differential equations,”
*International Journal of Computer Mathematics*, vol. 82, no. 6, pp. 709–719, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet