Investigation of exact traveling wave solution for the (2+1) dimensional nonlinear evolution equations via modified extended tanh-function method

  • Authors

    • Dipankar Kumar Assistant Professor
    • Prodip Sarker Lecturer
    2016-09-19
    https://doi.org/10.14419/ijpr.v4i2.6588
  • Modified Extended Tanh-Function Method, Riccati Equation, The General (2 1) Dimensional Nonlinear Evolution Equations, Traveling Wave Solutions.

  • In this study, we have implemented the modified extended tanh-function method to obtain the exact travelling wave solutions for the general (2+1)-dimensional nonlinear evolution equations. By using this method, some travelling wave solutions are successfully obtained and which have been expressed by the trigonometric, hyperbolic and rational functions. These obtained solutions are an appropriate and desirable for instructive specific nonlinear physical phenomena in genuinely nonlinear dynamical systems. The method is an efficient and reliable mathematical tool for solving many nonlinear evolution equations arising in science and engineering problems.

  • References

    1. [1] Wazwaz, A. M., 2004. A sine-cosine method for handling nonlinear wave equations. Mathematical and Computer modeling, 40(5): 499-508. http://dx.doi.org/10.1016/j.mcm.2003.12.010.

      [2] Wazwaz, A. M., 2004. The sine–cosine method for obtaining solutions with compact and noncompact structures. Applied Mathematics and Computation, 159(2): 559-576. http://dx.doi.org/10.1016/j.amc.2003.08.136.

      [3] Yusufoğlu, E., & Bekir, A., 2006. Solitons and periodic solutions of coupled nonlinear evolution equations by using the sine–cosine method. International Journal of Computer Mathematics, 83(12): 915-924. http://dx.doi.org/10.1080/00207160601138756.

      [4] Wang, M., Zhou, Y., & Li, Z., 1996. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Physics Letters A, 216(1): 67-75. http://dx.doi.org/10.1016/0375-9601(96)00283-6.

      [5] Wang, M., 1996. Exact solutions for a compound KdV-Burgers equation. Physics Letters A, 213(5): 279-287. http://dx.doi.org/10.1016/0375-9601(96)00103-X.

      [6] Zhang, S., & Xia, T., 2007. A generalized new auxiliary equation method and its applications to nonlinear partial differential equations. Physics Letters A, 363(5): 356-360. http://dx.doi.org/10.1016/j.physleta.2006.11.035.

      [7] Jiong, S., 2003. Auxiliary equation method for solving nonlinear partial differential equations. Physics Letters A, 309(5): 387-396.

      [8] Malfliet, W., 1992. Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7): 650-654. http://dx.doi.org/10.1119/1.17120.

      [9] Wazwaz, A. M., 2007. The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Applied Mathematics and Computation, 184(2): 1002-1014. http://dx.doi.org/10.1016/j.amc.2006.07.002.

      [10] Wazwaz, A. M., 2007. New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations. Applied Mathematics and Computation, 186(1): 130-141. http://dx.doi.org/10.1016/j.amc.2006.07.092.

      [11] Abdou, M. A., & Soliman, A. A., 2006. Modified extended tanh-function method and its application to nonlinear physical equations. Physics Letters A, 353(6): 487-492. http://dx.doi.org/10.1016/j.physleta.2006.01.013.

      [12] Elwakil, S. A., El-Labany, S. K., Zahran, M. A., & Sabry, R., 2005. Modified extended tanh-function method and its applications to nonlinear equations. Applied Mathematics and Computation, 161(2): 403-412. http://dx.doi.org/10.1016/j.amc.2003.12.035.

      [13] Zahran, E. H., & Khater, M. M., 2016. Modified extended tanh-function method and its applications to the Bogoyavlenskii equation. Applied Mathematical Modelling, 40(3): 1769-1775. http://dx.doi.org/10.1016/j.apm.2015.08.018.

      [14] Jawad, A. J. A. M., Petković, M. D., & Biswas, A., 2010. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation, 217(2): 869-877. http://dx.doi.org/10.1016/j.amc.2010.06.030.

      [15] Zayed, E. M., 2011. A note on the modified simple equation method applied to Sharma–Tasso–Olver equation. Applied Mathematics and Computation, 218(7): 3962-3964. http://dx.doi.org/10.1016/j.amc.2011.09.025.

      [16] Zayed, E. M. E., & Ibrahim, S. H., 2012. Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method. Chinese Physics Letters, 29(6): 060201. http://dx.doi.org/10.1088/0256-307X/29/6/060201.

      [17] Khan, K., & Akbar, M. A., 2014. Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method. Journal of the Association of Arab Universities for Basic and Applied Sciences, 15: 74-81. http://dx.doi.org/10.1016/j.jaubas.2013.05.001.

      [18] Al-Amr, M. O., 2015. Exact solutions of the generalized (2+1)-dimensional nonlinear evolution equations via the modified simple equation method. Computers & Mathematics with Applications, 69(5): 390-397. http://dx.doi.org/10.1016/j.camwa.2014.12.011.

      [19] Wang, M., Li, X., & Zhang, J., 2008. The (G’/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423. http://dx.doi.org/10.1016/j.physleta.2007.07.051.

      [20] Islam, S.R., 2015. The Traveling Wave Solutions of the Cubic Nonlinear Schrodinger Equation Using the Enhanced (G/G)-Expansion Method. World Applied Sciences Journal, 33(4), pp.659-667.

      [21] Khan, K., & Akbar, M. A., 2014. Traveling wave solutions of nonlinear evolution equations via the enhanced (G′/G)-expansion method. Journal of the Egyptian Mathematical Society, 22(2), 220-226. http://dx.doi.org/10.1016/j.joems.2013.07.009.

      [22] Akbar, M.A., Hj, N., Ali, M. and Mohyud-din, S.T., 2012. Some new exact traveling wave solutions to the (3+ 1)-dimensional Kadomtsev-Petviashvili equation. In World Appl. Sci. J.

      [23] Naher H, Abdullah F A, 2014. New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics. Journal of the Egyptian Mathematical Society, 22(3): 390-395. http://dx.doi.org/10.1016/j.joems.2013.11.008.

      [24] Wu, X. H. B., & He, J. H., 2008. Exp-function method and its application to nonlinear equations. Chaos, Solitons & Fractals, 38(3): 903-910. http://dx.doi.org/10.1016/j.chaos.2007.01.024.

      [25] Wu, X. H. B., & He, J. H., 2007. Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method. Computers & Mathematics with Applications, 54(7): 966-986. http://dx.doi.org/10.1016/j.camwa.2006.12.041.

      [26] Akbar, M. A., & Ali, N. H. M., 2014. Solitary wave solutions of the fourth order Boussinesq equation through the exp (–Ф (η))-expansion method. SpringerPlus, 3(1): 344. http://dx.doi.org/10.1186/2193-1801-3-344.

      [27] Hafez, M. G., Alam, M. N., & Akbar, M. A., 2015. Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. Journal of King Saud University-Science, 27(2): 105-112. http://dx.doi.org/10.1016/j.jksus.2014.09.001.

      [28] Hafez, M. G., & Akbar, M. A., 2015. An exponential expansion method and its application to the strain wave equation in microstructured solids. Ain Shams Engineering Journal, 6(2): 683-690. http://dx.doi.org/10.1016/j.asej.2014.11.011.

      [29] Wen-Hua, H., 2006. A generalized extended F-expansion method and its application in (2+ 1)-dimensional dispersive long wave equation. Communications in Theoretical Physics, 46(4): 580. http://dx.doi.org/10.1088/0253-6102/46/4/002.

      [30] Zhao, Y. M., 2013. F-expansion method and its application for finding new exact solutions to the Kudryashov-Sinelshchikov equation. Journal of Applied Mathematics. http://dx.doi.org/10.1155/2013/895760.

      [31] Islam, M. S., Khan, K., Akbar, M. A., & Mastroberardino, A., 2014. A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations. Royal Society open science, 1(2): 140038.

      [32] Akbar, M. A., & Ali, N. H. M., 2016. An ansatz for solving nonlinear partial differential equations in mathematical physics. SpringerPlus, 5(1): 1-13. http://dx.doi.org/10.1186/s40064-015-1652-9.

      [33] Triki, H., Jovanoski, Z., & Biswas, A., 2014. Shock wave solutions to the Bogoyavlensky–Konopelchenko equation. Indian Journal of Physics, 88(1): 71-74. http://dx.doi.org/10.1007/s12648-013-0380-7.

      [34] Singh, S.S., 2016. Solutions of Kudryashov-Sinelshchikov equation and generalized Radhakrishnan-Kundu-Lakshmanan equation by the first integral method. International Journal of Physical Research, 4(2), pp.37-42. http://dx.doi.org/10.14419/ijpr.v4i2.6202.

      [35] Najafi, M., Arbabi, S., & Najafi, M., 2013. New application of sine-cosine method for the generalized (2+ 1)-dimensional nonlinear evolution equations. International Journal of Advanced Mathematical Sciences, 1(2): 45-49. http://dx.doi.org/10.14419/ijams.v1i2.685.

      [36] Najafi, M., Najafi, M., & Arbabi, S., 2013. New Exact Solutions for the generalized (2+1)-dimensional Nonlinear Evolution Equations by Tanh-Coth Method. Int. J. Modern Theo. Physics, 2(2): 79-85.

      [37] Darvishi, M. T., Najafi, M., & Najafi, M., 2010. New application of EHTA for the generalized (2+ 1)-dimensional nonlinear evolution equations. International Journal of Mathematical and Computer Sciences, 6(3): 132-138.

      [38] Moatimid, G. M., El-Shiekh, R. M., & Al-Nowehy, A. G. A., 2013. Exact solutions for Calogero–Bogoyavlenskii–Schiff equation using symmetry method. Applied Mathematics and Computation, 220: 455-462. http://dx.doi.org/10.1016/j.amc.2013.06.034.

      [39] Wazwaz, A. M., 2008. Multiple-soliton solutions for the Calogero–Bogoyavlenskii–Schiff, Jimbo–Miwa and YTSF equations. Applied Mathematics and Computation, 203(2): 592-597. http://dx.doi.org/10.1016/j.amc.2008.05.004.

      [40] Bhrawy, A. H., Abdelkawy, M. A., & Biswas, A., 2013. Topological solitons and cnoidal waves to a few nonlinear wave equations in theoretical physics. Indian Journal of Physics, 87(11): 1125-1131. http://dx.doi.org/10.1007/s12648-013-0338-9.

  • Downloads

  • How to Cite

    Kumar, D., & Sarker, P. (2016). Investigation of exact traveling wave solution for the (2+1) dimensional nonlinear evolution equations via modified extended tanh-function method. International Journal of Physical Research, 4(2), 62-68. https://doi.org/10.14419/ijpr.v4i2.6588