Construction of Analytical Solutions to the Conformable New (3+1)-Dimensional Shallow Water Wave Equation

Abstract

This study investigates the new (3+1)-dimensional shallow water wave equation. To do so, the definitions of conformable derivatives and their descriptions are given. Using the Riccati equation and modified Kudryashov methods, exact solutions to this problem are discovered. The gathered data's contour plot surfaces and related 3D and 2D surfaces emphasize the result's physical nature. To monitor the problem's physical activity, exact and complete solutions are necessary. The results demonstrate the potential applicability of additional nonlinear physical models from mathematical physics and under-investigation in real-world settings. In order to solve fractional differential equations, it may prove helpful to use these methods in various situations.

Keywords

• Riccati equation method
• modified Kudryashov method
• new $(3+1)$-dimensional shallow water wave (SWW) equation
• conformable derivative

References

1. M. Şenol, O. Taşbozan, A. Kurt, \emph{Comparison of Two Reliable Methods to Solve Fractional Rosenau‐Hyman Equation}, Mathematical Methods in the Applied Sciences 44 (10) (2021) 7904--7914.
2. G. Jumarie, \emph{Fractional Partial Differential Equations and Modified Riemann-Liouville Derivative New Methods for Solution}, Journal of Applied Mathematics and Computing 24 (1-2) (2007) 31--48.
3. A. Atangana, D. Baleanu, \emph{Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer}, Journal of Engineering Mechanics 143 (5) (2017) 1--5.
4. Y. Khan, Q. Wu, N. Faraz, A. Yıldırım, M. Madani, \emph{A New Fractional Analytical Approach via a Modified Riemann–Liouville Derivative}, Applied Mathematics Letters 25 (10) (2012) 1340--1346.
5. A. Atangana, I. Koca, \emph{Chaos in a Simple Nonlinear System with Atangana–Baleanu Derivatives with Fractional Order}, Chaos, Solitons \& Fractals 89 (2016) 447--454.
6. F. M. Alharbi, D. Baleanu, A. Ebaid, \emph{Physical Properties of the Projectile Motion Using the Conformable Derivative}, Chinese Journal of Physics 58 (2019) 18--28.
7. M. Gençyiğit, M. Şenol, M. E. Köksal, \emph{Analytical Solutions of the Fractional (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation}, Computational Methods for Differential Equations (in press).
8. L. Akinyemi, M. Şenol, E. Az-Zo’bi, P. Veeresha, U. Akpan, \emph{Novel Soliton Solutions of Four Sets of Generalized (2+1)-Dimensional Boussinesq–Kadomtsev–Petviashvili-like Equations}, Mo\-dern Physics Letters B 36 (01) (2022) 1--30.

9. H. Durur, O. Taşbozan, A. Kurt, M. Şenol, \emph{New Wave Solutions of Time Fractional Kadomtsev-Petviashvili Equation Arising in the Evolution of Nonlinear Long Waves of Small Amplitude}, Erzincan University Journal of Science and Technology 12 (2) (2019) 807--815.
10. A. Kurt, M. Şenol, O. Taşbozan, M. Chand, \emph{Two Reliable Methods for the Solution of Fractional Coupled Burgers’ Equation Arising as a Model of Polydispersive Sedimentation}, Applied Mathematics and Nonlinear Sciences 4 (2) (2019) 523--534.
11. L. Akinyemi, M. Şenol, M. S. Osman, \emph{Analytical and Approximate Solutions of Nonlinear Schrödinger Equation with Higher Dimension in the Anomalous Dispersion Regime}, Journal of Ocean Engineering and Science 7 (2) (2022) 143--154.
12. \bibitem{Az-Zo’bi} E. Az-Zo’bi, W. A. Alzoubi, L. Akinyemi, M. Şenol, B. S. Masaedeh, \emph{A Variety of Wave Amplitudes for the Conformable Fractional (2+1)-Dimensional Ito Equation}, Modern Physics Letters B 35 (15) (2021) 1--13.
13. M. Mirzazadeh, A. Akbulut, F. Taşcan, L. Akinyemi, \emph{A Novel Integration Approach to Study the Perturbed Biswas-Milovic Equation with Kudryashov’s Law of Refractive Index}, Optik 252 (2022) 1--9.
14. S. S. Ray, S. Sahoo, \emph{A Novel Analytical Method with Fractional Complex Transform for New Exact Solutions of Time-Fractional Fifth-Order Sawada-Kotera Equation}, Reports on Mathematical Physics 75 (1) (2015) 63--72.
15. D. Ntiamoah, W. Ofori-Atta, L. Akinyemi, \emph{The Higher-Order Modified Korteweg-de Vries Equation: Its Soliton, Breather and Approximate Solutions}, Journal of Ocean Engineering and Science (in press).
16. H. Grosse, G. Opelt, \emph{Fractional Charges in External Field Problems and the Inverse Scattering Method}, Nuclear Physics B 285 (1987) 143--161.
17. K. R. Raslan, K. K. Ali, M. A. Shallal, \emph{The Modified Extended tanh Method with the Riccati Equation for Solving the Space-Time Fractional EW and MEW Equations}, Chaos, Solitons \& Fractals 103 (2017) 404--409.
18. Y. Gu, S. M. Zia, M. Isam, J. Manafian, A. Hajar, M. Abotaleb, \emph{Bilinear Method and Semi-Inverse Variational Principle Approach to the Generalized (2+1)-Dimensional Shallow Water Wave Equation}, Results in Physics 45 (2023) 1--12.
19. M. Mirzazadeh, L. Akinyemi, M. Şenol, K. Hosseini, \emph{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) 1--15.
20. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, \emph{A New Definition of Fractional Derivative}, Journal of Computational and Applied Mathematics 264 (2014) 65--70.
21. Y. Cenesiz, A. Kurt, \emph{New Fractional Complex Transform for Conformable Fractional Partial Differential Equations}, Journal of Applied Mathematics, Statistics and Informatics 12 (2) (2016) 41--47.
22. K. S. Nisar, L. Akinyemi, M. Inc, M. Şenol, M. Mirzazadeh, A. Houwe, S. Abbagari, H. Rezazadeh, \emph{New Perturbed Conformable Boussinesq-like Equation: Soliton and Other Solutions}, Results in Physics 33 (2022) 1--10.