Application of the Advanced exp(-φ(ξ))-Expansion Method to the Nonlinear Conformable Time-Fractional Partial Differential Equations

Year 2021, Volume: 13 Issue: 1, 68 - 80, 30.06.2021

Md. Habibul Bashar Tasnim Tahseen Nur Hasan Shahen

Abstract

With the assistance of representative calculation programming, the present paper examines the careful voyaging wave arrangements from the nonlinear time fractional modified Kawahara equation by utilizing the advanced $\text{ exp}( -\varphi ( \xi ) $-expansion strategy in-terms of hyperbolic, trigonometric and rational function with some appreciated parameters. The dynamics nonlinear wave solution is examined and demonstrated by maple18 in 3-D, 2-d plots and contour plot with specific values of the intricate parameters are plotted. The advanced $\text{ exp}( -\varphi ( \xi ) )$-expansion method is reliable treatment for searching essential nonlinear waves that enrich a variety of dynamic models that arises in engineering fields.

Keywords

Conformable fractional derivative, advanced $\text{exp}( -\varphi ( \xi ))$-expansion method, exact solution, time fractional modified Kawahara equation

References

• [1] Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279(2015), 57-69. doi: 10.1016/j.cam.2014.10.016
• [2] Aksoy, E., Kaplan, M., Bekir, A., Exponential rational function method for spaceOˆC¸ oˆtime fractional dierential equations, Waves Random Complex , 26(2)(2016), 142-151. doi: 10.1080/17455030.2015.1125037
• [3] Al-Amr, M., Exact solutions of the generalized (2+1)-dimensional nonlinear evolution equations via the modified simple equation method, Comput. Math. Appl., 69(5)(2015), 390-397. doi: 10.1016/j.camwa.2014.12.011
• [4] Amfilokhiev,, B., Voitkunskii, I., Mazaeva, P., Khodorkovskii, S., Flows of polymer solutions in the case of convective accelerations, Tr. Leningr. Korablestroit. Inst., 96(1975), 3-9.
• [5] Atangana, A., Bildik, N., Noutchie., S.C.O., New Iteration Methods for Time-Fractional Modified Nonlinear Kawahara Equation, Abstr. Appl. Anal., 2014. doi: 10.1155/2014/740248
• [6] Bashar, M.H., Islam, S.M.R., Kumar, D., Construction of Traveling Wave Solutions of the (2+1)-Dimensional Heisenberg Ferromagnetic Spin Chain Equation, Partial Differential Equations in Applied Mathematics, 2021. doi: 10.1016/j.padiff.2021.100040
• [7] Bashar, M.H., Islam, S.M.R., Exact solutions to the (2 + 1)-Dimensional Heisenberg ferromagnetic spin chain equation by using modified simple equation and improve F-expansion methods, Physics Open, 5(2020). doi: 10.1016/j.physo.2020.100027
• [8] Baskonus, H.M., Ghavi S., New singular soliton solutions to the longitudinal wave equation in a magneto-electro-elastic circular rod with MM-derivative, Modern Physics Letters B., 33(21)(2019), 1-16.
• [9] Baskonus, H.M., Complex Soliton Solutions to the Gilson-Pickering Model, Axioms, 8(1)(2019), 18. doi: 10.3390/axioms8010018
• [10] Biswas, A., Mirzazadeh, M., Eslami, M., Zhou, Q., Bhrawy, A., Belic, M.,Optical solitons in nano-fibers with spatio-temporal dispersion by trial solution method, Optik, 127(18)(2016), 7250-7257. doi: 10.1016/j.ijleo.2016.05.052
• [11] Eslami M., Rezazadeh H., The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo, 53(3)(2016), 475-485. doi: 10.1007/s10092-015-0158-8
• [12] Feng, L., Zhang, T., Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrodinger equation, Appl. Math. Lett, 78(2018), 133-140.
• [13] Gao W., Ismael, H.F., Bulut, H., Baskonus, H.M., Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media, Physica Scripta, 95(3)(2019). doi: 10.1016/j.rinp.2019.102555
• [14] Gao,W., Partohaghighi, M., Baskonus, H.M., Ghavi S., Regarding the group preserving scheme and method of line to the numerical simulations of Klein-Gordon model, Results in Physics, 15(2019), 1-7.
• [15] Gao W., Yel G., Baskonus, H.M., Cattani, C., Complex solitons in the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation, Aims Mathematics, 15(1)(2020), 507-521.
• [16] Guirao J.L.G., Baskonus H.M., Kumar A., Rawat M.S., Yel G., Complex patterns to the (3+1)-dimensional B-type Kadomtsev-Petviashvili- Boussinesq equation, Symmetry, 12(1)(2020), 17.
• [17] Guner, O., Hasan, A., Soliton solution of fractional-order nonlinear differential equations based on the exp-function method, Optik, 127(20)(2016), 10076-10083. doi: 10.1016/j.ijleo.2016.07.07
• [18] Guo, S., Mei, L.Q., Li, Y., Sun, Y.F., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys. Lett. A, 376(2012), 407-411. doi: 10.1016/j.physleta.2011.10.056
• [19] Heris, J., Zamanpour, I., Analytical treatment of the coupled Higgs equation and the Maccari system via exp-function method, Acta Universitatis Apulensis, 33(2013), 203-216.
• [20] Ilhan, O.A., Esen, A., Bulut, H., Baskonus, H.M., Singular solitons in the pseudo-parabolic model arising in nonlinear surface waves, Results in Physics, 12(2019), 1712-1715. doi: 10.1016/j.rinp.2019.01.059
• [21] Jumarie., G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 51(9)(2006), 1367-1376. doi: 10.1016/j.camwa.2006.02.001
• [22] Kadkhoda, N., Jafari, H., Analytical solutions of the Gerdjikov-Ivanov equation by using exp $( -\varphi ( \xi ) )$-expansion method, Optik, 139(2017), 72-76. doi: 10.1016/j.ijleo.2017.03.078
• [23] Khater, M.M.A. Kumar, D., Implementation of three reliable methods for finding the exact solutions of (2+1) dimensional generalized fractional evolution equations, Optical and Quantum Electronics, 49(12)(2017), 427. doi: 10.1007/s11082-017-1249-3
• [24] Khalil, R.M., Al Horani, A., Yousef, M., Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264(2014), 65-75. doi: 10.1016/j.cam.2014.01.002
• [25] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Dierential Equations, Elsevier, Amsterdam, 204, 2006.
• [26] Zhang, S., Zhang, H-Q., Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Letter A., 375(7)(2011), 1069-1073. doi: 10.1016/j.physleta.2011.01.029
• [27] Kumar, D., Ray, S.C., Application of the extended exp $( -\varphi ( \xi ) )$-expansion method to the nonlinear conformable time-fractional partial differential equations, 7(2)(2019), 81-93.
• [28] Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Fractional Dierential Equations, Wiley, New York, 4(1)(1993).
• [29] Najafi M., Najafi M., Arbabi S., New exact solutions for the generalized (2+1)-dimensional nonlinear evolution equations by Tanh-Coth method, Int. J. Modern Theoretical Phys., 2(2)(2013), 79-85.
• [30] Najafi M., Arbabi S., Najafi M., New application of sine-cosine method for the generalized (2+ 1) dimensional nonlinear evolution equations, Int. J. Adv. Math. Sci., 1(2)(2013), 45-49. doi: 10.1155/2013/746910
• [31] Podlubny, I., Fractional Differential Equations, Academic Press, California, 6(3)(1999).
• [32] Rahhman, M.M., Aktar, A., Roy, K.C., Analytical solutions of nonlinear coupled Schrodinger-KdV equation via advance exponential expansion, American Journal of Mathematical and Computer Modelling, 3(3)(2018), 46-51.
• [33] Ray, S.S., Sahoo, S., New exact solutions of fractional Zakharov-Kuznetsov and modified ZakharovOÇ oKuznetsov equations using fractional sub-equation method, Commun. Theor. Phys., 6325(1)(2015). doi: 10.1088/0253-6102/63/1/05
• [34] Roshid, H., Hoque, M., Akbar, M., New extended (G0=G)-expansion method for traveling wave solutions of nonlinear partial differential equations (NPDEs) in mathematical physics, Italian. J. Pure Appl. Math., 33(2014), 175-190.
• [35] Roshid, M., Roshid, H, Exact and explicit traveling wave solutions to two nonlinear evolution equations which describe incompressible viscoelastic Kelvin-Voigt fluid, Heliyon, 4(8)(2018). doi: 10.1016/j.heliyon.2018.e00756
• [36] Shuwei, X., Jingsong, H., The rogue wave and breather solution of the Gerdjikov-Ivanov equation, Journal of Mathematical Physics, 53(2012). doi: 10.1063/1.4726510
• [37] Turgut A., Aydemir T., Saha A., Kara A. ,Propagation of nonlinear shock waves for the generalised Oskolkov equation and its dynamic motions in the presence of an external periodic perturbation, Pramana-J. Phys., 78-90, 2018.
• [38] Wang, G.W., Xu, T.Z., The improved fractional sub-equation method and its applications to nonlinear fractional partial differential equations, Rom. Rep. Phys., 6(3)(2014), 595-602.
• [39] Yasar E., Yildirim Y., Adem A., Perturbed optical solitons with spatio-temporal dispersion in (2 + 1)-dimensions by extended Kudryashov method, Optik, 158(2018), 1-14. doi: 10.1016/j.ijleo.2017.11.205
• [40] Yel, G., Baskonus, H.M., Bulut, H., Regarding some novel exponential travelling wave solutions to the Wu-Zhang system arising in nonlinear water wave model, Indian Journal of Physics, 93(8)(2019), 1031-1039.
• [41] Yokus, A., Numerical solution for space and time fractional order Burger type equation, Alexandria engineering J., 57(3)(2018), 2085-2091. doi: 10.1016/j.aej.2017.05.028
• [42] Zhao, Y., New exact solutions for a higher-order wave equation of KdV type using the multiple simplest equation method, Journal of Applied Mathematics, (2014), 1-13.