Document Type : Original Article
Author
Giulan University
Abstract
In past years, various methods were applied to achieve the analytical solution of the fractional partial differential equation (FPDE). However, most have not been applied directly to manipulate the system of equations. In the current paper, the modified exp function method is expanded to attain the answer of a fractional two-dimensional system, without reducing fractional equations. Furthermore, the proposed methods have been used to achieve the analytical solutions of the coupled space Time-Fractional Boussinesq-Burgers System and coupled Time-Fractional Long System. The proposed methods are highly accurate, flexible, effective, and programmable to solve nonlinear evolution equations. Moreover, the plots of obtained solutions have been illustrated for some parameters.
Keywords
[1] Bhatter, S., Mathur, A., Kumar, D., & Singh, J. (2020). A new analysis of fractional Drinfeld–Sokolov–Wilson model with exponential memory. Physica A: Statistical Mechanics and its Applications, 537, 122578.
[2] Singh, Y., Gill, V., Singh, J., Kumar, D., & Khan, I. (2021). Computable generalization of fractional kinetic equation with special functions. Journal of King Saud University-Science, 33(1), 101221.
[3] Singh, J., Ahmadian, A., Rathore, S., Kumar, D., Baleanu, D., Salimi, M., & Salahshour, S. (2021). An efficient computational approach for local fractional Poisson equation in fractal media. Numerical Methods for Partial Differential Equations, 37(2), 1439-1448.
[4] Goswami, A., Singh, J., & Kumar, D. (2020). Numerical computation of fractional Kersten-Krasil'shchik coupled KdV-mKdV system occurring in multi-component plasmas. AIMS Mathematics, 5(3), 2346-2369.
[5] Djilali, S., & Ghanbari, B. (2021). Dynamical behavior of two predators–one prey model with generalized functional response and time-fractional derivative. Advances in difference Equations, 2021(1), 235.
[6] Darvishi, M. T., Najafi, M., & Najafi, M. (2012). Traveling wave solutions for the (3+ 1)-dimensional breaking soliton equation by (G’/G)-expansion method and modified F-expansion method. International Journal of Computational and Mathematical Sciences, 6(2), 64-69.
[7] Guner, O., Atik, H., & Kayyrzhanovich, A. A. (2017). New exact solution for space-time fractional differential equations via (G′/G)-expansion method. Optik, 130, 696-701.
[8] Guner, O., & Bekir, A. (2017). The Exp-function method for solving nonlinear space–time fractional differential equations in mathematical physics. Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, 277-282.
[9] Zheng, B. (2013). Exp‐function method for solving fractional partial differential equations. The Scientific World Journal, 2013(1), 465723.
[10] Guner, O. (2017). Exp-function method and fractional complex transform for space-time fractional KP-BBM equation. Communications in Theoretical Physics, 68(2), 149.
[11] Bekir, A., Guner, O., & Cevikel, A. (2016). The exp-function method for some time-fractional differential equations. IEEE/CAA Journal of Automatica Sinica, 4(2), 315-321.
[12] He, J. H., & Abdou, M. A. (2007). New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos, Solitons & Fractals, 34(5), 1421-1429.
[13] Khani, F., Hamedi-Nezhad, S., Darvishi, M. T., & Ryu, S. W. (2009). New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method. Nonlinear analysis: Real world applications, 10(3), 1904-1911.
[14] Shin, B. C., Darvishi, M. T., & Barati, A. (2009). Some exact and new solutions of the Nizhnik–Novikov–Vesselov equation using the Exp-function method. Computers & Mathematics with Applications, 58(11-12), 2147-2151.
[15] Wu, X. H. B., & He, J. H. (2008). Exp-function method and its application to nonlinear equations. Chaos, Solitons & Fractals, 38(3), 903-910.
[16] Darvishi, M. T., Najafi, M., & Najafi, M. (2011). Some new exact solutions of the (3+ 1)-dimensional breaking soliton equation by the Exp-function method. Nonlinear Sci. Lett. A, 2(4), 221-232.
[17] 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.
[18] Zhang, S. (2008). Application of Exp-function method to high-dimensional nonlinear evolution equation. Chaos, Solitons & Fractals, 38(1), 270-276.
[19] Darvishi, M. T., Najafi, M., & Najafi, M. (2011). Application of multiple exp-function method to obtain multi-soliton solutions of (2+ 1)-and (3+ 1)-dimensional breaking soliton equations. Am. J. Comput. Appl. Math, 1(2), 41-47.
[20] Ma, W. X., & Zhu, Z. (2012). Solving the (3+ 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Applied Mathematics and Computation, 218(24), 11871-11879.
[21] Zayed, E. M., & Al-Nowehy, A. G. (2015). The multiple exp-function method and the linear superposition principle for solving the (2+ 1)-dimensional Calogero–Bogoyavlenskii–Schiff equation. Zeitschrift für Naturforschung A, 70(9), 775-779.
[22] Adem, A. R. (2016). A (2+ 1)-dimensional Korteweg–de Vries type equation in water waves: Lie symmetry analysis; multiple exp-function method; conservation laws. International Journal of Modern Physics B, 30(28n29), 1640001.
[23] Zayed, E. M., Amer, Y. A., & Al-Nowehy, A. G. (2016). The modified simple equation method and the multiple exp-function method for solving nonlinear fractional Sharma-Tasso-Olver equation. Acta Mathematicae Applicatae Sinica, English Series, 32, 793-812.
[24] 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.
[25] Yildirim, Y., Yasar, E., & Adem, A. R. (2017). A multiple exp-function method for the three model equations of shallow water waves. Nonlinear Dynamics, 89, 2291-2297.
[26] Yıldırım, Y., & Yaşar, E. (2017). Multiple exp-function method for soliton solutions of nonlinear evolution equations. Chinese Physics B, 26(7), 070201.
[27] Liu, J. G., Zhou, L., & He, Y. (2018). Multiple soliton solutions for the new (2+ 1)-dimensional Korteweg–de Vries equation by multiple exp-function method. Applied Mathematics Letters, 80, 71-78.
[28] Bhatter, S., Mathur, A., Kumar, D., & Singh, J. (2020). A new analysis of fractional Drinfeld–Sokolov–Wilson model with exponential memory. Physica A: Statistical Mechanics and its Applications, 537, 122578.
[29] Saad, K. M., AL-Shareef, E. H., Alomari, A. K., Baleanu, D., & Gómez-Aguilar, J. F. (2020). On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger’s equations using homotopy analysis transform method. Chinese Journal of Physics, 63, 149-162.
[30] Liao, S. (2004). On the homotopy analysis method for nonlinear problems. Applied mathematics and computation, 147(2), 499-513.
[31] Rashidi, M. M., Domairry, G., DoostHosseini, A., & Dinarvand, S. (2008). Explicit approximate solution of the coupled KdV equations by using the homotopy analysis method. International Journal of Mathematical Analysis, 2(9-12), 581-589.
[32] Gao, Y. T., & Tian, B. (2001). Ion-acoustic shocks in space and laboratory dusty plasmas: Two-dimensional and non-traveling-wave observable effects. Physics of Plasmas, 8(7), 3146-3149.
[33] Hirota, R., & Satsuma, J. (1981). Soliton solutions of a coupled Korteweg-de Vries equation. Physics Letters A, 85(8-9), 407-408.
[34] Das, G. C., & Sarma, J. (1999). Response to" Comment on'A new mathematical approach for finding the solitary waves in dusty plasma'"[Phys. Plasmas 6, 4392 (1999)]. Physics of Plasmas, 6(11), 4394.
[35] Hirota, R. (2004). The direct method in soliton theory (No. 155). Cambridge university press.
[36] Tahami, M., & Najafi, M. (2017). Multi-wave solutions for the generalized (2+ 1)-dimensional nonlinear evolution equations. Optik, 136, 228-236.
[37] Ayub, K., Khan, M. Y., & Mahmood-Ul-Hassan, Q. (2017). Solitary and periodic wave solutions of Calogero–Bogoyavlenskii–Schiff equation via exp-function methods. Computers & mathematics with applications, 74(12), 3231-3241.
[38] Wazwaz, A. M. (2008). Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota’s method, tanh–coth method and Exp-function method. Applied Mathematics and Computation, 202(1), 275-286.
[39] Cheng, L., & Zhang, Y. (2015). Multiple wave solutions and auto-Bäcklund transformation for the (3+ 1)-dimensional generalized B-type Kadomtsev–Petviashvili equation. Computers & Mathematics with Applications, 70(5), 765-775.
[40] Yu, S. (2012). N-soliton solutions of the KP equation by Exp-function method. Applied Mathematics and Computation, 219(8), 3420-3424.
[41] Hamid, M., Usman, M., Zubair, T., Haq, R. U., & Shafee, A. (2019). An efficient analysis for N-soliton, Lump and lump–kink solutions of time-fractional (2+ 1)-Kadomtsev–Petviashvili equation. Physica A: Statistical Mechanics and its Applications, 528, 121320.
[42] Jumarie, G. (2007). Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution. Journal of Applied Mathematics and Computing, 24, 31-48.
[43] Momani, S., & Odibat, Z. (2007). Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Computers & Mathematics with Applications, 54(7-8), 910-919.
[44] Ayati, Z. (2023). Exact solutions of (2+ 1)-dimentional Sakovich equation using two well known methods. Computational Sciences and Engineering, 3(1), 163-175.
)