Investigation of partial differential equations with fractional order by G'/G method

In this study, one of the basic definitions of fractional derivative operators, which has an important role in applied sciences and can model the real life problem in the most accurate way, has been examined with the Atangana(beta) fractional derivative operator G'/G method defined with the Mitta-Leffler function without having a local and singular kernel. This method is a generalization of the G'/G classical method. With this developed method, nonlinear differential equations containing the fractional derivative Atangana(beta) have been handled in order to find their exact solutions. This method has been applied to the Boussinesq equation with the non-linear Sharma-Tasso-Olver equation including the Atangana(beta) fractional derivative. In this way, new exact solutions of the equations were found. In order to understand the physical behavior of these new exact solutions, three-dimensional graphs were drawn according to different parameter values.