THE SOLUTION OF COUPLED MODIFIED KDV SYSTEM BY THE ...

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M. GHOREISHI, A.I.B.MD. ISMAIL, A. RASHID: COUPLED MODIFIED KDV SYSTEM ... 133 Table 5: Absolute error E 4 for variables x and t at = −1, for v(x, t). x/t 0.1 0.2 0.3 0.4 -100 9.89160×10 −13 1.97793×10 −12 2.96632×10 −12 3.95434×10 −12 -40 1.60774×10 −7 3.21486×10 −7 4.82137×10 −7 6.42728×10 −7 -10 3.83856×10 −5 7.67786×10 −5 1.15179×10 −4 1.53586×10 −4 10 3.83784×10 −5 7.67495×10 −5 1.15113×10 −4 1.53470×10 −4 40 1.60838×10 −7 3.21742×10 −7 4.82713×10 −7 6.43752×10 −7 100 9.89547×10 −13 1.97951×10 −12 2.96988×10 −12 3.96067×10 −12 In figures 7 and 8, we have displayed the absolute error between solution obtained using HAM with four terms and the exact solution for = −1, k = 0.1, λ = 0.5 at x = −100 and x = 0, respectively. The -curves have been plotted in figure 9 for u t (0, 0) and v t (0.5, t). As we have expressed, it is easy to check that the HAM solution is convergent to the exact solutions when −1.4 < < −0.6. 4. Conclusion In this paper, we have illustrated how HAM can be used for the new coupled modification of KdV system. The method was tested on two examples. The HAM solution contains the auxiliary parameter ≠ 0 provides a method to adjust and control the convergence region of the infinite series for large value of t. The obtained results show that the HAM is a very accurate and effective technique for the solution of the new MKdV system. References [1] Alomari, A.K., Noorani, M.S.M., Nazar, R., Li, C.P., (2010), Homotopy analysis method for solving fractional Lorenz system, Communications in Nonlinear Science and Numerical Simulation, 15, pp.1864–1872. [2] Alomari, A. K., Noorani, M. S. M., Nazar, R.,(2009), Comparison between the homotopy analysis method and homotopy paerturbation method to solve coupled Schrodinger-KdV equation, Journal of Applied Mathematics and Computation, 31, pp.1–12. [3] Alomari, A.K., Noorani, M.S.M., Nazar, R., (2008), The homotopy analysis method for the exact solutions of the K(2,2), Burgers and coupled Burgers equations, Applied Mathematical Sciences, 2(40), pp.1963–1977. [4] Awawdeh, F., Adawi, A., Mustafa, Z., (2009), Solutions of the SIR models of epidemics using HAM, Chaos, Solitons and Fractals, 42, pp.3047–3052. [5] Bataineh, A.S., Noorani, M.S.M., Hashim, I., (2009), On a new reliable modification of homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14, pp.409–423. [6] Bataineh, A.S., Noorani, M.S.M., Hashim, I., (2009), Modified homotopy analysis method for solving systems of second-order BVPs, Communications in Nonlinear Science and Numerical Simulation, 14, pp.430–442. [7] Fan, E., (2002), Using symbolic computation to exactly solve a new coupled MKdV system, Physics Letter A, 299(1), pp.46–48. [8] Fan, E., (2001), Soliton solutions for a generalized HirotaSatsuma coupled KdV equation and a coupled MKdV equation, Physics Letters A, 282, pp.18–22. [9] Inc, M., Cavlak, E., (2008), On numerical solutions of a new coupled MKdV system by using the Adomian decomposition method and He’s variational iteration method, Physica Scripta, 78, pp.1–7. [10] Liao, S.-J., (1992), The Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems, Ph.D.dissertation, Shanghai Jiao Tong University, Shanghai, China. [11] Liao, S.-J., (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton, Chapman and Hall. [12] Rashid, A., Izani, A.M.I., (2010), A Chebyshev spectral collocation method for the coupled nonlinear Schrdinger equations, International Journal of Applied and Computational Mathematics, 9(1), pp.104–115 [13] Raslan, K.R., (2004), The decomposition method for a Hirota-Satsuma coupled KdV equation and a coupled MKdV equation, International Journal of Computer Mathematics, 81(12), pp.1497–1505.
134 TWMS JOUR. PURE APPL. MATH., V.3, N.1, 2012 [14] Wu, Y.T., Geng, X.G., Hu, X.B., Zhu, S., (1999), A generalized HirotaSatsuma coupled Kortewegde Vries equation and Miura transformations, Physics Letters A, 255(4-6), pp.259–264. Seyed Mohammad Ghoreishi was born in Tehran-Iran. He obtained his B.Sc. from University of Sistan and Baluchestan in 1994. He received his M. Sc. from University of Science and Technology, Iran in 1997. He worked as a lecturer at several Universities of Iran in 1997 to 2007. He received his Ph.D. in Applied Mathematics from School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia in 2011. He studied the numerical and approximate analytical solution of parabolic equations with nonlocal boundary conditions. Ahmad Izani Md. Ismail joined the School of Mathematical Sciences, Universiti Sains Malaysia, in 1984 and was appointed a Professor in 2011. He has taught undergraduate and graduate courses on numerical analysis, mathematical modeling, fluid mechanics and computer programming. His primary research interest is in mathematical modeling and he is presently supervising and co-supervising 8 Ph.D. students. Abdur Rashid joined the Department of Mathematics, Gomal University, Dera Ismail Khan, Pakistan in 1983. He completed Ph.D. from Shanghai University of Science Technology, Shanghai, P. R. China, in 1993. He was appointed a Professor at Department of Mathematics, Gomal University, Dera Ismail Khan, Pakistan, in 2008. His research interest is Numerical Solution of Partial Differential Equations by Spectral methods.
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