Applying the Modified F-Expansion Method to Find the Exact Solutions of the Bogoyavlenskii Equation

Öz

The aim of this study is to obtain the new exact solutions of the Bogoyavlenskii equation (BE) using the modified F-expansion method. With the aid of symbolic computation, this method has been successfully implemented in the BE and the exact solutions obtained have been expressed by the hyperbolic functions, trigonometric functions, and rational functions. To the best of our knowledge, the BE has not been previously investigated by the modified F-expansion method. The findings of this study demonstrate that the suggested method is highly effective, powerful, and practical for obtaining the exact solutions of one dimensional and higher-dimensional nonlinear partial differential equations arising in mathematical physics and engineering.

Anahtar Kelimeler

• Bogoyavlenskii equation
• exact solution
• modified F-expansion method

Kaynakça

1. Jawad, A.J.M., Petkovic, M.D., Biswas, A., “Modified simple equation method for nonlinear evolution equations”, Appl. Math. Comput. 217 (2010) : 869-877.
2. Zayed, E.M.E., “A note on the modified simple equation method applied to Sharma–Tasso–Olver equation”, Appl. Math. Comput. 218 (2011) : 3962-3964.
3. Zayed, E.M.E., Al-Nowehy, A.G., “The modified simple equation method, the exp-function method and the method of soliton ansatz for solving the long-short wave resonance equations”, Z. Naturforsch. 71a (2016) : 103-112.
4. Alfalqi, S.H., Alzaidi, J.F., Lu, D., Khater, M.M.A., “On exact and approximate solutions of (2+1)-dimensional Konopelchenko-Dubrovsky equation via modified simplest equation and cubic B-spline schemes”, Therm Sci. 23 (2019) : 1889-1899.
5. Zhou, Y., Wang, M., Miao, T., “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations”, Phys. Lett. A 323 (2004) : 77-88.
6. Wang, M., Li, X., “Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation”, Chaos, Solitons & Fractals 24(5) (2005) : 1257-1268.
7. Zhang, J.L., Wang, M.L., Wang, Y.M., Fang, Z.D., “The improved F-expansion method and its applications”, Physics Letters A 350 (2006) : 103-109.
8. Zhang, S., “New exact solutions of the KdV-Burgwers-Kuramoto equation”, Phys. Lett. A 358 (2006) : 414-420.

9. Zhang, S., Xia, T., “A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations”, Appl. Math. Comput. 183 (2006) : 1190-1200.
10. Abdou, M.A., “An improved generalized F-expansion method and its applications”, Journal of Computational and Applied Mathematics 214(1) (2008) : 202-208 .
11. Wang, M., Li X., Zhang, J., “The (G^'/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics”, Physics Letters A 372(4) (2008) : 417-423.
12. Zhang, S., Wang, W., Tong, J.L., “A generalized (G^'/G) -expansion method and its application to the (2+1)-dimensional Broer-Kaup equations”, Applied Mathematics and Computation 209(2) (2009) : 399-404.
13. Zhang, J., Jiang, F., Zhao, X., “An improved (G'/G)-expansion method for solving nonlinear evolution equations”, International Journal of Computer Mathematics 87(8) (2010) : 1716-1725.
14. Feng, J., Wan, W., Li, Q., “Using (G'/G)-expansion method to seek traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation”, Applied Mathematics and Computation 217(12) (2011) : 5860-5865.
15. Kim, H., Sakthivel, R., “New exact traveling wave solutions of some nonlinear higher-dimensional physics models”, Reports on Mathematical Physics 70 (2012) : 39-50.
16. Alam, M.N., Li, X., “Exact traveling wave solutions to higher order nonlinear equations”, Journal of Ocean Engineering and Science 4(3) (2019) : 276-288.
17. Yokus, A., Durur, H., Ahmad, H., Yao, S.W., “Construction of different types analytic solutions for the Zhiber-Shabat equation”, Mathematics 8(6) (2020) : 908.
18. Akbar, M.A., Abdullah, F.A., Kumar, S., Gepreel, K.A., “Assorted soliton solutions to the nonlinear dispersive wave models in inhomogeneous media”, Results in Physics 39 (2022) : 105720.
19. He, J.H., Wu, X.H., “Exp-function method for nonlinear wave equations”, Chaos & Solitons and Fractals 30 (2006) : 700–708.
20. Abdou, M.A., “Generalized solitonary and periodic solutions for nonlinear partial differential equations by the exp-function method”, Nonlinear Dynamics 52 (2008) : 1-9.
21. Ma, W.X., Huang, T., Zhang, Y., “A multiple Exp-function method for nonlinear differential equations and its application”, Physica Scripta 82(6) (2010) : 065003.
22. Hosseini, K., Ansari, R., Samadani, F., Zabihi, A., Shafaroody A., Mirzazadeh M., “High-order dispersive cubic-quintic Schrödinger equation and its exact solutions”, Acta Physica Polonica A 136 (2019) : 203-207.
23. Hosseini, K., Osman, M.S., Mirzazadeh, M., Rabiei, F., “Investigation of different wave structures to the generalized third-order nonlinear Schrödinger equation”, Optik 206 (2020) : 164259.
24. Kudryashov, N.A., “One method for finding exact solutions of nonlinear differential equations”, Commun. Nonlinear Sci. Numer. Simul. 17(6) (2012) : 2248-2253.
25. Bashar, M.H., Islam, S.M.R., Kumar, D., “Construction of traveling wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation”, PDE in Applied Mathematics 4 (2021) : 100040.
26. Başhan, A., Esen, A., “Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times”, Numerical Methods for Partial Differential Equations 37 (2021) : 1561–1582.
27. Başhan, A., Yağmurlu N.M., Uçar Y., Esen A.” A new perspective for the numerical solution of the Modified Equal Width wave equation”, Mathematical Methods in the Applied Sciences 44 (2021) : 8925-8939.
28. Akram, G., Sadaf, M., Dawood, M., Abbas, M., Baleanu, D., “Solitary wave solutions to Gardner equation using improved tan((Ω(Υ))/2)-expansion method”, AIMS Mathematics 8(2) (2023) : 4390-4406.
29. Yildirim, O., “On the unique weak solvability of second-orderunconditionally stable difference scheme for the system of sine-Gordon equations”, Nonlinear Analysis: Modelling and Control 29(2) (2024) : 244-264.
30. Peng, Y. Z., Shen, M., “On exact solutions of the Bogoyavlenskii equation”, Pramana 67 (2006) : 449-456.
31. Malik, A., Chand, F., Kumar, H., Mishra, S.C., “Exact solutions of the Bogoyavlenskii equation using the multiple 𝐺′𝐺-expansion method”, Computers & Mathematics with Applications 64 (2012) :2850-2859.
32. Zahran, E.H.M., Khater, M.M.A., “Modified extended tanh-function method and its applications to the Bogoyavlenskii equation”, Applied Mathematical Modelling 40 (2016) : 1769-1775.
33. Alam, M.N., Tunc, C., “An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system”, Alexandria Engineering Journal 55 (2016) : 1855-1865.
34. Yu, J., Sun, Y., “Modified method of simplest equation and its applications to the Bogoyavlenskii equation”, Computers & Mathematics with Applications 72 (2016) : 1943-1955.
35. Inc, M., Hashemi, M.S., Aliyu, A.I., “Exact solutions and conservation laws of the Bogoyavlenskii Equation”, Acta Physica Polonica A 33 (2018) : 1133-1137.
36. Yokus, A., Durur, H., Ahmad, H., Thounthong P., Zhang Y.F., “Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G,1/G)-expansion and (1/G′)-expansion techniques” Results in Physics 19 (2020) : 103409.
37. Leta, T.D., Liu, W., Achab A.E., Rezazadeh H., Bekir A., “Dynamical Behavior of Traveling Wave Solutions for a (2+1)-Dimensional Bogoyavlenskii Coupled System”, Qualitative Theory of Dynamical Systems 20 (2021) : 14.
38. Zahran, E.H.M., Bekir, A., “New impressive vision solitary wave solutions of the Bogoyavlenskii equation in comparison with its numerical solutions”, Optical and Quantum Electronics 54 (2022) : 743.
39. Yue, Y., Huang, L., “On the similarity reduction solutions of the Bogoyavlenskii equation”, Applied Mathematics Letters 131 (2022) : 108050.
40. Zhang, G., Qi, Ji, Zhu, Q., “On the study of solutions of Bogoyavlenskii equation via improved 𝐺′/G^2 method and simplified tan(𝜙(𝜉)/2) method”, AIMS Mathematics 7 (2023) : 19649-19663.
41. Bogoyavlenskii, O.I., “Breaking solitons in (2+1)-dimensional integrable equations”, Russian Math. Surveys 45 (1990) : 1-86.
42. Ozisik, M., Secer, A., Bayram, M., “On solitary wave solutions for the extended nonlinear Schrödinger equation via the modified F‑expansion method”, Optical and Quantum Electronics 55 (2023) : 215.