New Solitary and Periodic Wave Solutions of (n + 1)-Dimensional Fractional Order Equations Modeling Fluid Dynamics

Sadullah Bulut,Hijaz Ahmad,Mesut Karabacak,Sameh Askar

doi:10.3390/sym13112017

Abstract

In this study, first, fractional derivative definitions in the literature are examined and their disadvantages are explained in detail. Then, it seems appropriate to apply the (G′G)-expansion method under Atangana’s definition of β-conformable fractional derivative to obtain the exact solutions of the space–time fractional differential equations, which have attracted the attention of many researchers recently. The method is applied to different versions of (n+1)-dimensional Kadomtsev–Petviashvili equations and new exact solutions of these equations depending on the β parameter are acquired. If the parameter values in the new solutions obtained are selected appropriately, 2D and 3D graphs are plotted. Thus, the decay and symmetry properties of solitary wave solutions in a nonlocal shallow water wave model are investigated. It is also shown that all such solitary wave solutions are symmetrical on both sides of the apex. In addition, a close relationship is established between symmetric and propagated wave solutions.

Highlights

The effectiveness of integer-ordered derivatives of known mathematical models, including nonlinear models, is discussed in most cases
When previous studies are examined, it is observed that there is no single method for finding exact solutions of nonlinear differential equations and each method has advantages and disadvantages depending on the experience of the researchers
It is important to apply some of the methods commonly used in the literature to previous unsolved nonlinear partial differential equations to search for possible new exact solutions or validate existing solutions with a different approach

Summary

Introduction

The effectiveness of integer-ordered derivatives of known mathematical models, including nonlinear models, is discussed in most cases. It occurs as a reduction in a quadratic nonlinear system accepting weakly dispersed waves in the non-parallel wave approach It is currently used as a classic model for the development and checking of new mathematical techniques, e.g., in applications of dynamical system methods for water waves [20], variational theory of existence, stability of energy minimizers [21] and nonclassical function spaces [22]. It has been widely used as a model for two-dimensional shallow water waves [23,24,25] and ion-acoustic waves in plasmas, for example [26]. 0 [40], homoclinic breath limit approach [41], solitary wave ansatz [42], exp-function, GG -methods [43] and Lie group method [44]

Mathematical Background

Clairaut’s β-derivatives

Applications

Discussion and Conclusions