Periodic wave analysis to the time-fractional phi-four and (2+1) dimensional CBS equations

  • Authors

    • Samsun Nahar Ananna Hohai University, Nanjing, China
    • Abdulla Al – Mamun Hohai University, Nanjing, China
    • Tianqing An Hohai University, Nanjing, China
    2021-11-19
    https://doi.org/10.14419/ijpr.v9i2.31824
  • The ( )-Expansion Technique, Travelling Wave Solution, Phi-Four Equation, Periodic Wave, Calogero-Bogoyavlanskil Schilf Equation.

  • In this paper, we investigate some innovative and exact travelling wave solutions to the time-fractional phi-four equation and the (2+1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equation using the ( )-expansion method. Periodic solutions are displayed in hyperbolic, trigonometric, and rational function solutions. Subsequently, we construct some new solutions connecting the free parameters of the phi-four equation and the CBS equation, which are characterized into three complete forms: rational function, trigonometric function, and hyperbolic functions. Graphical representations of some attained solutions are also presented in this article. Hence, this study shows the efficiency and the easiness of the ( )-expansion technique with the assistance of emblematically computational software MATLAB and Mathematica.

     

     

  • References

    1. [1] Miller, M., Kenneth, S., & Bertram, R. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley.

      [2] Alam, L. M. B., Jiang, X., & Mamun, A. A. (2021). Exact and explicit travelling traveling wave solution to the time-fractional phi-four and (2+1) dimensional CBS equations using the modified extended tanh-function method in mathematical physics. Partial Differential Equations in Applied Mathematics, 4, 100039. https://doi.org/10.1016/j.padiff.2021.100039.

      [3] Mamun, A. A., An, T., Shahen, N. H. M., Ananna, S. N., Foyjonnesa, Hossain, M. F., & Muazu, T. (2020). Exact and explicit travelling-wave solutions to the family of new 3D fractional WBBM equations in mathematical physics. Results in Physics, 19, 103517. https://doi.org/10.1016/j.rinp.2020.103517.

      [4] Mamun, A. A., Ananna, S. N., An, T., Shahen, N. H. M., Asaduzzaman, M., & Foyjonnesa. (2021). Dynamical behaviour of travelling wave solutions to the conformable time-fractional modified Liouville and mRLW equations in water wave mechanics. Heliyon, 7(8), e07704. https://doi.org/10.1016/j.heliyon.2021.e07704.

      [5] Shahen, N. H. M., Foyjonnesa, Ali, M. S., Mamun, A. A., & Rahman, M. (2021). Interaction among lump, periodic, and kink solutions with dynamical analysis to the conformable time-fractional Phi-four equation. Partial Differential Equations in Applied Mathematics, 4, 100038. https://doi.org/10.1016/j.padiff.2021.100038.

      [6] Shahen, N. H. M., Foyjonnesa, Bashar, M. H., Ali, M. S., & Mamun, A. A. (2020). Dynamical analysis of long-wave phenomena for the nonlinear conformable space-time fractional (2+1)-dimensional AKNS equation in water wave mechanics. Heliyon, 6(10), e05276. https://doi.org/10.1016/j.heliyon.2020.e05276.

      [7] Alam, L. M. B., Xingfang, J., Mamun, A. A., & Ananna, S. N. (2021). Investigation of lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional CBS equations in mathematical physics. Partial Differential Equations in Applied Mathematics, 4, 100122. https://doi.org/10.1016/j.padiff.2021.100122.

      [8] Mamun, A. A., Ananna, S. N., An, T., Shahen, N. H. M., & Foyjonnesa. (2021). Periodic and solitary wave solutions to a family of new 3D fractional WBBM equations using the two-variable method. Partial Differential Equations in Applied Mathematics, 3, 100033. https://doi.org/10.1016/j.padiff.2021.100033.

      [9] Mamun, A.-A.-, Ali, M. S., & Miah, M. M. (2018). A study on an analytic solution 1D heat equation of a parabolic partial differential equation and implement in computer programming. International Journal of Scientific & Engineering Research, 9(9), 913–921.

      [10] Ananna, S. N., & Mamun, A.-A.-. (2020). Solution of Volterra’s integro-differential equations by using variational iteration method. International Journal of Scientific & Engineering Research, 11(3), 1–9.

      [11] Mamun, A.-A.-, Asaduzzaman, M., & Ananna, S. N. (2019). Solution of eighth order boundary value problem by using variational iteration method. International Journal of Mathematics and Computational Science, 5(2), 13–23.

      [12] Mamun, A.-A.-, & Asaduzzaman, M. (2019). Solution of seventh order boundary value problem by using variational iteration method. International Journal of Mathematics and Computational Science, 5(1), 6–12.

      [13] Yaşar, E., Yıldırım, Y., Zhou, Q., Moshokoa, S. P., Ullah, M. Z., Triki, H., Biswas, A., & Belic, M. (2017). Perturbed dark and singular optical solitons in polarization preserving fibers by modified simple equation method. Superlattices and Microstructures, 111, 487–498. https://doi.org/10.1016/j.spmi.2017.07.004.

      [14] Khater, M. M. A., & Kumar, D. (2017). New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water. Journal of Ocean Engineering and Science, 2(3), 223–228. https://doi.org/10.1016/j.joes.2017.07.001.

      [15] Ma, W.-X., Huang, T., & Zhang, Y. (2010). A multiple exp-function method for nonlinear differential equations and its application. Physica Scripta, 82(6), 065003. https://doi.org/10.1088/0031-8949/82/06/065003.

      [16] Khatun, M. S., Hoque, M. F., & Rahman, M. A. (2017). Multisoliton solutions, completely elastic collisions and non-elastic fusion phenomena of two PDEs. Pramana, 88(6), 1. https://doi.org/10.1007/s12043-017-1390-3.

      [17] Manafian, J., & Lakestani, M. (2016). The Classification of the Single Traveling Wave Solutions to the Modified Fornberg–Whitham Equation. International Journal of Applied and Computational Mathematics, 3(4), 3241–3252. https://doi.org/10.1007/s40819-016-0288-y.

      [18] Lakestani, M., & Manafian, J. (2017). Application of the ITEM for the modified dispersive water-wave system. Optical and Quantum Electronics, 49(4), 1. https://doi.org/10.1007/s11082-017-0967-x.

      [19] Li, Y., Ma, W.-X., & Zhang, J. E. (2000). Darboux transformations of classical Boussinesq system and its new solutions. Physics Letters A, 275(1–2), 60–66. https://doi.org/10.1016/S0375-9601(00)00583-1.

      [20] Rezazadeh, H., Osman, M. S., Eslami, M., Mirzazadeh, M., Zhou, Q., Badri, S. A., & Korkmaz, A. (2019). Hyperbolic rational solutions to a variety of conformable fractional Boussinesq-Like equations. Nonlinear Engineering, 8(1), 224–230. https://doi.org/10.1515/nleng-2018-0033.

      [21] Lu, D., Seadawy, A., & Arshad, M. (2017). Applications of extended simple equation method on unstable nonlinear Schrödinger equations. Optik, 140, 136–144. https://doi.org/10.1016/j.ijleo.2017.04.032.

      [22] Younis, M., & Zafar, A. (2013). The modified simple equation method for solving nonlinear Phi-Four equation. International Journal of Innovation and Applied Studies, 2(4), 661–664.

      [23] MA, W. E. N.-X. I. U., ZHOU, R. U. G. U. A. N. G., & GAO, L. I. A. N. G. (2009). EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS. Modern Physics Letters A, 24(21), 1677–1688. https://doi.org/10.1142/S0217732309030096.

      [24] Eslami, M., & Rezazadeh, H. (2015). The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo, 53(3), 475–485. https://doi.org/10.1007/s10092-015-0158-8.

      [25] Mahmood, S., Mahmood, S., Mahmood, S., & Mahmood, S. (2014). Mathematical analysis of the Generalized Benjamin and Burger-Kdv Equations via the Extended Trial Equation Method. Journal of the Association of Arab Universities for Basic and Applied Sciences, 16(1), 91–100. https://doi.org/10.1016/j.jaubas.2013.07.005.

      [26] Ma, W.-X. (2020). Symbolic Computation of Lump Solutions to a Combined Equation Involving Three Types of Nonlinear Terms. East Asian Journal on Applied Mathematics, 10(4), 732–745. https://doi.org/10.4208/eajam.151019.110420.

      [27] Yang, J.-Y., Ma, W.-X., & Khalique, C. M. (2020). Determining lump solutions for a combined soliton equation in (2+1)-dimensions. The European Physical Journal Plus, 135(6), 1. https://doi.org/10.1140/epjp/s13360-020-00463-z.

      [28] Ma, W.-X., & Lee, J.-H. (2009). A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation. Chaos, Solitons & Fractals, 42(3), 1356–1363. https://doi.org/10.1016/j.chaos.2009.03.043.

      [29] Zhou, Q., Mirzazadeh, M., Zerrad, E., Biswas, A., & Belic, M. (2015). Bright, dark, and singular solitons in optical fibers with spatio-temporal dispersion and spatially dependent coefficients. Journal of Modern Optics, 63(10), 950–954. https://doi.org/10.1080/09500340.2015.1111456.

      [30] Bekir, A. (2008). New solitons and periodic wave solutions for some nonlinear physical models by using the sine–cosine method. Physica Scripta, 77(4), 045008. https://doi.org/10.1088/0031-8949/77/04/045008.

      [31] Rezazadeh, H. (2018). New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Optik, 167, 218–227. https://doi.org/10.1016/j.ijleo.2018.04.026.

      [32] Younis, M., & Zafar, A. (2014). Exact Solution to Nonlinear Differential Equations of Fractional Order via ( )-Expansion Method. Applied Mathematics, 05(01), 1–6. https://doi.org/10.4236/am.2014.51001.

      [33] Mamun, A. A., Shahen, N. H. M., Ananna, S. N., Asaduzzaman, M., & Foyjonnesa. (2021). Solitary and periodic wave solutions to the family of new 3D fractional WBBM equations in mathematical physics. Heliyon, 7(7), e07483. https://doi.org/10.1016/j.heliyon.2021.e07483.

      [34] Rezazadeh, H., Korkmaz, A., Eslami, M., Vahidi, J., & Asghari, R. (2018). Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method. Optical and Quantum Electronics, 50(3), 150. https://doi.org/10.1007/s11082-018-1416-1.

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  • How to Cite

    Nahar Ananna, S., Al – Mamun, A., & An, T. (2021). Periodic wave analysis to the time-fractional phi-four and (2+1) dimensional CBS equations. International Journal of Physical Research, 9(2), 98-104. https://doi.org/10.14419/ijpr.v9i2.31824