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Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems

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  • Abdel-Halim Hassan, I.H.

Abstract

In this paper, we will compare the differential transformation method DTM and Adomian decomposition method ADM to solve partial differential equations (PDEs). The definition and operations of differential transform method was introduced by Zhou [Zhou JK. Differential transformation and its application for electrical circuits. Wuuhahn, China: Huarjung University Press; 1986 [in Chinese]]. Adomian decomposition method which is given by Adomian [Adomian G. Convergent series solution of nonlinear equation. J Comput Appl Math 1984;11:113–7; Adomian G. Solutions of nonlinear PDE. Appl Math Lett 1989;11:121–3; Adomian G. Solving Frontier problems of physics. The decomposition method, Boston, 1994] for approximate solution of linear and nonlinear differential equations and to the solutions of various scientific models see [Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals 2006;29:313–30; El-Danaf TS, Ramadan MA, Abd Alaal FEI. The use of Adomian decomposition method for solving the regularized long-wave equation. Chaos, Solitons & Fractals 2005;26:747–57; El-Sayed SM. The decomposition method for studying the Klein–Gordon equation. Chaos, Solitions & Fractals 2003;18:1025–30; Helal MA, Mehanna MS. A comparison between two different methods for solving KdV-Burgers equation. Chaos, Solitons & Fractals 2006;28:320–6, Hashim I, Noorani MSM, Ahmad R, Bakar SA, Ismail ES, Zakaria AM. Accuracy of the decomposition method applied to the Lorenz system. Chaos, Solitons & Fractals 2006;28:1149–58; Kaya D, El-Sayed SM. An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals 2003;17:869–77; Lesnic D. Blow-up solutions obtained using the decomposition method. Chaos, Solitons & Fractals 2006;28:776–87; Wazwaz AM. Construction of solitary wave solutions and rational solutions for KdV equation by Adomian decomposition method. Chaos, Solitons & Fractals 2001;12:2283–93; Wazwaz AM. Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, Solitons & Fractals 2001;12:1549–56]. A distinctive practical feature of the differential transformation method DTM is ability to solve linear or nonlinear differential equations. Higher-order dimensional differential transformation are applied to a few some initial value problems to show that the solutions obtained by the proposed method DTM coincide with the approximate solution ADM and the analytic solutions.

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  • Abdel-Halim Hassan, I.H., 2008. "Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 53-65.
    • Handle: RePEc:eee:chsofr:v:36:y:2008:i:1:p:53-65
    DOI: 10.1016/j.chaos.2006.06.040
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    References listed on IDEAS

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    1. Helal, M.A. & Mehanna, M.S., 2006. "A comparison between two different methods for solving KdV–Burgers equation," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 320-326.
    2. Kamdem, J. Sadefo & Qiao, Zhijun, 2007. "Decomposition method for the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 437-447.
    3. Abbasbandy, S., 2007. "A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 31(1), pages 257-260.
    4. Abulwafa, E.M. & Abdou, M.A. & Mahmoud, A.A., 2006. "The solution of nonlinear coagulation problem with mass loss," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 313-330.
    5. El-Danaf, Talaat S. & Ramadan, Mohamed A. & Abd Alaal, Faysal E.I., 2005. "The use of adomian decomposition method for solving the regularized long-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 747-757.
    6. C. L. Chen & Y. C. Liu, 1998. "Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 23-35, October.
    7. Lesnic, D., 2006. "Blow-up solutions obtained using the decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 776-787.
    8. Hashim, I. & Noorani, M.S.M. & Ahmad, R. & Bakar, S.A. & Ismail, E.S. & Zakaria, A.M., 2006. "Accuracy of the Adomian decomposition method applied to the Lorenz system," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1149-1158.
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    2. Dehghan, Mehdi & Shakourifar, Mohammad & Hamidi, Asgar, 2009. "The solution of linear and nonlinear systems of Volterra functional equations using Adomian–Pade technique," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2509-2521.
    3. Kangalgil, Figen & Ayaz, Fatma, 2009. "Solitary wave solutions for the KdV and mKdV equations by differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 464-472.

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