Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method

ABSTRACT

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

KEYWORDS

Homotopy Analysis Method; Approximate Analytical Solution; Fractional Coupled mKdV Equation

Homotopy Analysis Method; Approximate Analytical Solution; Fractional Coupled mKdV Equation

Cite this paper

nullO. Tasbozan, A. Esen and N. Yagmurlu, "Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method,"*Open Journal of Applied Sciences*, Vol. 2 No. 3, 2012, pp. 193-197. doi: 10.4236/ojapps.2012.23029.

nullO. Tasbozan, A. Esen and N. Yagmurlu, "Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method,"

References

[1] L. Podlubny, “Fractional Differantial Equations,”, Academic Press, London, 1999.

[2] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent,” Geopyhsical Journal International, Volume. 13, No. 5, 1967, pp. 529-539. doi:10.1111/j.1365-246X.1967.tb02303.x

[3] M. Caputo, “Elasticità e Dissipazione,”, Zanichelli, Bologna, 1969.

[4] S.J. Liao, “The proposed homotopy analysis tecnique for the solution of nonlinear problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[5] S.J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman and Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164

[6] S.J. Liao, “Homotopy analysis method: A new analytical technique for nonlinear problems,” Communications in Nonlinear Science and Numerical. Simulations, Vol. 2, No. 2, 1997, pp. 95-100. doi:10.1016/S1007-5704(97)90047-2

[7] S.J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, No. 147, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7

[8] S.J. Liao, “Notes on the homotopy analysis method: Some de_nitions and theorems,” Communications in Nonlinear Science and Numerical. Simulations, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[9] S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letter A., Vol. 361, No. 6, 2007, pp. 478-483. doi: 10.1016/ j.physleta.2006.09.105

[10] E. Babolian and J. Saeidian, “Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 5, 2009, pp. 1984-1992. doi:10.1116/j.cnsns.2008.07.019

[11] A. Fakhari, G. Domairry and Ebrahimpour, “Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution,” Physics Letters A., No .368, No. 1-2., 2007, pp. 64-68. doi: 10.1116/j.physleta.2007.03.062

[12] M.M. Rashidi, G. Domairry, A. Doosthosseini and S. Dinarvand, “Explicit Approximate Solution of the Coupled KdV Equations by using the Homotopy Analysis Method,” International Journal of Mathematical Analysis, Vol.2 No. 12, 2008, pp. 581-589.

[13] M. Inc, “On numerical solution of Burgers’ equation by homotopy analysis method,” Physics Letters A., Volume. 372, No. 4, 2008, pp. 356-360. doi:10.1016/j.physleta.2007.07.057

[14] A.S. Bataineh, M.S.M. Noorani and I. Hashim, “Approximate analytical solutions of systems of PDEs by homotopy analysis method,” Computers and Mathematics with Applications, Vol. 55, No.12, 2008, pp. 2913-2923. doi:10.1016/j.camwa.2007.11.022

[15] S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, Vol. 360, No. 1, 2006, pp. 109-113. doi:10.1016/j.physleta.2006.07.065

[16] T. Hayat and M. Sajid, “On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder,” Physics Letters A, Vol. 361, No. 4-5, 2007, pp. 316-322. doi:10.1016/j.physleta.2006.09.060

[17] H. Xu and J. Cang, “Analysis of a time fractional wave-like equation with the homotopy analysis method,” Physics Letters A, Vol. 372, No. 8, 2008, pp. 1250-1255. doi:10.1016/j.physleta.2007.09.039

[18] L. Song and H.Q. Zhang, “Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation,” Physics Letters A, Vol. 367, No. 1-2, 2007, pp. 88-94. doi:10.1016/j.physleta.2007.02.083

[19] D.B. Cao, J.R. Yanb and Y. Zhangc, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Physics Letters A, Vol. 279, No. 1-2, 2002, pp. 68-74. doi:10.1016/S0375-9601(02)00376-6

[1] L. Podlubny, “Fractional Differantial Equations,”, Academic Press, London, 1999.

[2] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent,” Geopyhsical Journal International, Volume. 13, No. 5, 1967, pp. 529-539. doi:10.1111/j.1365-246X.1967.tb02303.x

[3] M. Caputo, “Elasticità e Dissipazione,”, Zanichelli, Bologna, 1969.

[4] S.J. Liao, “The proposed homotopy analysis tecnique for the solution of nonlinear problems,” Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.

[5] S.J. Liao, “Beyond Perturbation: Introduction to the Homotopy Analysis Method,” Chapman and Hall/CRC Press, Boca Raton, 2003. doi:10.1201/9780203491164

[6] S.J. Liao, “Homotopy analysis method: A new analytical technique for nonlinear problems,” Communications in Nonlinear Science and Numerical. Simulations, Vol. 2, No. 2, 1997, pp. 95-100. doi:10.1016/S1007-5704(97)90047-2

[7] S.J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, No. 147, 2004, pp. 499-513. doi:10.1016/S0096-3003(02)00790-7

[8] S.J. Liao, “Notes on the homotopy analysis method: Some de_nitions and theorems,” Communications in Nonlinear Science and Numerical. Simulations, Vol. 14, No. 4, 2009, pp. 983-997. doi:10.1016/j.cnsns.2008.04.013

[9] S. Abbasbandy, “The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation,” Physics Letter A., Vol. 361, No. 6, 2007, pp. 478-483. doi: 10.1016/ j.physleta.2006.09.105

[10] E. Babolian and J. Saeidian, “Analytic approximate solutions to Burgers, Fisher, Huxley equations and two combined forms of these equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 5, 2009, pp. 1984-1992. doi:10.1116/j.cnsns.2008.07.019

[11] A. Fakhari, G. Domairry and Ebrahimpour, “Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution,” Physics Letters A., No .368, No. 1-2., 2007, pp. 64-68. doi: 10.1116/j.physleta.2007.03.062

[12] M.M. Rashidi, G. Domairry, A. Doosthosseini and S. Dinarvand, “Explicit Approximate Solution of the Coupled KdV Equations by using the Homotopy Analysis Method,” International Journal of Mathematical Analysis, Vol.2 No. 12, 2008, pp. 581-589.

[13] M. Inc, “On numerical solution of Burgers’ equation by homotopy analysis method,” Physics Letters A., Volume. 372, No. 4, 2008, pp. 356-360. doi:10.1016/j.physleta.2007.07.057

[14] A.S. Bataineh, M.S.M. Noorani and I. Hashim, “Approximate analytical solutions of systems of PDEs by homotopy analysis method,” Computers and Mathematics with Applications, Vol. 55, No.12, 2008, pp. 2913-2923. doi:10.1016/j.camwa.2007.11.022

[15] S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,” Physics Letters A, Vol. 360, No. 1, 2006, pp. 109-113. doi:10.1016/j.physleta.2006.07.065

[16] T. Hayat and M. Sajid, “On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder,” Physics Letters A, Vol. 361, No. 4-5, 2007, pp. 316-322. doi:10.1016/j.physleta.2006.09.060

[17] H. Xu and J. Cang, “Analysis of a time fractional wave-like equation with the homotopy analysis method,” Physics Letters A, Vol. 372, No. 8, 2008, pp. 1250-1255. doi:10.1016/j.physleta.2007.09.039

[18] L. Song and H.Q. Zhang, “Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation,” Physics Letters A, Vol. 367, No. 1-2, 2007, pp. 88-94. doi:10.1016/j.physleta.2007.02.083

[19] D.B. Cao, J.R. Yanb and Y. Zhangc, “Exact solutions for a new coupled MKdV equations and a coupled KdV equations,” Physics Letters A, Vol. 279, No. 1-2, 2002, pp. 68-74. doi:10.1016/S0375-9601(02)00376-6