Analytical and numerical study on the solutions of a new (2+1)-dimensional conformable shallow water wave equation

Abstract

The (2+1)-dimensional conformable nonlinear shallow water wave equation is examined in this work. Initially, definitions and properties of suitable derivatives are presented. Subsequently, exact solutions to this equation are derived using the exp(–ϕ(ξ))-expansion and the modified extended tanh function methods. Then, a numerical method, namely the residual power series method, is utilized to obtain approximate solutions. The interplay between analytical and numerical approaches is explored to validate the solutions. This study fills a gap in the literature on fractional shallow water models, particularly in (2+1) dimensions, and offers new insights into wave dynamics governed by fractional derivatives. The physical implications of the findings are illustrated through 3D and 2D contour surfaces of some obtained data, offering insight into the physical interpretation of geometric structures. A table is also presented to compare the obtained results. These solutions highlight the practical uses of the investigated model and other nonlinear models in applied sciences. These techniques can potentially yield significant results in solving various fractional differential equations.

Keywords

• The exp(–φ(ξ))-expansion method
• modified extended tanh-function method
• residual power series method
• shallow water wave equation
• conformable derivative

References

1. Akbar, M. A., Mohd Ali, N. H., Solitary wave solutions of the fourth order Boussinesq equation through the exp(−ϕ(η))-expansion method, SpringerPlus, 3(1) (2014), 1-6. https://doi.org/10.1186/2193-1801-3-344
2. Akinyemi, L., Veeresha, P., Şenol, M., Rezazadeh, H., An efficient technique for generalized conformable Pochhammer-Chree models of longitudinal wave propagation of elastic rod, Indian J. Phys., 96(14) (2022), 4209-4218. https://doi.org/10.1007/s12648-022-02324-0
3. Akinyemi, L., Shallow ocean soliton and localized waves in extended (2+1)-dimensional nonlinear evolution equations, Phys. Lett. A, 463 (2023), 128668. https://doi.org/10.1016/j.physleta.2023.128668
4. Alam, L. M. B., Jiang, X., Exact and explicit traveling wave solution to the time-fractional phi-four and (2+1)-dimensional CBS equations using the modified extended tanh-function method in mathematical physics, PDE Appl. Math., 4 (2021), 100039. https://doi.org/10.1016/j.padiff.2021.100039
5. Alam, M. N., Iqbal, M., Hassan, M., Fayz-Al-Asad, M., Hossain, M. S., Tunç, C., Bifurcation, phase plane analysis and exact soliton solutions in the nonlinear Schrodinger equation with Atangana’s conformable derivative, Chaos Solit. Fractals, 182 (2024), 114724. https://doi.org/10.1016/j.chaos.2024.114724
6. Alquran, M., Analytical solutions of fractional foam drainage equation by residual power series method, Math. Sci., 8(4) (2014), 153-160. https://doi.org/10.1007/s40096-015-0141-1
7. Alquran, M., Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method, J. Appl. Anal. Comput., 5(4) (2015), 589-599. https://doi.org/10.11948/2015046
8. Arqub, O. A., Series solution of fuzzy differential equations under strongly generalized differentiability, J. Adv. Res. Appl. Math., 5(1) (2013), 31-52. 10.5373/jaram.1447.051912

9. Arqub, O. A., El-Ajou, A., Bataineh, A. S., Hashim, I., A representation of the exact solution of generalized Lane-Emden equations using a new analytical method, Abstr. Appl. Anal., 2013, 365725. https://doi.org/10.1155/2013/378593
10. Arqub, O. A., El-Ajou, A., Momani, S., Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, J. Comput. Phys., 293 (2015), 385-399. https://doi.org/10.1016/j.jcp.2014.09.034
11. Az-Zobi, E. A., Alzoubi, W. A., Akinyemi, L., Şenol, M., Masaedeh, B. S., A variety of wave amplitudes for the conformable fractional (2+1)-dimensional Ito equation, Mod. Phys. Lett. B, 35(15) (2021), 2150254. https://doi.org/10.1142/S0217984921502547
12. Cenesiz, Y., Kurt, A., New fractional complex transform for conformable fractional partial differential equations, J. Appl. Math., Statistics and Informatics, 12(2) (2016), 41-47. https://doi.org/10.1515/jamsi-2016-0007
13. Duran, S., Durur, H., Yavuz, M., Yokus, A., Discussion of numerical and analytical techniques for the emerging fractional order Murnaghan model in materials science, Opt. Quantum Electron., 55(6) (2023), 571. https://doi.org/10.1007/s11082-023-04838-1
14. Duran, S., Yokus, A., Kilinc, G., A study on solitary wave solutions for the Zoomeron equation supported by two dimensional dynamics, Phys. Scripta, 98(12) (2023), 125265. https://doi.org/10.1088/1402-4896/ad0c3c
15. Fang, J., Nadeem, M., Habib, M., Akgül, A., Numerical investigation of nonlinear shock wave equations with fractional order in propagating disturbance, Symmetry, 14(6) (2022), 1179. https://doi.org/10.3390/sym14061179
16. Gencyigit, M., Senol, M., Koksal, M. E., Analytical solutions of the fractional (3+1)-dimensional Boiti-Leon-Manna- Pempinelli equation, CMDE, 11(3) (2023), 564-575. https://doi.org/10.22034/cmde.2023.54758.2278
17. Hilfer, R., Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B, 104(16) (2000), 3914-3917. https://doi.org/10.1021/jp9936289
18. Iqbal, M., Alam, M. N., Lu, D., Seadawy, A. R., Alsubaie, N. E., Ibrahim, S., Applications of nonlinear longitudinal wave equation with periodic optical solitons wave structure in magneto electro elastic circular rod, Opt. Quantum Electron., 56(6) (2024), 1-22. 10.1007/s11082-024-06671-6
19. Iqbal, M., Nur Alam, M., Lu, D., Seadawy, A. R., Alsubaie, N. E., Ibrahim, S., On the exploration of dynamical optical solitons to the modify unstable nonlinear Schr¨odinger equation arising in optical fibers, Opt. Quantum Electron., 56(5) (2024), 765. https://doi.org/10.1007/s11082-024-06468-7
20. Isah, M. A., Yokus, A., Optical solitons of the complex Ginzburg-Landau equation having dual power nonlinear form using $φ^6$-model expansion approach, Math. Model. Numer. Simul. Appl., 3(3) (2023), 188-215. https://doi.org/10.53391/mmnsa.1337648
21. Islam, M. R., Application of Exp(ϕ(ξ))-expansion method for Tzitzeica type nonlinear evolution equations, J. Found. Appl. Phys., 4(1) (2016), 8-18.
22. Ismael, H. F., The (3+1)-dimensional Boussinesq equation: Novel multi-wave solutions, Results Phys., 53 (2023), 106965. https://doi.org/10.1016/j.rinp.2023.106965
23. Kadkhoda, N., Jafari, H., Analytical solutions of the Gerdjikov-Ivanov equation by using exp(−ϕ(ξ))-expansion method, Optik, 139 (2017), 72-76. https://doi.org/10.1016/j.ijleo.2017.03.078
24. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. https://doi.org/10.1016/j.cam.2014.01.002
25. Liu, J., Nadeem, M., Habib, M., Akgül, A., Approximate solution of nonlinear time-fractional Klein-Gordon equations using Yang transform, Symmetry, 14(5) (2022), 907. https://doi.org/10.3390/sym14050907
26. Luo, X., Nadeem, M., Inc, M., Dawood, S., Fractional complex transform and homotopy perturbation method for the approximate solution of Keller-Segel model, J. Funct. Spaces, 2022(1) (2022), 9637098. https://doi.org/10.1155/2022/9637098
27. Ma, Z., Chen, B., Bi, Q., Multiple rogue wave solutions for a modified (2+1)-dimensional nonlinear evolution equation, Nonlinear Dyn., 1-11 (2024). https://doi.org/10.1007/s11071-024-10076-1
28. Mirzazadeh, M., Akinyemi, L., Senol, M., Hosseini, K., A variety of solitons to the sixth-order dispersive (3+1)- dimensional nonlinear time-fractional Schrödinger equation with cubic-quintic-septic nonlinearities, Optik, 241 (2021), 166318. https://doi.org/10.1016/j.ijleo.2021.166318
29. Nisar, K. S., Akinyemi, L., Inc, M., S¸enol, M., Mirzazadeh, M., Houwe, A., Abbagari, S., Rezazadeh, H., New perturbed conformable Boussinesq-like equation: Soliton and other solutions, Results Phys., 33 (2022), 105200. https://doi.org/10.1016/j.rinp.2022.105200
30. Raslan, K. R., Ali, K. K., Shallal, M. A., The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations, Chaos Solit. Fractals, 103 (2017), 404-409. https://doi.org/10.1016/j.chaos.2017.06.029
31. Razzaq, W., Habib, M., Nadeem, M., Zafar, A., Khan, I., Mwanakatwea, P. K., Solitary wave solutions of conformable time fractional equations using modified simplest equation method, Complexity, 2022(1) (2022), 8705388. https://doi.org/10.1155/2022/8705388
32. Roshid, M. M., Alam, M. N., İlhan, O. A., Rahim, M. A., Tuhin, M. M. H., Rahman, M. M., Modulation instability and comparative observation of the effect of fractional parameters on new optical soliton solutions of the paraxial wave model, Opt. Quantum Electron., 56(6) (2024), 1010. https://doi.org/10.1007/s11082-024-06921-7
33. Senol, M., New analytical solutions of fractional symmetric regularized-long-wave equation, Revista Mexicana de Fisica, 66(3) (2020), 297-307. https://doi.org/10.31349/revmexfis.66.297
34. Senol, M., Analytical and approximate solutions of (2+1)-dimensional time-fractional Burgers Kadomtsev Petviashvili equation, Commun. Theor. Phys., 72(5) (2020), 055003. https://doi.org/10.1088/15729494/ab7707
35. Yel, G., On the new travelling wave solution of a neural communication model, Balikesir Univ. J. Sci. Eng., 21(2) (2019), 666-678. https://doi.org/10.25092/baunfbed.636782
36. Yel, G., Sulaiman, T. A., Baskonus, H. M., On the complex solutions to the (3+1)-dimensional conformable fractional modified KdV-Zakharov-Kuznetsov equation, Mod. Phys. Lett. B, 34(05) (2020), 2050069. https://doi.org/10.1142/S0217984920500694
37. Yel, G., Baskonus, H. M., Gao, W., New dark-bright soliton in the shallow water wave model, Aims Math., 5(4) (2020), 4027-4044. https://doi.org/10.3934/math.2020259
38. Yel, G., Aktürk, T., A new approach to (3+1) dimensional Boiti-Leon-Manna-Pempinelli equation, Appl. Math. Nonlinear Sci., 5(1) (2020), 309-316. https://doi.org/10.2478/amns.2020.1.00029
39. Yel, G., Bulut, H., Ilhan, E., A new analytical method to the conformable chiral nonlinear Schr¨odinger equation in the quantum Hall effect, Pramana J. Phys., 96(1) (2022), 54. https://doi.org/10.1007/s12043-022-02292-4
40. Yokus, A., Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schr¨odinger equation, Math. Model. Numer. Simul. Appl., 1(1) (2021), 24-31. https://doi.org/10.1007/10.53391/mmnsa.2021.01.003
41. Zahran, E. H. M., Khater, M. M. A., Modified extended tanh-function method and its applications to the Bogoyavlenskii equation, Appl. Math. Model., 40(3) (2016), 1769-1775. https://doi.org/10.1016/j.apm.2015.08.018