Journal Menu
- About this Journal
- Abstracting and Indexing
- 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
doi:10.1155/2012/965367
Research Article
Numerical Simulation of Fractional Fornberg-Whitham Equation by Differential Transformation Method
1Department of Mathematics Engineering, Gümüşhane University, 29100 Gümüşhane, Turkey
2Department of Mathematics, Ege University, 35000 İzmir, Turkey
3Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
4HITEC 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.
Linked 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.
- 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. 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.
- 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. 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.
- 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 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 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.
- 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