IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v36y2008i1p53-65.html
   My bibliography  Save this article

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

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077906005790
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2006.06.040?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. Kamdem, J. Sadefo & Qiao, Zhijun, 2007. "Decomposition method for the Camassa–Holm equation," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 437-447.
    5. Lesnic, D., 2006. "Blow-up solutions obtained using the decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 776-787.
    6. 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.
    7. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Lan, Heng-you & Cui, Yi-Shun, 2009. "A neural network method for solving a system of linear variational inequalities," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1245-1252.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Memarbashi, Reza, 2008. "Numerical solution of the Laplace equation in annulus by Adomian decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 138-143.
    2. Elgazery, Nasser S., 2008. "Numerical solution for the Falkner–Skan equation," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 738-746.
    3. Tajvidi, T. & Razzaghi, M. & Dehghan, M., 2008. "Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 59-66.
    4. Ramos, J.I., 2009. "Piecewise-adaptive decomposition methods," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1623-1636.
    5. Jules Sadefo-Kamdem, 2011. "Integral Transforms With The Homotopy Perturbation Method And Some Applications," Working Papers hal-00580023, HAL.
    6. Mossa Al-sawalha, M. & Noorani, M.S.M., 2009. "A numeric–analytic method for approximating the chaotic Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1784-1791.
    7. Goh, S.M. & Noorani, M.S.M. & Hashim, I., 2009. "A new application of variational iteration method for the chaotic Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1604-1610.
    8. 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.
    9. Ramos, J.I., 2009. "Generalized decomposition methods for nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1078-1084.
    10. Ramos, J.I., 2007. "Solitary waves of the EW and RLW equations," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1498-1518.
    11. Al-Sawalha, M. Mossa & Noorani, M.S.M. & Hashim, I., 2009. "On accuracy of Adomian decomposition method for hyperchaotic Rössler system," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1801-1807.
    12. Hammad, D.A. & El-Azab, M.S., 2016. "Chebyshev–Chebyshev spectral collocation method for solving the generalized regularized long wave (GRLW) equation," Applied Mathematics and Computation, Elsevier, vol. 285(C), pages 228-240.
    13. Lozi, René & Pogonin, Vasiliy A. & Pchelintsev, Alexander N., 2016. "A new accurate numerical method of approximation of chaotic solutions of dynamical model equations with quadratic nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 108-114.
    14. Lv, Na & Mei, Jian-Qin & Zhang, Hong-Qing, 2012. "Differential form method for finding symmetries of a (2+1)-dimensional Camassa–Holm system based on its Lax pair," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 503-506.
    15. Golbabai, A. & Javidi, M., 2009. "A spectral domain decomposition approach for the generalized Burger’s–Fisher equation," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 385-392.
    16. Noorani, M.S.M. & Hashim, I. & Ahmad, R. & Bakar, S.A. & Ismail, E.S. & Zakaria, A.M., 2007. "Comparing numerical methods for the solutions of the Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1296-1304.
    17. Hashim, I. & Chowdhury, M.S.H. & Mawa, S., 2008. "On multistage homotopy-perturbation method applied to nonlinear biochemical reaction model," Chaos, Solitons & Fractals, Elsevier, vol. 36(4), pages 823-827.
    18. Chein-Shan Liu, 2012. "The Lie-Group Shooting Method for Solving Multi-dimensional Nonlinear Boundary Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 468-495, February.
    19. Ramos, J.I., 2007. "Solitary wave interactions of the GRLW equation," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 479-491.
    20. Javidi, M. & Golbabai, A., 2009. "A new domain decomposition algorithm for generalized Burger’s–Huxley equation based on Chebyshev polynomials and preconditioning," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 849-857.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:36:y:2008:i:1:p:53-65. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.