IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v41y2009i2p716-726.html

Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials

Author

Listed:
  • Abourabia, A.M.
  • El-Danaf, T.S.
  • Morad, A.M.

Abstract

The problems under consideration are related to wave propagation in microstructured materials, characterized by higher-order nonlinear and higher-order dispersive effects; particularly, the wave propagation in dilatant granular materials. In the present paper the model equation is solved analytically by exact methods. The types of solutions are defined and discussed over a wide range of material parameters (two dispersion parameters and one microstructure parameter). The dispersion properties and the relation between group and phase velocities of the model equation are studied. The diagrams are drawn to illustrate the physical properties of the exact solutions.

Suggested Citation

  • Abourabia, A.M. & El-Danaf, T.S. & Morad, A.M., 2009. "Exact solutions of the hierarchical Korteweg–de Vries equation of microstructured granular materials," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 716-726.
    • Handle: RePEc:eee:chsofr:v:41:y:2009:i:2:p:716-726
    DOI: 10.1016/j.chaos.2008.03.015
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077908001379
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2008.03.015?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

    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. Zayed, E.M.E. & Abourabia, A.M. & Gepreel, Khaled A. & El Horbaty, M.M., 2007. "Travelling solitary wave solutions for the nonlinear coupled Korteweg–de Vries system," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 292-306.
    3. Ilison, O. & Salupere, A., 2006. "On the propagation of solitary pulses in microstructured materials," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 202-214.
    4. Soliman, A.A., 2006. "The modified extended tanh-function method for solving Burgers-type equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(2), pages 394-404.
    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. Abourabia, A.M. & Morad, A.M., 2015. "Exact traveling wave solutions of the van der Waals normal form for fluidized granular matter," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 333-350.
    2. Abourabia, Aly M. & Hassan, Kawsar M. & Morad, Adel M., 2009. "Analytical solutions of the magma equations for molten rocks in a granular matrix," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1170-1180.

    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. Abourabia, Aly M. & Hassan, Kawsar M. & Morad, Adel M., 2009. "Analytical solutions of the magma equations for molten rocks in a granular matrix," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1170-1180.
    2. Ramos, J.I., 2007. "Solitary waves of the EW and RLW equations," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1498-1518.
    3. Soliman, A.A., 2009. "Exact solutions of KdV–Burgers’ equation by Exp-function method," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 1034-1039.
    4. 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.
    5. 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.
    6. Mohanty, R.K. & Dai, Weizhong & Han, Fei, 2015. "Compact operator method of accuracy two in time and four in space for the numerical solution of coupled viscous Burgers’ equations," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 381-393.
    7. Başhan, Ali, 2020. "A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    8. Wang, Yue-yue & Dai, Chao-qing & Wu, Lei & Zhang, Jie-fang, 2007. "Exact and numerical solitary wave solutions of generalized Zakharov equation by the Adomian decomposition method," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1208-1214.
    9. 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.
    10. Attaullah & Muhammad Shakeel & Nehad Ali Shah & Jae Dong Chung, 2022. "Modified Exp-Function Method to Find Exact Solutions of Ionic Currents along Microtubules," Mathematics, MDPI, vol. 10(6), pages 1-10, March.
    11. Kavitha, L. & Prabhu, A. & Gopi, D., 2009. "New exact shape changing solitary solutions of a generalized Hirota equation with nonlinear inhomogeneities," Chaos, Solitons & Fractals, Elsevier, vol. 42(4), pages 2322-2329.
    12. Ramos, J.I., 2007. "Solitary wave interactions of the GRLW equation," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 479-491.
    13. Zayed, E.M.E. & Gepreel, Khaled A. & El Horbaty, M.M., 2009. "Extended proposed algorithm with symbolic computation to construct exact solutions for some nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 436-452.
    14. Yuzhen Chai & Tingting Jia & Huiqin Hao & Jianwen Zhang, 2014. "Exp-Function Method for a Generalized MKdV Equation," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-8, May.
    15. Goswami, Amit & Singh, Jagdev & Kumar, Devendra & Sushila,, 2019. "An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 563-575.
    16. Lai, Huilin & Ma, Changfeng, 2014. "A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 445-457.
    17. Elgazery, Nasser S., 2008. "Numerical solution for the Falkner–Skan equation," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 738-746.
    18. Hassani, Hossein & Naraghirad, Eskandar, 2019. "A new computational method based on optimization scheme for solving variable-order time fractional Burgers’ equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 162(C), pages 1-17.
    19. Piao, Xiangfan & Kim, Philsu, 2021. "Comment on: “The modified extended tanh-function method for solving Burgers-type equations” [Physica A 361 (2006) 394–404]," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 569(C).
    20. Li, Qianhuan & Chai, Zhenhua & Shi, Baochang, 2015. "A novel lattice Boltzmann model for the coupled viscous Burgers’ equations," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 948-957.

    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:41:y:2009:i:2:p:716-726. 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.