New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method

Abstract

Our concern in the present paper is to generate a few new explicit and exact solutions for the time-fractional Cahn–Allen and Cahn–Hilliard equations in the context of the conformable fractional derivative. A new version of Kudryashov method with the help of the Maple package is utilized to carry out this purpose. It is believed that the modified Kudryashov method is practically well suited; such that it can be adopted to a wide range of fractional differential equations (FDEs).

Introduction

Fractional differential equations have attracted special interest during the past two decades owing to their ability to model many phenomena in the areas of physics, biology, engineering, signal processing, control theory, finance, and fractal dynamics. A number of powerful methods, such as $(G\prime /G)$-expansion method [1], [2], [3], [4], first integral method [5], [6], [7], [8], sub-equation method [9], [10], [11], [12], exp-function method [13], [14], [15], [16], trial equation method [17], [18], [19], [20], and Kudryashov method [21], [22], [23], [24] have been utilized to construct the exact solutions of nonlinear fractional differential equations. One of the most effective approaches that has recently gained considerable attention because of its capacity in extracting new exact solutions of FDEs is a modified version of Kudryashov method. Here, some recent applications of this well-established technique are listed. Ray and Sahoo [25] used the modified Kudryashov method to look for new exact solutions for the time-fractional fifth-order modified Sawada–Kotera equation. Bulut et al. [26] utilized the modified Kudryashov method to search the exact solutions of generalized Fisher equation with fractional order. Ege and Misirli [27] adopted the modified Kudryashov method to find the analytical exact solutions of the space-time fractional modified Benjamin–Bona–Mahony and potential Kadomtsev–Petviashvili equations. Saha [28] employed the modified Kudryashov method to seek new analytical exact solutions of the time-fractional KdV–KZK equation. Interested readers are referred to [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44]. In this study, a few new exact solutions for the time-fractional Cahn–Allen and Cahn–Hilliard equations in the context of the conformable fractional derivative are obtained by making use of a new version of Kudryashov method. The time-fractional Cahn–Allen and Cahn–Hilliard equations can be expressed as follows:

• The time-fractional Cahn–Allen equation [45]

$$D_t^{\alpha }u-u_{xx}+u^3-u=0,t>\text{0, 0}<\alpha \le 1\text{.}$$

• The time-fractional Cahn–Hilliard equation [46]

$$D_t^{\alpha }u-u_x-6u{\left(u_x\right)}^2-\left(3u^2-1\right)u_{xx}+u_{xxxx}=0,t>\text{0, 0}<\alpha \le 1\text{.}$$

These nonlinear time-fractional equations were studied using various techniques, for example the sub-equation method [47], the fractional Fan sub-equation method of the fractional Riccati equation [48], the extended fractional Riccati expansion method [49], and the generalized tanh-coth method [50]. The outline of this article is as follows: In Section 2, a brief exposition of the conformable fractional derivative is given. In Section 3, the key ideas of the modified Kudryashov method are presented. In Section 4, a few new exact solutions for the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations are extracted via the modified Kudryashov method. Finally, a brief conclusion is provided in the last section.