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Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method
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.
A homogeneous nonlinear fractional Fornberg-Whitham equation  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 , fractional convection-diffusion equation with nonlinear source term by Momani and Yıldırım , space-time fractional advection-dispersion equation by Yıldırım and Koçak , fractional Zakharov-Kuznetsov equations by Yıldırım and Gülkanat , boundary value problems by He , integro-differential equation by El-Shahed , non-Newtonian flow by Siddiqui et al. , fractional PDEs in fluid mechanics by Yıldırım , fractional Schrödinger equation [17, 18] and nonlinear fractional predator-prey model  by HPM, linear PDEs of fractional order by He , Momani, and Odibat , and so forth. In 2009, Tian and Gao  studied the proof of the existence of the attractor for the one-dimensional viscous Fornberg-Whitham equation. Abidi and Omrani  have solved the Fornberg-Whitham equation by the homotopy analysis method. Recently, Gupta and Singh  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
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. .
Definition 2.3. The fractional derivative of in the Caputo  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 . 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.
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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
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