Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations

This work aims to construct exact solutions for the space-time fractional(2+1)-dimensional dispersive longwave(DLW)equation and approximate long water wave equation(ALW)utilizing the twovariable(G′/G,1/G)-expansion method and the modified Riemann-Liouville fractional derivative.The recommended equations play a significant role to describe the travel of the shallow water wave.The fractional complex transform is used to convert fractional differential equations into ordinary differential equations.Several wave solutions have been successfully achieved using the proposed approach and the symbolic computer Maple package.The Maple package program was used to set up and validate all of the computations in this investigation.By choosing particular values of the embedded parameters,we pro-duce multiple periodic solutions,periodic wave solutions,single soliton solutions,kink wave solutions,and more forms of soliton solutions.The achieved solutions might be useful to comprehend nonlinear phenomena.It is worth noting that the implemented method for solving nonlinear fractional partial dif-ferential equations(NLFPDEs)is efficient,and simple to find further and new-fangled solutions in the arena of mathematical physics and coastal engineering.

Analytical behavior of weakly dispersive surface and internal waves in the ocean

The(2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis(CD)and fractional poten-tial Kadomstev-Petviashvili(PKP)equation.It can be modeled according to the Hamiltonian structure,the lax pair with the non-isospectral problem,and the pain level property.The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves,demonstrate the propagation of waves in dissipative and nonlinear media,and reveal the propagation of waves in dissipative and nonlinear media.In this paper,we have established further exact solutions to the nonlinear fractional partial differential equation(NLFPDEs),namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable(G/G,1/G)-expansion method.As far as trigonometric,hyperbolic,and rational function so-lutions containing parameters are concerned,solutions are acquired when unique characteristics are as-signed to the parameters.Subsequently,the solitary wave solutions are generated from the solutions of the traveling wave.It is important to observe that this method is a realistic,convenient,well-organized,and ground-breaking strategy for solving various types of NLFPDEs.