Analytical approximate solutions of fractional nonlinear Drinfeld–Sokolov–Wilson model using modified Mittag–Leffler function

Abstract

In this article, we introduce a modification of the generalized Mittag–Leffler function method (GMLFM) in order to provide appropriate analytical-approximate solutions for time-fractional nonlinear partial differential equations (PDEs). Moreover, we implement the proposed method on the Drinfeld–Sokolov–Wilson (DSW) model which is considered one of the most important mathematical models describing mathematical physics and dispersive water waves. We use Caputo fractional derivative (CFD) as a time-fractional operator for the mathematical formulation of this model. We introduce the results obtained by modified generalized Mittag Leffler function method (MGMLFM) when the fractional operator takes different values to show the impact of fractional order on the solution of the fractional DSW model, as well, we compare our results when α=1 with the exact solution to prove the efficiency of the used method. Furthermore, a simulation of our outcomes in some graphical and tabular forms is presented to be clearer for the reader. This simulation detects that by decreasing the value of time t the obtained solutions of the DSW system are remarkably close to the given exact solutions. Furthermore, decreasing the fractional operator from α=1 rapidly increases the amplitude of the solitary wave of the model. Also, results reveal that the absolute error decreases as spatial variable x increases at small-time t. The outcomes of this article confirm that this modification made to the GMLFM leads us to an effective, easy and accurate method to solve linear and nonlinear fractional PDEs which need little effort for computations.