Communications in Numerical Analysis
Volume 2015, No. 2 (2015), Pages 115-124
Article ID cna-00239, 10 Pages
Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform
Yasser. S. Hamed1,3 *, Mohamed S. Mohamed2,3, E. R. El-Zahar4,5
1Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt
2Department of Mathematics, Faculty of Science, Al Azhar University, Cairo 11884, Egypt
3Department of Mathematics and Statistics, Faculty of Science, Taif University,21974, Kingdom of Saudi Arabia
4Department of Mathematics, College of Sciences and Humanities, Salman Bin Abdulaziz University, P.O. Box 83, Alkharj 11942, KSA
5Department of Basic Engineering Science, Faculty of Engineering, Shebin El-Kom, Menofia University, Egypt
* Corresponding author. Email address: email@example.com
Received: 31 March 2015; Revised: 28 June 2015; Accepted: 07 July 2015
Copyright © 2015 Yasser. S. Hamed, Mohamed S. Mohamed and E. R. El-Zahar. 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.
In this article, fractional complex transform with optimal homotopy analysis method (OHAM) is used to obtain numerical and analytical solutions for the nonlinear time-space fractional Fornberg-Whitham. Fractional complex transform is proposed to convert time-space fractional Fornberg-Whitham equation to the nonlinear ordinary differential equations and then applied OHAM to the new obtained equations. This optimal approach has general meaning and can be used to get fast convergent series solutions of the different type of nonlinear fractional differential equation. The results reveal that the method is very effective, simple and the OHAM contains a certain auxiliary parameter $h$ which provides us with a simple way to adjust and control the convergence region of convergence of the series solution.
Keywords: Homotopy analysis method; Optimal value; Fractional complex transform; Time-space fractional Fornberg-Whitham equation.
K. B. Oldham, J. Spanier, The fractional calculus, Academic press, New York (1974).
K. S. Miller, B. Ross, An introduction to the fractional and fractional differential equations, John Wiley and Sons, New York, (1993).
Y. Luchko, R. Gorenflo, The initial-value problem for some fractional differential equations with Caputo derivative, Preprint Series A08-98, Fachbereich Mathematik and Informatic, Freie Universitat, Berlin, (1998).
I. Podlubny, Fractional differential equations, Academic press, New York, (1999).
V. Daftardar-Gejji, S. Bhalekar, Solving multi-term linear and non-linear diffusion wave equations of fractional order by Adomian decomposition method, Applied Mathematics and Computation, 202 (2008) 113-120.
V. Daftardar-Gejji, H. Jafari, Solving a multi-order fractional differential equation using Adomian decomposition method, Applied Mathematics and Computation, 189 (2007) 541-548.
Mohamed A. E. Herzallah, Khaled A. Gepreel, Approximate solution to the time-space fractional cubic nonlinear Schrodinger equation, Appl. Math. Modeling, 36 (11) (2012) 5678-5685.
N. H. Sweilam, M. M. Khader, R.F. Al-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A, 371 (2007) 26-33.
A. Golbabai, K. Sayevand, Fractional calculus- a new approach to the analysis of generalized fourth-order diffusion-wave equations, Comput. Math. Appl, 61 (2011) 2227-2231.
A. Golbabai, k. Sayevand, The homotopy perturbation method for multi-order time fractional differential equations, Nonlinear Sci. Lett. A, (2010) 147-154.
K. Gepreel, The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations, Appl. Math. Letters, 4 (2011) 1428-1434.
WH Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM journal on numerical analysis, 47 (1) (2008) 204-226.
WH Deng, Finite difference method and their physical constraints for the fractional Klein-Kramers equation, Numerical methods for the partial differential equations, 27 (6) (2011) 1561-1583.
SJ Liao, The proposed homotopy analysis technique for the solution of nonlinear problem, Ph.D thesis, Shanghai Jiao Tong University; (1992).
SJ Liao, An approximate solution technique which does not depend upon small parameters: a special example, Int. J. Nonlinear Mech, 30 (1995) 371-380.
K. A. Gepreel, M. S. Mohamed, S. M. Abo-Dahab, Optimal Homotopy Analysis method for nonlinear partial fractional differential Fisher's equation, Journal of Computational and Theoretical Nanoscience, 12 (6) (2015) 965-970.
M. S. Mohamed, Analytical approximate solutions for the nonlinear differential-difference equations arising in nanotechnology, Journal of Computational and Theoretical Nanoscience, 12 (6) (2015) 1040-1044.
K. A. Gepreel, M. S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation, Chinese physics B, 22 (1) (2013) 010201-6.
S. M. Abo-Dahab, M. S. Mohamed, T. A. Nofal, Homotopy Analysis Method for propagation of harmonic waves in nonlinear generalized magneto-thermoelasticity with two relaxation times under influence of rotation, Journal of in Abstract and Applied Analysis, Article ID 614874, (2013) 14 pages.
K. M. Hemida, Khaled A. Gepreel, Mohamed S. Mohamed, Analytical approximate solution to the time-space nonlinear partial fractional differential equations, International Journal of Pure and Applied Mathematics, 78 (2) (2012) 233-244.
M. S. Mohamed, F. AL-Malki, R. Talib, An analytic algorithm for the time-space-fractional Newell-Whitehead equation, International Review Physics, 6 (4) (2012) 337-343.
H. A. Ghany, Mohamed S. Mohamed, White noise functional solutions for the wick-type stochastic fractional Kdv-Burgers-Kuramoto Equations, Journal of the Chinese Journal of Physics, 50 (4) (2012) 52-66.
B. Ghazanfari, A. G. Ghazanfari, Solving system of fractional differential equations by fractional complex transform method, Asian Journal of Applied Sciences, 5 (6) (2012) 438-444.
Z. Li, J. H. He, Application of the fractional complex transform to fractional differential equations, Nonlinear Sci.Lett. A, 2 (3) (2011) 121-126.
Z. Li, J. H. He, Fractional Complex transform for fractional differential equations, Mathematical and Computational Applications, 15 (5) (2010) 970-973.
Z. Li, J. H. He, An Extended Fractional Complex Transform, Journal of Non-linear Science and Numerical Simulation, 11 (2010) 335-337.
MG. Saker, F. Erdogan, A. Yildirim, Variational iteration method for the time fractional Fornberg-Whitham equation, Comput. Math. Appl. 63 (9) (2012) 1382-1388.
S. Jagdev, K. Devendra, K. Sunil, New treatment of fractional Fornberg-Whitham equation via Laplace transform, Ain Shams Engineering Journal, 4 (2013) 557-562.
M. S. Mohamed, Analytical treatment of Abel integral equations by optimal homotopy analysis transform method, Journal of Information and Computing Science, 10 (1) (2015) 19-28.
M. S. Mohamed, Muteb R. Alharthi, Refah A. Alotabi, Analytic approximate solution for some integral equations by optimal homotopy analysis transform method, Communications in Numerical Analysis, 2015 (1) (2015) 51-61.
Khaled A. Gepreel, Mohamed S. Mohamed, An optimal homotopy analysis method nonlinear fractional differential equation, Journal of Advanced Research Dynamical and Control Systems, 6 (2014) 1-10.
H. N. Hassan, M. S. Semary, Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method, Communications in Numerical Analysis, 2013 (2013) 1-14.