New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

Abstract

The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel ( G ′ G ) -expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel ( G ′ G ) -expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.

General Traveling Wave Solutions of Nonlinear Conformable Fractional Sharma-Tasso-Olever Equations and Discussing the Effects of the Fractional Derivatives

“…As an effective mathematical modeling tool, it is widely used in the mathematical modeling of nonlinear phenomena in biology, physics, signal processing, control theory, system recognition, and other scientific fields. The widely studied fractional Sharma-Tasso-Olever (STO) equation in space and time [1][2][3][4]…”

“…In literature [3], the author used fractional complex transformation on equation (1) and then used the novel ðG′/GÞ-expansion method to obtain the exact traveling wave solution of equation (1); the generalized Kudryashov method was also used to obtain the precise traveling wave solution of equation (1).…”

“…In literature [7], the author used fractional complex transformation in equation ( 3) and then used a new generalized ðG′/GÞ-expansion method to obtain the precise traveling wave solution of the fractal equation (3).…”

“…It can be seen from literature [1][2][3][4][5][6][7] that the first step of these authors was to reduce the STO equation of the fractional order to a nonlinear ordinary differential equation by using fractional complex transformation and then to solve the reduced equation by various methods to obtain the solution of the original fractional-order equation. In addition to the methods provided in literature [1][2][3][4][5][6][7], there are other techniques that can be used to obtain wave solutions of different structures [14][15][16][17][18][19]. It can be seen from equations ( 1)-( 3) that the most general equation is equation ( 1), so we only discuss the accurate general traveling wave solution of equation (1).…”

“…In literature [1][2][3][4][5][6][7], some articles integrate equation ( 13) once and then assume that the integral constant is zero,…”