Fractional order Hirota-Satsuma coupled KdV equation by Homotopy perturbation transforms method

	International Journal of Mathematics Trends and Technology (IJMTT)
	© 2016 by IJMTT Journal
	Volume-33 Number-3
	Year of Publication : 2016
	Authors : R.N Prajapati, Rakesh Mohan ,Pankaj Kumar

R.N Prajapati, Rakesh Mohan ,Pankaj Kumar "Fractional order Hirota-Satsuma coupled KdV equation by Homotopy perturbation transforms method", International Journal of Mathematics Trends and Technology (IJMTT). V33(3):148-155 May 2016. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

Abstract
In this paper, we have used homotopy perturbation method and Laplace transformation to determine approximate solutions which converge to exact solution of generalized Hirota−Satsuma coupled KdV equation. The nonlinear terms handled by the use of He’s polynomial.

References
[1] J.H. Homotopy Perturbation Technique, Computer Methods in Applied Mechanics and Engineering, 179(1999) 257-262, doi:10.1016/S0045-925(99)00019-3.
[2] J.H. He A Coupling Method of a Homotopy Technique and Perturbation Technique for Non-Linear problems, International Journal of Nonlinear Mechanics, 35(2000) 37-43, doi: 10.1016/S0020-7462(99)00095-7.
[3] J.H.He, Homotopy Perturbation Method: A New Non-Linear Analytical Technique, Applied Mathematics and Computation, 135(2003)73-79,doi:10.1016/S0096- 3003(01)00312-5.
[4] K. Batiha, Approximate Analytical Solutions for Time- Dependent Emden-Fowler-Type Equations by Variational Iteration Method, American Journal of Applied Sciences, 4(2007) 439-443, doi:10.3944/ajassp.2007.439.443.
[5] J. Biazar and H. Ghazvini, He’s Variational Iteration Method for Solving Linear and Non-Linear Systems of Ordinary Differential Equations, Applied Mathematics and Computation, 191(2007) 297-297, doi:10.1016/j.amc.2007.02.153.
[6] Ahmet Yıldırım, Solving Differential-Difference Equation, Mathematical Problems in Engineering, Article ID 869614,7 pageshttp://dx.doi.org/10.1155/2008/869614
[7] A. M. Wazwaz, Analytic Treatment for Varcient Fourth-Order Parabolic Partial Differential Equations, Applied Mathematics and Computation, 123(2001) 219-227,doi:10.1016/S0096-3003(00)00070-9.
[8] J. H. He, An Elementary Introduction Perturbation Method, Computers and Mathematics with Applications, 57(2009) 410-412, doi:10.1016/j.camwa.2009.06.003.
[9] Z. M. Odibat, A New Modification of the Homotopy Perturbation Method for Linear and Nonlinear Operators, Applied Mathematics and Computation, 199(2007),746-753, doi:10.1016/j.amc.2006.11.199.
[10] E. Yusufoglu, Improved Homotopy Perturbation Method for Solving Fredholm type Integro-Differential Equations, Chaos Solitons and Fractals, 41(2009) 29-37. doi:10.1016/j.chaos.2007.11.005.
[11] M. A. Jafari and A. Aminataei, “Improved Homotopy Perturbation Method,” International Mathematical Forum, 5(2010) 1567- 1579.
[12] M. Ghasemi, M. T. Kajani and E. Babolian, Application of He’S Homotopy Perturbation Method to Nonlinear In- tegro-Differential Equations, Applied Mathematics and Computation, 199(2007) 539-549, doi:10.1016/j.amc.2006.10.016.
[13] H. Jafari, M. Zabihi and M. Saidy, Application of Homotopy Perturbation Method for Solving Gas Dynamics Equation, Applied Mathematical Sciences, 2(2009) 2393-2396,
[14] Y. Yu and H. Li, Application of the Multistage Homotopy Perturbation Method to Solve a Class of Hypercha- otic Systems, Ghaos Solitons and Fratals, 42(2009) 2330-2337,doi:10.1016/j.chaos.2009.064
[15] P. Roul, Application of Homotopy Perturbation Biological Population Model, Application and Mathematics: An International Journal, 5(2010)13969-1379.
[16] M. Meran, Homotopy Perturbation Method for Solving a Model for HIV Infection of CD4+ Cell, Istanbul Ticaret Universitesi Fen Bilimleri Dergisi, 12(2007) 39-52.
[17] Y. X. Wang, H. Y. Si and L. F. Mo, Homotopy Perturbation Method for Solving Reaction-Diffusion Equations, Mathematical Problem in Engineering, Vol. 2009, and Article ID: 795939. doi:10.1155/2009/795939.
[18] S. Kumar, Y. Khan, A. Yildirim, A mathematical modeling arising in the chemical system and its approximate numerical solution, Asia Pacific J. Chem. Eng. 7(2012) 835–840.
[19] N. Faraz, Y. Khan, Analytical solution of electrically conducted rotating flow of a second grade fluid over a shrinking surface, Ain Shams Eng. J. ,2(2011) 221–226.
[20] S. Kumar, Om P. Singh, Numerical inversion of the Abe integral equation using homotopy, Z. Naturforsch, 65( 2010) 677–682.
[21] S.kumar, A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alexandria Engineering Journal, 52(2013) 813–819.
[22] S. Kumar, H. Kocak, A. Yildirim, A fractional model of Gasdynamics equation and its analytical approximate solution by using Laplace transform, Z. Naturforsch, 67a(2012) 389–396.
[23] S. Kumar, A. Yildirim, Y. Khan, L. Wei, A fractional model of the diffusion equation and its analytical solution using Laplace transform, Sci. Iran. B, 19(2012) 1117–1123.
[24] K.S. Miller, B. Ross,An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[25] Abbaoui, K., Cherruault, Y.,New ideas for proving convergence of decomposition methods, Computers & Mathematics with Applications, 29(1995) 103–108.
[26] Reza Abazari , Malek Abazari ,Numerical simulation of generalized Hirota–Satsuma coupled KdV equation by RDTM and comparison with DTM, 17(2012) 619–629

Keywords
Homotopy perturbation method, Laplace transform method, Generalized Hirota-Satsuma Coupled KdV Equation.