Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

Abstract

In this paper, the ( G ′ / G , 1 / G ) -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, solitary wave solutions of kink type, and so on. The method is very efficient, powerful, and reliable for solving the proposed equations and other nonlinear fractional partial differential equations with the aid of a symbolic software package.

A solitary wave solution to the generalized Burgers-Fisher's equation using an improved differential transform method: A hybrid scheme approach

“…Several researches have been carried out in proffering an exact solution to nonlinear partial differential equations including the Burger-Fisher's equation [9] , [10] , [51] , [52] , [53] . Majority of these solutions obtained to nonlinear partial differential equations are usually in multivariate series form converging rapidly to the exact solution of the problem.…”

“…Numerous researchers have studied the nonlinear ordinary and partial differential equations over the years [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] , but consistent findings still yield highly pertinent results and recommendations, as there is no best method or algorithm in providing exact solutions to an equation. Nonlinear PDEs can be classified as the integrable and non-integrable [6] , [7] depending the nature of the equation in question.…”

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

“…Definition 1. Let f be a function such that f : [0, ∞) → R. Then, the conformable derivative of f of order α, where 0 < α ≤ 1, is defined as [52][53][54][55][56][57][58]…”

“…Let α ∈ (0, 1], and f (t), g(t) be α-conformable differentiable functions at a point t > 0. Then, the important properties of the conformable derivative are as follows [52,53,[55][56][57]59]: 2 .…”

“…Using the definition in (3) and the above properties, the conformable derivatives of certain interesting functions are defined as follows [52,53,[55][56][57]:…”