Travelling Wave Solutions for Some Time-Fractional Nonlinear Differential Equations

Abstract

This study employs the powerful generalized Kudryashov method to address the challenges posed by fractional differential equations in mathematical physics. The main objective is to obtain new exact solutions for three important equations: the (3+1)-dimensional time fractional Jimbo-Miwa equation, the (3+1)-dimensional time fractional modified KdV-Zakharov-Kuznetsov equation, and the (2+1)-dimensional time fractional Drinfeld-Sokolov-Satsuma-Hirota equation. The generalized Kudryashov method is highly versatile and effective in addressing nonlinear problems, making it a pivotal component in our research. Its adaptability makes it useful in diverse scientific disciplines. The method simplifies complex equations, improving our analytical capabilities and deepening our understanding of system dynamics. Additionally, we define fractional derivatives using the conformable fractional derivative framework, providing a strong foundation for our mathematical investigations. This paper examines the effectiveness of the generalized Kudryashov method in solving complex challenges presented by fractional differential equations and aims to provide guidance for future studies.

Keywords

• Kudryashov method
• Time-fractional Jimbo-Miwa equation
• KdV-Zakharov-Kuznetsov equation
• Drinfeld-Sokolov-Satsuma-Hirota equation
• Conformable fractional derivative

References

1. Abdeljawad T. 2015. On conformable fractional calculus. J Comput Appl Math, 279: 57-66.
2. Alabedalhadi M, Al-Omari S, Al-Smadi M, Alhazmi S. 2023. Traveling wave solutions for time-fractional mKdV-ZK equation of weakly nonlinear ion-acoustic waves in magnetized electron–positron plasma. Symmetry, 15(2): 361.
3. Ali HS, Miah MM, Akbar MA. 2018. Study of abundant explicit wave solutions of the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) equation and the shallow water wave equation. Propuls Power Res, 7(4): 320-328.
4. Arafa AAM, Rida SZ, Mohamed H. 2011. Homotopy analysis method for solving biological population model. Commun Theor Phys, 56(5): 797.
5. Bulut H, Sulaiman TA, Baskonus HM. 2018. Dark, bright and other soliton solutions to the Heisenberg ferromagnetic spin chain equation. Superlattices Microstruct, 123: 12-19.
6. Ding S, Feng Q. 2014. New exact solutions for the DSSH equation. Int J Appl Sci Res Rev, 19(3): 194.
7. Ekici M, Ayaz F. 2017. Solution of model equation of completely passive natural convection by improved differential transform method. Res Eng Struct Mater, 3(1): 1-10.
8. Ekici M, Ünal M. 2020. Application of the exponential rational function method to some fractional soliton equations. Emerging Applications of Differential equations and Game Theory. IGI Global, Pennsylvania, US, pp: 13-32.

9. Ekici M, Ünal M. 2022. Application of the rational (G'/G)-expansion method for solving some coupled and combined wave equations. Commun Fac Sci Univ, 71(1): 116-132.
10. Ekici M. 2023. Exact solutions to some nonlinear time-fractional evolution equations using the generalized Kudryashov method in mathematical physics. Symmetry, 15(10): 1961.
11. He JH, Wu XH. 2006. Exp-function method for nonlinear wave equations. Chaos Solit Fractals, 30(3): 700-708.
12. Jiang X, Wang J, Wang W, Zhang H. 2023. A predictor–corrector compact difference scheme for a nonlinear fractional differential equation. Fractal Fract, 7(7): 521.
13. Kaplan M, Bekir A, Akbulut A. 2016. A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics. Nonlinear Dyn, 85: 2843-2850.
14. Khater MM. 2022. Abundant stable and accurate solutions of the three-dimensional magnetized electron-positron plasma equations. J Ocean Eng Sci, (In Press, Corrected Proof). https://doi.org/10.1016/j.joes.2022.03.001.
15. Korkmaz A. 2017. Exact solutions to (3+ 1) conformable time fractional Jimbo–Miwa, Zakharov–Kuznetsov and modified Zakharov–Kuznetsov equations. Commun Theor Phys, 67(5): 479.
16. Lazarus IJ, Bharuthram R, Hellberg MA. 2008. Modified Korteweg–de Vries–Zakharov-Kuznetsov solitons in symmetric twotemperature electron–positron plasmas. J Plasma Phys, 74: 519-529.
17. Mace RL, Hellberg MA. 2001. The Korteweg–de Vries–Zakharov–Kuznetsov equation for electron-acoustic waves. Phys Plasmas, 8(6): 2649-2656.
18. Naher H, Abdullah FA, Akbar MA, Mohyud-Din ST. 2012. Some new solutions of the higher-order Sawada-Kotera equation via the exp-function method. Middle-East J Sci Res, 11(12): 1659-1667.
19. Odibat Z, Momani S. 2008. A generalized differential transform method for linear partial differential equations of fractional order. Appl Math Lett, 21(2): 194-199.
20. Onder I, Secer A, Bayram M. 2023. Soliton solutions of time-fractional modified Korteweg-de-Vries Zakharov-Kuznetsov equation and modulation instability analysis. Phys Scr, 99: 015213.
21. Osman MS. 2019. New analytical study of water waves described by coupled fractional variant Boussinesq equation in fluid dynamics. Pramana, 93(2): 26.
22. Rehman H, Seadawy AR, Younis M, Rizvi S, Anwar I, Baber M, Althobaiti A. 2022. Weakly nonlinear electron-acoustic waves in the fluid ions propagated via a (3+1)-dimensional generalized Korteweg–de-Vries–Zakharov-Kuznetsov equation in plasma physics. Results Phys, 33: 105069.
23. Roshid HO, Hoque MF, Alam MN, Akbar MA. 2014. New extended (G’/G)-expansion method and its application in the (3+ 1)-dimensional equation to find new exact traveling wave solutions. J Comput Maths, 2: 32-37.
24. Senol M, Az-Zobi E, Akinyemi L, Alleddawi A. 2021. Novel soliton solutions of the generalized (3+ 1)-dimensional conformable KP and KP–BBM equations. Comput Sci Eng, 1(1): 1-29.
25. Tian Q, Yang X, Zhang H, Xu D. 2023. An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties. Comput Appl Math, 42(6): 246.
26. Tuluce Demiray S, Pandir Y, Bulut H. 2014. Generalized Kudryashov method for time-fractional differential equations. Abstr Appl Anal, 2014: 901540.
27. Ünal M, Ekici M. 2021. The Double (G'/G, 1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential equations. J Sci Technol, 11(1): 599-608.
28. Wang M, Zhou Y, Li Z. 1996. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys Lett A, 216(1-5): 67-75.
29. Yang X, Wu L, Zhang H. 2023. A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl Math Comput, 457: 128192.
30. Younas U, Ren J, Baber MZ, Yasin MW. Shahzad, T. 2023. Ion-acoustic wave structures in the fluid ions modeled by higher dimensional generalized Korteweg-de Vries–Zakharov–Kuznetsov equation. J Ocean Eng Sci, 8(6): 623-635.
31. Zhang S, Tong JL, Wang W. 2008. A generalized (G'/G)-expansion method for the mKdV equation with variable coefficients. Phys Lett A, 372(13): 2254-2257.
32. Zhou TY, Tian B, Zhang CR, Liu SH. 2022. Auto-Bäcklund transformations, bilinear forms, multiple-soliton, quasi-soliton and hybrid solutions of a (3+ 1)-dimensional modified Korteweg-de Vries Zakharov-Kuznetsov equation in an electron-positron plasma. Eur Phys J Plus, 137: 912.