Modified simple equation method (MSEM) for solving nonlinear (3+1)-dimensional space-time fractional equations

Document Type : Research Paper


Department of Mathematics, Takestan branch, Islamic Azad University, Takestan, Iran


In the present paper, modified simple equation method (MSEM) is implemented for obtaining exact solutions of three nonlinear (3 + 1)-dimensional space-time fractional equation, namely three types of modified Korteweg-de-Vries (mKdV) equations. Here, the derivatives are of the type of conformable fractional derivatives. The solving process produces a system of algebraic equations which is possible to be easily with no need of using software for determining unknown coefficients. Results show that this method can supply a powerful mathematical tool to construct exact solutions of mKdV equations and it can be employed for other nonlinear (3 + 1) - dimensional space-time fractional equations.


Main Subjects

  • [1] P. Agarwal, S. Deniz,and S. Jain, AA. Alderremy ,and S. Aly., A new analysis of a partial differential equation arising in biology and population genetics via semi analytical techniques, Physica A: Statistical Mechanics and its Applications, 15(542) (2020) , 122769.
  • [2] M. S. Ahmed, A. A. Zaghrout, and H. M. Ahmed, Travelling wave solutions for the doubly dispersive equation using improved modified extended tanh-function method, Alexandria Engineering Journal, 61(10) (2022), 7987– 7994.
  • [3] G. Akram, M. Sadaf, S. Arshed, and F. Sameen, Bright, dark, kink singular and periodic soliton solutions of Lakshmanan-Porsezian-Daniel model by generalized projective Riccati equations method, Optik, 241 (2021), 167051.
  • [4] G. Akram, M. Sadaf, and I. Zainab, The dynamical study of Biswas-Arshed equation via modified auxiliary equation method, Optik, 255 (2022), 168614.
  • [5] N. H. Aljahdaly and F. O. Al Zobidi, On the Schro¨dinger equation for deep water waves using the Pad´e-Adomian decomposition method, Journal of Ocean Engineering and Science, (2022).
  • [6] J. Biazar and Z. Ayati, Improved G’/G-expansion method and comparing with tanh-coth method, Applications and Applied Mathematics: An International Journal (AAM), 6(1) (2011), 20.
  • [7] N. Bildik and S. Deniz, A comparative study on solving fractional cubic isothermal autocatalytic chemical system via new efficient technique, Chaos, Solitons & Fractals, 132 (2020), 109555.
  • [8] E. Bonyah, A. K. Sagoe, D. Kumar, and S. Deniz, Fractional optimal control dynamics of coronavirus model with Mittag-Leffler law, Ecological Complexity, 45 (2021), 100880.
  • [9] J. L. Brent, E. N. Onder, and A. P. Andrew, Chapter 12 - Nonlinear differential equations, Advanced Mathematics for Engineering Students, Butterworth-Heinemann, (2022), 329–347.
  • [10] C. Chen, Singular solitons of Biswas-Arshed equation by the modified simple equation method, Optik, 184 (2019), 412–420.
  • [11] M. De la Sen, S. Deniz, and H. S¨ozen, A new efficient technique for solving modified Chua’s circuit model with a new fractional operator., Advances in Difference Equations, 1 (2021), 1–16.
  • [12] S. Deniz, A. Konuralp, and M. De la Sen, Optimal perturbation iteration method for solving fractional model of damped Burgers’ equation, Symmetry, 12(6) (2020), 958.
  • [13] S. Deniz and M. Sezer, Rational Chebyshev collocation method for solving nonlinear heat transfer equations, International Communications in Heat and Mass Transfer, 114 (2020), 104595.
  • [14] A. El Achab, Constructing of exact solutions to the nonlinear Schr¨odinger equation (NLSE) with power-law nonlinearity by the Weierstrass elliptic function method, Optik, 127(3) (2016), 1229–1232.
  • [15] E. M. Eskandari and N. Taghizadeh, Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method., Applications and Applied Mathematics: An International Journal (AAM), 15(2) (2020), 15.
  • [16] M. Eslami and M. Mirzazadeh, Exact solutions of modified Zakharov-Kuznetsov equation by the homogeneous balance method, Ain Shams Engineering Journal, 5(1) (2014), 221–225.
  • [17] O. Gonzalez-Gaxiola, A. Biswas, M. Ekici, and S. Khan, Highly dispersive optical solitons with quadratic-cubic law of refractive index by the variational iteration method, Journal of Optics, 51(1) (2022), 29–36.
  • [18] W. Hereman, R. Martino, J. Miller, L. Hong, S. Formenac, and A. Menz, Exact solutions of nonlinear partial differential equations The tanh/sech method, Math. Visiting Scholar Grant Program Wolfram Research Inc., Champaign, Illinois, 25 (2000), 1–26.
  • [19] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, Journal of Mathematical Physics, 14(7) (1973), 805–809.
  • [20] A. J. A. M. Jawad, M. D. Petkovi´c, and A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217(2) (2010), 869–877.
  • [21] T. A. Khalil, N. Badra, H. M. Ahmed, and W. B. Rabie, Optical solitons and other solutions for coupled system of nonlinear Biswas-Milovic equation with Kudryashov’s law of refractive index by Jacobi elliptic function expansion method, Optik, (2022), 168540.
  • [22] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, Journal of computational and applied mathematics, 264 (2014), 65–70.
  • [23] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, 204 (2006). elsevier.
  • [24] W. X. Ma, Soliton solutions by means of Hirota bilinear forms, Partial Differential Equations in Applied Mathematics, 5 (2022), 100220.
  • [25] S. K. Mohanty, S. Kumar, A. N. Dev, M. K. Deka, D. V. Churikov, and O. V. Kravchenko, An efficient technique of G0G-expansion method for modified KdV and Burgers equations with variable coefficients, Results in Physics, (2022), 105504.
  • [26] R. I. Nuruddeen, Multiple soliton solutions for the (3 + 1) conformable space-time fractional modified kortewegde-vries equations, Journal of Ocean Engineering and Science, 3(1) (2018), 11–18.
  • [27] H. M. Srivastava, S. Deniz, and K. M. Saad, An efficient semi-analytical method for solving the generalized regularized long wave equations with a new fractional derivative operator, Journal of King Saud University-Science, 33(2) (2021), 101345.
  • [28] N. Taghizadeh, M. Mirzazadeh, and F. Farahrooz, Exact soliton solutions for second-order Benjamin-Ono equation, Applications and Applied Mathematics: An International Journal (AAM), 6(1) (2011), 31.
  • [29] N. Taghizadeh, M. Mirzazadeh, AS. Paghaleh, and J. Vahidi, Exact solutions of nonlinear evolution equations by using the modified simple equation method, Ain Shams Engineering Journal, 3(3) (2012), 321–5.
  • [30] A. C. Varsoliwala and T. R. Singh, Mathematical modeling of atmospheric internal waves phenomenon and its solution by Elzaki Adomian decomposition method, Journal of Ocean Engineering and Science, 7(3) (2022), 203– 212.
  • [31] A. M. Wazwaz, Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation, Open Engineering, 7(1) (2017), 169–174.
  • [32] A. Zulfiqar and J. Ahmad, Soliton solutions of fractional modified unstable Schr¨odinger equation using Expfunction method, Results in Physics, 19 (2020), 103476.