Novel inquiry of the fractional solutions of the optics equation and the surface tension equation by Riccati sub-equation method

Finding analytical solutions to nonlinear space and time fractional equations is an important task in mathematical physics, and the Riccati sub-equation method is an appropriate instrument for this purpose. In this research, we present the Riccati sub-equation approach to solve the nonlinear space and time fractional equations: (3+1)-dimensional Jimbo-Miwa equation and (1+1)-dimensional Benney-Luke equation, which describe the optics equation and the surface tension equation, respectively, and lead to new analytical solutions with parameters. Each equation that uses the fractional analytical solution method of the Riccati sub-equation method yields 15 different solutions, which fall into three forms: rational function, generalized triangular functions, and generalized hyperbolic functions. The solution graphs, when plotted in contour, two-dimensional, and three-dimensional space, produce periodic waves and kink waves.