Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach

Abstract

It is known that the Laplace transform method is used to solve only a finite class of linear differential equations. In this paper, we suggest a new method that relies on a new fractional expansion in the Laplace transform space and residual power series method to construct exact solitary solutions to the nonlinear time-fractional dispersive PDEs. The main idea of this method is to convert the original fractional differential equation into the Laplace space, and then, solve the newly obtained equation using a new approach of the residual power series method. Moreover, four important and attractive applications are also presented and discussed based on our proposed method to reveal the efficiency, effectiveness, and applicability of this method for constructing solitary solutions to the target equations. The results confirmed that the proposed method is a simple, understandable, and very fast method to obtain solitary solutions compared with other methods such as homotopy perturbation method, variation iteration method, sine–cosine method, and residual power series method.