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Saha Ray, S.; Sahoo, S.

Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein-Gordon-Zakharov equations in plasma physics. (English) Zbl 1432.35230

Comput. Math. Math. Phys. 56, No. 7, 1319-1335 (2016).
Summary: In this paper, homotopy perturbation transform method and modified homotopy analysis method have been applied to obtain the approximate solutions of the time fractional coupled Klein-Gordon-Zakharov equations. We consider fractional coupled Klein-Gordon-Zakharov equation with appropriate initial values using homotopy perturbation transform method and modified homotopy analysis method. Here we obtain the solution of fractional coupled Klein-Gordon-Zakharov equation, which is obtained by replacing the time derivatives with a fractional derivatives of order \(\alpha\in(1, 2]\), \(\beta\in(1, 2]\). Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present methods homotopy perturbation transform method and modified homotopy analysis method. The fractional derivatives here are described in Caputo sense.

Cited in 6 Documents

MSC:

35R11 Fractional partial differential equations
35Q82 PDEs in connection with statistical mechanics

Keywords:

fractional coupled Klein-Gordon-Zakharov (KGZ) equation; homotopy perturbation transform method (HPTM); modified homotopy analysis method (MHAM); Caputo fractional derivative

Software:

BVPh
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Full Text: DOI

References:

[1] T. Wang, J. Chen, and L. Zhang, “Conservative difference methods for the Klein-Gordon-Zakharov equations,” J. Comput. Appl. Math. 205, 430-452 (2007). · Zbl 1123.65091 · doi:10.1016/j.cam.2006.05.008
[2] M. Dehghan and A. Nikpour, “The solitary wave solution of coupled Klein-Gordon-Zakharov equations via two different numerical methods,” Comput. Phys. Commun. 184, 2145-2158 (2013). · Zbl 1344.82041 · doi:10.1016/j.cpc.2013.04.010
[3] M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim, “On the coupling of the homotopy perturbation method and Laplace transformation,” Math. Comput. Model. 53 (9-10), 1937-1945 (2011). · Zbl 1219.65121 · doi:10.1016/j.mcm.2011.01.023
[4] Y. Khan and Q. Wu, “Homotopy perturbation transform method for nonlinear equations using He’s polynomial,” Comput. Math. Appl. 61, 1963-1967 (2011). · Zbl 1219.65119 · doi:10.1016/j.camwa.2010.08.022
[5] Liu, Y., Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation method, 14 (2012) · Zbl 1242.65136
[6] S. Liao, Beyond Perturbation: Introduction to the Modified Homotopy Analysis Method (Chapman and Hall/CRC, Boca Raton, 2003). · Zbl 1051.76001 · doi:10.1201/9780203491164
[7] S. Liao, “The proposed homotopy analysis techniques for the solution of nonlinear problems,” Ph. D. Thesis (Shanghai Jiao Tong Univ., Shanghai, 1992).
[8] S. Liao, Homotopy Analysis Method in Nonlinear Differential Equations (Springer, Heidelberg, 2012). · Zbl 1253.35001 · doi:10.1007/978-3-642-25132-0
[9] S. Liao, “On the homotopy analysis method for nonlinear problems,” Appl. Math. Comput. 147, 499-513 (2004). · Zbl 1086.35005
[10] M. Zurigat, S. Momani, Z. Odibat, and A. Alawneh, “The homotopy analysis method for handling systems of fractional differential equations,” Appl. Math. Model. 34, 24-35 (2010). · Zbl 1185.65140 · doi:10.1016/j.apm.2009.03.024
[11] S. Saha Ray and A. Patra, “Application of homotopy analysis method and Adomian decomposition method for the solution of neutron diffusion equation in the hemisphere and cylindrical reactors,” J. Nuclear Eng. Technol. 1 (2-3), 1-14 (2011).
[12] M. G. Sakar and F. Erdogan, “The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian’s decomposition method,” Appl. Math. Model. 37, 8876-8885 (2013). · Zbl 1427.65323 · doi:10.1016/j.apm.2013.03.074
[13] J. Wang, L. Biao, and Y. Wang-Chuan, “Approximate solution for the Klein-Gordon-Schrodinger equation by the homotopy analysis method,” Chin. Phys. B 19 (3), 030401-(1-7) (2010). · doi:10.1088/1674-1056/19/3/030401
[14] I. Podlubny, Fractional Differential Equations (Academic, New York, 1999). · Zbl 0924.34008
[15] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Taylor and Francis, London, 2002). · Zbl 0818.26003
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