The GDTM-Padé technique for the nonlinear lattice equations. (English) Zbl 1242.65278

Summary: The GDTM-Padé technique is a combination of the generalized differential transform method and the Padé approximation. We apply this technique to solve the two nonlinear lattice equations, which results in the high accuracy of the GDTM-Padé solutions. Numerical results are presented to show its efficiency by comparing the GDTM-Padé solutions, the solutions obtained by the generalized differential transform method, and the exact solutions.


65Q10 Numerical methods for difference equations
39A10 Additive difference equations
Full Text: DOI


[1] V. E. Adler, S. I. Svinolupov, and R. I. Yamilov, “Multi-component Volterra and Toda type integrable equations,” Physics Letters A, vol. 254, no. 1-2, pp. 24-36, 1999. · Zbl 0983.37082 · doi:10.1016/S0375-9601(99)00087-0
[2] D. Baldwin, Ü. Gökta\cs, and W. Hereman, “Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equations,” Computer Physics Communications, vol. 162, no. 3, pp. 203-217, 2004. · Zbl 1196.68324 · doi:10.1016/j.cpc.2004.07.002
[3] L. Bonora and C. S. Xiong, “An alternative approach to KP hierarchy in matrix models,” Physics Letters B, vol. 285, no. 3, pp. 191-198, 1992. · Zbl 0946.81518 · doi:10.1016/0370-2693(92)91451-E
[4] E. Fermi, J. Pasta, and S. Ulam, The Collected Papers of Enrico Fermi, The University of Chicago Press, Chicago, Ill, USA, 1965. · Zbl 0353.70028
[5] R. Hirota and M. Iwao, “Time-discretization of soliton equations,” in SIDE III-Symmetries and Integrability of Difference Equations, vol. 25 of CRM Proceedings & Lecture Notes, pp. 217-229, American Mathematical Society, Providence, RI, USA, 2000. · Zbl 0961.35135
[6] D. Levi and O. Ragnisco, “Extension of the spectral-transform method for solving nonlinear differential difference equations,” Lettere al Nuovo Cimento, vol. 22, no. 17, pp. 691-696, 1978.
[7] D. Levi and R. Yamilov, “Conditions for the existence of higher symmetries of evolutionary equations on the lattice,” Journal of Mathematical Physics, vol. 38, no. 12, pp. 6648-6674, 1997. · Zbl 0896.34057 · doi:10.1063/1.532230
[8] Y. B. Suris, “New integrable systems related to the relativistic Toda lattice,” Journal of Physics A, vol. 30, no. 5, pp. 1745-1761, 1997. · Zbl 1001.37508 · doi:10.1088/0305-4470/30/5/035
[9] Y. B. Suris, “On some integrable systems related to the Toda lattice,” Journal of Physics A, vol. 30, no. 6, pp. 2235-2249, 1997. · Zbl 0935.37037 · doi:10.1088/0305-4470/30/6/041
[10] Y. B. Suris, “Integrable discretizations for lattice system: local equations of motion and their Hamiltonian properties,” Reviews in Mathematical Physics, vol. 11, no. 6, pp. 727-822, 1999. · Zbl 0965.37058 · doi:10.1142/S0129055X99000258
[11] R. I. Yamilov, “Construction scheme for discrete Miura transformations,” Journal of Physics A, vol. 27, no. 20, pp. 6839-6851, 1994. · Zbl 0844.65090 · doi:10.1088/0305-4470/27/20/020
[12] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. · Zbl 0802.65122
[13] G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501-544, 1988. · Zbl 0671.34053 · doi:10.1016/0022-247X(88)90170-9
[14] C. Dai and J. Zhang, “Jacobian elliptic function method for nonlinear differential-difference equations,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 1042-1047, 2006. · Zbl 1091.34538 · doi:10.1016/j.chaos.2005.04.071
[15] J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[16] M. Wang, X. Li, and J. Zhang, “The G\(^{\prime}\)/G-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417-423, 2008. · Zbl 1217.76023 · doi:10.1016/j.physleta.2007.07.051
[17] S. Zhang and H.-Q. Zhang, “Variable-coefficient discrete tanh method and its application to (2+1)-dimensional Toda equation,” Physics Letters A, vol. 373, no. 33, pp. 2905-2910, 2009. · Zbl 1233.37046 · doi:10.1016/j.physleta.2009.06.016
[18] V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642-1654, 2008. · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[19] S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters A, vol. 370, no. 5-6, pp. 379-387, 2007. · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[20] Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467-477, 2008. · Zbl 1221.34022 · doi:10.1016/j.cnsns.2007.02.006
[21] Z. Li, W. Zhen, and Z. Zhi, “Generalized differential transform method to differential-difference equation,” Physics Letters A, vol. 373, no. 45, pp. 4142-4151, 2009. · Zbl 1234.35235 · doi:10.1016/j.physleta.2009.09.036
[22] H. Aratyn, L. A. Ferreira, J. F. Gomes, and A. H. Zimerman, “On two-current realization of KP hierarchy,” Nuclear Physics B, vol. 402, no. 1-2, pp. 85-117, 1993. · Zbl 0941.37525 · doi:10.1016/0550-3213(93)90637-5
[23] G. A. Baker, Essential of Padé Approximants, Academic Press, London, UK, 1975. · Zbl 0315.41014
[24] G. A. Baker and P. Graves-Morris, Encyclopedia of Mathematics and its Application 13, Parts I and II: Padé Approximants, Addison-Wesley Publishing Company, New York, NY, USA, 1981.
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