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

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 ′ ∕G 2 )-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 ′ ∕G 2 )-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.

Dynamical Analysis of Nonlocalized Wave Solutions of 2 + 1 -Dimensional CBS and RLW Equation with the Impact of Fractionality and Free Parameters

“…Assume the constants' determination be attained as one or more results by determining the mathematical conditions in phase 3. Putting the constant designs laterally with the measures for Equation (7), from the evaluation equation, we can attain up-to-date, extensive, and exhaustive dynamic wave propagation (6).…”

“…In the recent past, nonlinear fractional partial differential equations (FPDEs) have been inventing a potential platform for the researchers to interpret the tangible phenomena. As a result, some significant, concise, and original methods have been explored and explain to exact closed solutions of nonlinear FPDEs, videlicet, namely, the Hirota bilinear method [5,6], modified extended tanh-function method [7], ðG ′ /G Þ-expansion method [8], fractional subequation method [9], modified trial equation method [10], advanced exp ð−φ ðξÞÞ method [11], tan ð−φðξÞÞ-expansion method [12], and improved Kudryashov method [13]. Recently, some researchers, like, Ferdous et al [14], attained exact wave solutions to the extended Zakharov-Kuzetsov equation by implementing the generalized exp ð−φðξÞÞ-expansion method.…”

Dynamical simulations of 3D fractional WBBM model: Mathematical and graphical analysis with the impact of fractionality and unrestricted parameters

“…To solve the complex WBBM equations, different methods have been used. These methods include the Sine-Gordon expansion method 37 , the two-variable method 2 , the Improved ( 𝑮 ′ 𝑮 𝟐 ) 34 , the improved extended 𝐭𝐚𝐧𝐡 function method 5 , the improved auxiliary equation technique 38, etc.…”

A new implementation of a novel analytical method for finding the analytical solutions of the (2+1)-dimensional KP-BBM equation

“…There are several methods to solve and find the exact solutions to the evolution equations involving non-linearity. For example, the -expansion method [6] , [7] , [8] , [9] , the exp-function method [10] , the modified exp-function method [11] , [12] , the tanh–coth expansion method [13] , [14] , [15] , the improved method [16] , [17] , the -expansion method [18] , [19] , [20] , [21] , the simple equation method (SEM) [22] , [23] , the Lie symmetry approach [24] , [25] , the sine-Gordon method [26] , the modified Sardar sub-equation method [27] , [28] , the generalized Kudryashov method [29] , [30] , the Riccati-Bernoulli sub-ODE method [31] , [32] , the improved generalized Riccati mapping method [33] , the modified double sub-equation method [34] , the generalized exponential rational function (GERF) method [35] , [36] and there are many more. Beside these integer order PDEs there are lots of techniques for investigating fractional order PDEs such as [37] , [38] etc.…”