Adomian decomposition and Padé approximate for solving differential-difference equation. (English) Zbl 1229.65116

Summary: We apply the Adomian decomposition method and Padé-approximate to solving the differential-difference equations (DDEs) for the first time. A simple but typical example is used to illustrate the validity and the great potential of the Adomian decomposition method (ADM) in solving DDEs. Comparisons are made between the results of the proposed method and exact solutions. The results show that ADM is an attractive method in solving the differential-difference equations.


65L05 Numerical methods for initial value problems involving ordinary differential equations
65L03 Numerical methods for functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
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[1] Fermi, E.; Pasta, J.; Ulam, S., Collected papers of enrico Fermi, (1965), Chicago Press Chicago, IL, pp. 978
[2] Levi, D.; Ragnisco, O., Extension of the spectral transform method for solving nonlinear differential- difference equations, Lett. nuovo. cimento, 22, 691, (1978)
[3] Levi, D.; Yamilov, RI., Conditions for the existence of higher symmetries of evolutionary equations on a lattice, J. math. phys., 38, 6648, (1997) · Zbl 0896.34057
[4] Yamilov, R.I., Construction scheme for discrete miura transformations, J. phys. A: gen., 27, 6839, (1994) · Zbl 0844.65090
[5] Adler, V.E.; Svinolupov, S.I.; Yamilov, R.L., Multi-component Volterra and Toda type integrable equations, Phys. lett. A, 254, 24, (1999) · Zbl 0983.37082
[6] Cherdantsev, I.Y.; Yamilov, R.I., Master symmetries for differential-difference equations of the Volterra type, Physica D, 87, 140, (1995) · Zbl 1194.35485
[7] Shabat, A.B.; Yamilov, R.I., Symmetries of nonlinear lattices, Leningrad math. J., 2, 377, (1991) · Zbl 0722.35006
[8] Zhang, D.J., Singular solutions in Casoratian form for two differential-difference equations, Chaos soliton fract., 23, 1333, (2005) · Zbl 1078.39018
[9] Narita, K., Soliton solution for a highly nonlinear difference-differential equation, Chaos soliton fract., 3, 279, (1993) · Zbl 0771.35060
[10] Wang, Z.; Zhang, H.Q., New exact solutions to some difference differential equations, Chin. phys., 15, 2210, (2006)
[11] Wang, Z.; Zhang, H.Q., A symbolic computational method for constructing exact solutions to difference differential equations, Appl. math. comput., 178, 431, (2006) · Zbl 1100.65083
[12] Suris, Yu.B., New integrable systems related to the relativistic Toda lattice, J. phys. A: math. gen., 30, 1745, (1997) · Zbl 1001.37508
[13] Suris, Yu.B., Integrable discretizations for lattice systems: local equations of motion and their Hamiltonian properties, Rev. math. phys., 11, 727-822, (1999) · Zbl 0965.37058
[14] Suris, Yu.B., The problem of integrable discretization: Hamiltonian approach, progress in mathematics, 219, (2003), Birkhäuser Verlag Basel
[15] Zou, L.; Zong, Z.; Dong, G.H., Generalizing homotopy analysis method to solve Lotka-Volterra equation, Comput. math. appl., 56, 2289, (2008) · Zbl 1165.34305
[16] Wang, Z.; Zou, L.; Zhang, H.Q., Applying homotopy analysis method for solving differential-difference equation, Phys. lett. A, 369, 77, (2007) · Zbl 1209.65119
[17] Adomian, G., The decomposition method, (1994), Kluwer Acad. Publ. Boston · Zbl 0803.35020
[18] Adomian, G.; Serrano, S.E., Stochastic contaminant transport equation in porous media, Appl. math. lett., 11, 53-55, (1998) · Zbl 1075.76672
[19] Adomian, G.; Rach, R.; Shawagfeh, N.T., On the analytic solution of the. Lane-Emden equation, Phys. lett., 2, 161-181, (1995)
[20] Kaya, D.; El-Sayed, S.M., On the solution of the coupled schroinger-KdV equation by the decomposition method, Phys. lett. A, 313, 82-88, (2003) · Zbl 1040.35099
[21] Badredine, T.; Abbaoui, K.; Cherruault, Y., Convergence of adomian’s method applied to integral equations, Kybernetes, 28, 557-564, (1999) · Zbl 0938.93023
[22] Adomian, G., Solutions of nonlinear PDE, Appl. math. lett., 11, 121-123, (1998) · Zbl 0933.65121
[23] Wu, L.; Xie, L.D.; Zhang, J.F., Adomian decomposition method for nonlinear differential-difference equations, Commun. nonlinear sci. numer. simul., 14, 1, 12-18, (2007)
[24] Baker, G.A., Essential of Padé approximants, (1975), Academic New York
[25] Chrysos, M.; Lefebvre, R.; Atabek, O., On the self-generation of asymptotic boundary conditions in energy quantization, J. phys B: at. mol. opt. phys., 27, 3005, (1994)
[26] Ryaboy, V.; Lefebvre, R.; Moiseyev, N., Cumulative reaction probabilities using pade analytical continuation procedures, J. chem. phys., 99, 3509, (1993)
[27] Wang, Z., Discrete tanh method for nonlinear difference-differential equations, Comput. phys. commun., 180, 1104, (2009) · Zbl 1198.65157
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