Solitary and periodic wave solutions to the family of new 3D fractional WBBM equations in mathematical physics

Abdulla - Al Mamun,Nur Hasan Mahmud Shahen,Samsun Nahar Ananna,Md Asaduzzaman,Foyjonnesa Foyjonnesa

doi:10.1016/j.heliyon.2021.e07483

Abdulla - Al Mamun, Nur Hasan Mahmud Shahen + Show 3 more

Open Access

https://doi.org/10.1016/j.heliyon.2021.e07483

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Abstract

For the newly implemented 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equation family, the present study explores the exact singular, solitary, and periodic singular wave solutions via the (G′/G2)-expansion process. In the sense of conformable derivatives, the equations considered are translated into ordinary differential equations. In spite with many trigonometric, complex hyperbolic, and rational functions, some fresh exact singular, solitary, and periodic wave solutions to the deliberated equations in fractional systems are attained by the implementation of the (G′/G2)-expansion technique through the computational software Mathematica. The unique solutions derived by the process defined are articulated with the arrangement of the functions tanh, sech; tan, sec; coth, cosech, and cot, cosec. With three-dimensional (3D), two dimensional (2D) and contour graphics, some of the latest solutions created have been envisaged, selecting appropriate arbitrary constraints to illustrate their physical representation. The outcomes were obtained to determine the power of the completed technique to calculate the exact solutions of the equations of the WBBM that can be used to apply the nonlinear water model in the ocean and coastal engineering. All the solutions given have been certified by replacing their corresponding equations with the computational software Mathematica.

Highlights

Consider the succeeding fractional kind of the WBBM equations [1]: Dγt u + Dγxu + Dγyu3 − D3xzγt u = 0, (1)Dγt u + Dγz u + Dγxu3 − D3xγyt u = 0, (2)Dγt u + Dγyu + Dγz u3 − D3xγxt u = 0, (3)u(x, y, z, t) is a differentiable function in the above equations with four independent variables x, y, z, and t, and Dγxu, Dγyu, Dγz u, and Dγt u denote the corresponding u derivatives of order γ with respect to x, y, z, and t respectively, where 0 < γ ≤ 1, t ≥ 0
Hyperbolic secant type solitary wave solutions and several invariants are reported in the same work
The (G′∕G2) is a useful technique for conniving the traveling wave solutions of nonlinear partial single, coupled and system of equations arising in several expanses of fluid mechanics, physics, water wave mechanics, wave propagation problems, etc

Summary

Introduction

This research aims to generate precise solitary wave solutions expending the (G′∕G2)-expansion technique for a deeper appreciation of the physical significance of a diversity of WBBM equations. The joint solutions created specify solitary wave, singular periodic, and singular joint solutions Among these mentioned approaches, the new investigative process (G′∕G2)-expansion method has been utilized to build exact and explicit solution of time and space-time fractional differential equations. The (G′∕G2) is a useful technique for conniving the traveling wave solutions of nonlinear partial single, coupled and system of equations arising in several expanses of fluid mechanics, physics, water wave mechanics, wave propagation problems, etc. The solutions gained using the mentioned technique can be articulated in trigonometric, hyperbolic, and rational functions These forms of solutions are satisfactory for reviewing certain nonlinear physical treatment.

Conformable derivative

The first 3D fractional WBBM equation

Physical explanation

Conclusion