Further results about the exact solutions of conformable space–time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation

Abstract

Seeking for the exact solutions of fractional nonlinear partial differential equations (FNPDE) has penetrated into almost every discipline of the natural, engineering, mathematics, and social sciences. In this article, with the aid of Maple software, using the (G′G2)-expansion method and the unified F-expansion method, we successfully investigate the exact solutions about conformable space–time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation. Consequently, three types traveling wave solutions are found such as Jacobian double periodic elliptic, simply periodic and the rational function solutions. The (G′G2)-expansion and unified F-expansion methods are relatively complex, but we can find more solutions that have not been seen before. We give the some specific examples solutions such as double-periodic Jacobian elliptic functions which can be molted into single-periodic functions. Also we point the bridge linking among the Weierstrass elliptic, hyperbolic, trigonometric functions and Jacobian elliptic functions. The connection and distinction between these three formal solutions are also indicated in our paper.