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.

The main intension of this paper is to extract new and further general analytical wave solutions to the (2 + 1)-dimensional fractional Ablowitz-Kaup-Newell-Segur (AKNS) equation in the sense of conformable derivative by implementing the advanced exp(-ϕ(ξ))-expansion method. This method is a particular invention of the generalized exp(-ϕ(ξ))-expansion method. By the virtue of the advanced exp(-ϕ(ξ))-expansion method, a series of kink, singular kink, soliton, combined soliton, and periodic wave solutions are constructed to our preferred space time-fractional (2 + 1)- dimensional AKNS equation. An extensive class of new exact traveling wave solutions are transpired in terms of, hyperbolic, trigonometric, and rational functions. To express the underlying propagated features, some attained solutions are exhibited by making their three-dimensional (3D), two-dimensional (2D) combined, and 2D line plot with the help of computational packages MATLAB. All plots are given to show the proper wave features through the founded solutions to the studied equation with particular preferring of the selected parameters. Moreover, it may conclude that the attained solutions and their physical features might be helpful to comprehend the water wave propagation in water wave mechanics.

In this current study, we described a modified extended tanh-function (mETF) method to find the new and efficient exact travelling and solitary wave solutions to the modified Liouville equation and modified regularized long wave (mRLW) equation in water wave mechanics. Travelling wave transformation decreases the leading equation to traditional ordinary differential equations (ODEs). The standardized balance technique provides the instruction of the portended polynomial related result stimulated from the mETF method. The substitution of this result follows the preceding step. Balancing the coefficients of the like powers of the portended solution leads to a system of algebraic equations (SAE). The solution of that SAE for coefficients provides the essential connection between the coefficients and the parameters to build the exact solution. Here the acquired solutions are hyperbolic, rational, and trigonometric function solutions. Our mentioned method is straightforward, succinct, efficient, and powerful and can be emphasized to establish the new exact solutions of different types of nonlinear conformable fractional equations in engineering and further nonlinear treatments.

Utilizing of illustrative scheming programming, the study inspects the careful voyaging wave engagements from the nonlinear time fractional modified Kawahara equation (mKE) by employing the advanced -expansion policy in terms of trigonometric, hyperbolic, and rational function through some treasured fractional order derivative and free parameters. The undercurrents of nonlinear wave answer are scrutinized and confirmed by MATLAB in 3D and 2D plots, and density plot by specific values of the convoluted parameters is designed. Our preferred advanced -expansion technique which is parallel to ( ) expansion technique is trustworthy dealing for searching significant nonlinear waves that progress a modification of dynamic depictions that ascend in mathematical physics and engineering grounds.