Nowadays,  nonlinear fractional partial differential equations have been highly using for modelling of physical phenomena. Therefore, it is very important to achieve exact solutions of fractional differential equations for understanding complex phenomena in mathematical physics. In this study,  new exact traveling wave solutions are reached of space-time fractional Phi-4 equation indicated by Atangana’s conformable derivative using two powerful different techniques. These are the functional variable method and the first integral method. Obtaining new solutions of this equation show that method is effective to understanding other nonlinear complex problems in particle and nuclear physics.

In recent years, many authors have researched about fractional partial differential equations. Physical phenomena, which arise in engineering and applied science, can be defined more accurately by using FPDEs. Thus, obtaining exact solutions of the FPDEs equations have become more important to understand physical problems. In this article, we have reached the new traveling wave solutions of the conformable fractional modified Camassa -Holm equation via two efficient methods such as first integral method and the functional variable method. The wave transformation and conformable fractional derivative have been used to convert FPDE to the ordinary differential equation. The Camassa -Holm equation is physical model of shallow water waves with non-hydrostatic pressure. Thanks to these powerful methods, some comparisons, such as type of solutions and physical behaviours, have been made. Additionally, mathematica program have been used with the aim of checking of solutions. Investigating results of the fractional differential equations can help understanding complex phenomena in applied mathematics and physics.

Recently, many successful methods have been developed to achieve analytical solutions of nonlinear partial differential equations. In this study, some new exact solutions of the non-linear coupled Klein- Gordon system and non-linear modified Benjamin-Bona-Mahony equation have been obtained by using functional variable method (FVM). Additionally, all solutions have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Mathematica program. These equations have been used in various fields such as plasma physics, biophysics, and fluid dynamics. The main advantage of FVM is generate more solutions than other analytical methods and therefore, FVM is an effective and powerful method to solve evolution equations in engineering and mathematical physics.