The multistep homotopy analysis method for solving the JaulentMiodek equations
DOI:
https://doi.org/10.4067/S071609172015000100004Keywords:
Differential algebraic equations, multistep homotopy analysis method, Numerical solutions.Abstract
In this work, the multistep homotopy analysis method (MHAM) is applied to obtain the explicit analytical solutions for system of the Jaulent Miodek equations. The proposed scheme is only a simple modification of the homotopy analysis method (HAM), in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. Thus, it is valid for both weakly and strongly nonlinear problems. this work verifies the validity and the potential of the MHAM for the study of nonlinear systems. A comparative study between the new algorithm and the exact solution is presented graphically. convenient.References
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[2] J. L. Zhang, M. L. Wang, and X. R. Li, The subsidiary ellipticlike equation and the exact solutions of the higherorder nonlinear Schr¨odinger equation, Chaos Solitons Fractals, 33, pp. 1450—1457, (2007).
[3] E. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Solitons Fractals, 16, pp. 819—839, (2003).
[4] J. B. Chen and X. G. Geng, Decomposition to the modified Jaulent— Miodek hierarchy, Chaos Solitons Fractals, 30, pp. 797—803, (2006).
[5] A. M. Wazwaz, The tanhcoth and the sech methods for exact solutions of the Jaulent—Miodek equation, Phys Lett A, 366, pp. 85—90, (2007).
[6] D. D. Ganji, M. Jannatabadi, and E. Mohseni, Application of He’s variational iteration method to nonlinear Jaulent—Miodek equations and comparing it with ADM, J Comput Appl Math, 207, pp. 35—45, (2007).
[7] J. Cang, Y. Tan, H. Xu, S. Liao, Series solutions of nonlinear Riccati differential equations with fractional order, Chaos, Solitons & Fractals, 40 (1), pp. 19, (2009).
[8] M. Zurigat, S. Momani, Z. odibat, A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling, 34 (1), pp. 2435, (2010).
[9] M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraicdifferential equations by homotopy analysis method, Computers and Mathematics with Applications, 59 (3), pp. 12271235, (2010).
[10] A. K. Alomari, M. S. M. Noorani, R. Nazar, C. P. Li, Homotopy analysis method for solving fractional Lorenz system, Commun Nonliear Sci Numer Simult, 15 (7), pp. 18641872, (2010).
[11] A. Rafiq, M. Rafiullah, Some multistep iterative methods for solving nonlinear equations, Computers & Mathematics with Applications, 58 (8), pp. 15891597, (2009).
[12] M. M. Rashidi, G. Domairry and S. Dinarvand, The Homotopy Analysis Method for Explicit Analytical Solutions of Jaulent—Miodek Equations, Numerical Methods for Partial Differential Equations, 25 (2), pp. 430439, (2008).
[13] H. T. Ozer and S. Salihoglu, Nonlinear Schr¨odinger equations and N = 1 superconformal algebra, Chaos Solitons Fractals 33, pp. 1417—1423, (2007).
[14] J. L. Zhang, M. L. Wang, and X. R. Li, The subsidiary ellipticlike equation and the exact solutions of the higherorder nonlinear Schr¨odinger equation, Chaos Solitons Fractals 33, pp. 1450—1457, (2007).
[15] J. M. Zhu and Z. Y. Ma, Exact solutions for the cubicquintic nonlinear Schr¨odinger equation, Chaos Solitons Fractals 33, pp. 958—964, (2007)
[16] A. Yildirim and A. Kelleci, Numerical Simulation of the Jaulentmiodek Equation byHe’s Homotopy Perturbation Method,World Applied Sciences Journal 7 (Special Issue for Applied Math), (2009).
[2] J. L. Zhang, M. L. Wang, and X. R. Li, The subsidiary ellipticlike equation and the exact solutions of the higherorder nonlinear Schr¨odinger equation, Chaos Solitons Fractals, 33, pp. 1450—1457, (2007).
[3] E. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics, Chaos Solitons Fractals, 16, pp. 819—839, (2003).
[4] J. B. Chen and X. G. Geng, Decomposition to the modified Jaulent— Miodek hierarchy, Chaos Solitons Fractals, 30, pp. 797—803, (2006).
[5] A. M. Wazwaz, The tanhcoth and the sech methods for exact solutions of the Jaulent—Miodek equation, Phys Lett A, 366, pp. 85—90, (2007).
[6] D. D. Ganji, M. Jannatabadi, and E. Mohseni, Application of He’s variational iteration method to nonlinear Jaulent—Miodek equations and comparing it with ADM, J Comput Appl Math, 207, pp. 35—45, (2007).
[7] J. Cang, Y. Tan, H. Xu, S. Liao, Series solutions of nonlinear Riccati differential equations with fractional order, Chaos, Solitons & Fractals, 40 (1), pp. 19, (2009).
[8] M. Zurigat, S. Momani, Z. odibat, A. Alawneh, The homotopy analysis method for handling systems of fractional differential equations, Applied Mathematical Modelling, 34 (1), pp. 2435, (2010).
[9] M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraicdifferential equations by homotopy analysis method, Computers and Mathematics with Applications, 59 (3), pp. 12271235, (2010).
[10] A. K. Alomari, M. S. M. Noorani, R. Nazar, C. P. Li, Homotopy analysis method for solving fractional Lorenz system, Commun Nonliear Sci Numer Simult, 15 (7), pp. 18641872, (2010).
[11] A. Rafiq, M. Rafiullah, Some multistep iterative methods for solving nonlinear equations, Computers & Mathematics with Applications, 58 (8), pp. 15891597, (2009).
[12] M. M. Rashidi, G. Domairry and S. Dinarvand, The Homotopy Analysis Method for Explicit Analytical Solutions of Jaulent—Miodek Equations, Numerical Methods for Partial Differential Equations, 25 (2), pp. 430439, (2008).
[13] H. T. Ozer and S. Salihoglu, Nonlinear Schr¨odinger equations and N = 1 superconformal algebra, Chaos Solitons Fractals 33, pp. 1417—1423, (2007).
[14] J. L. Zhang, M. L. Wang, and X. R. Li, The subsidiary ellipticlike equation and the exact solutions of the higherorder nonlinear Schr¨odinger equation, Chaos Solitons Fractals 33, pp. 1450—1457, (2007).
[15] J. M. Zhu and Z. Y. Ma, Exact solutions for the cubicquintic nonlinear Schr¨odinger equation, Chaos Solitons Fractals 33, pp. 958—964, (2007)
[16] A. Yildirim and A. Kelleci, Numerical Simulation of the Jaulentmiodek Equation byHe’s Homotopy Perturbation Method,World Applied Sciences Journal 7 (Special Issue for Applied Math), (2009).
How to Cite
[1]
M. Zurigat, A. A. Freihat, and A. H. Handam, “The multistep homotopy analysis method for solving the JaulentMiodek equations”, Proyecciones (Antofagasta, On line), vol. 34, no. 1, pp. 4554, 1.
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