Abstract
The current work deals with the fractional forms of EW and modified EW equations in the conformable sense and their exact solutions. In this respect, by utilizing a traveling wave transformation, the governing space-time fractional models are converted to the nonlinear ordinary differential equations (NLODEs); and then, the resulting NLODEs are solved through an effective method called the exp(−ϕ(ϵ))-expansion method. As a consequence, a number of exact solutions to the fractional forms of EW and modified EW equations are generated.
1 Introduction
These days, the fractional differential equations (FDEs) have been the subject of a lot of research, owing to their frequent appearance in various areas from physics and chemistry to biology and engineering. A variety of useful methods, such as sub-equation method [1,2,3,4], modified trial equation method [5,6,7,8], (G′/G)-expansion method [9,10,11,12], exp-function method [13,14,15,16], Kudryashov method [17,18,19,20], and first integral method [21,22,23,24] have been applied to find the exact solutions of FDEs. One capable technique which has newly gained special interest is the exp(−ϕ(ϵ)) method. For instance, Hosseini et al. [25] exerted the exp(−ϕ(ϵ)) method to produce the exact solutions of the density-dependent conformable fractional diffusion-reaction equation and Raza et al. [26] adopted the exp(−ϕ(ϵ)) method to extract the explicit solutions of higher dimensional equations with fractional temporal evolution. For more articles, see [27,28,29,30,31,32,33,34,35,36,37,38,39,40]. In this paper, the exact solutions of the fractional forms of EW and modified EW equations in the conformable sense are achieved by means of the exp(−ϕ(ϵ)) method. The mathematical modelings of these fractional differential equations are presented below:
The fractional EW equation [41]
The fractional modified EW equation [41]
The EW equations describe the propagation of the wave in nonlinear media [42]. Several schemes have already been exerted for studying the above models; for example, Kudryashov method [17], ansatz method [41], and Fan sub-equation method [43]. This paper is organized as below: In Section 2, we will introduce the conformable derivative and its properties. In Section 3, we will explain the ideas of the exp(−ϕ(ϵ)) method. In Section 4, we will employ the exp(−ϕ(ϵ)) method to solve the fractional forms of EW and modified EW equations; and at the end, we will provide the results.
2 Conformable fractional derivative
Recently, a new version of fractional derivatives “the conformable fractional derivative” was proposed in [44] which obeys some classical properties that cannot be satisfied by the other definitions [45]. The conformable fractional derivative of f of order α is defined as [44]
where f : (0, ∞) → R, t > 0, and α ∈ (0, 1]. Some worthwhile features of the conformable derivative are as follows
3 Basic ideas of exp(−ϕ(ϵ))-expansion method
Let’s consider a nonlinear space-time FDE in the conformable sense as follows
By using the transformation
Eq. (3) changes into an ODE of integer order as
where G is a function in the unknown function f and its derivatives.
Let present the solution of Eq. (4) as below
where an, n = 0, 1, …, N (aN ≠ 0) are unknown constants and ϕ(ϵ) satisfies a nonlinear ordinary differential equation as
Now, various cases can be considered:
If λ2 − 4μ > 0 and μ ≠ 0, then
If λ2 − 4μ > 0, μ = 0, and λ ≠ 0, then
If λ2 − 4μ < 0 and μ ≠ 0, then
To procure the positive integer N in Eq. (5), we balance the terms in Eq. (4). Setting Eq. (5) in Eq. (4), results in
Through equating all the coefficients in (6) to zero, we will attain an algebraic set, which can be easily solved for determining the unknowns. Substituting them into (5), finally yields the exact solutions of original Eq. (3).
4 Applications
Now, we adopt the exp(−ϕ(ϵ)) method to seek the exact solutions of the fractional forms of EW and modified EW equations in the conformable sense.
4.1 The conformable space-time fractional EW equation
By using the transformation
Eq. (1) can be changed into the following ODE
Integrating (7) once with respect to ϵ, gives
where integrating constant is supposed to be zero.
4.1.1 Applying the exp(−ϕ(ϵ))-expansion method
By balancing f2 and f″ in Eq. (8), we obtain N = 2. Hence, Eq. (8) has the following formal solution
Through setting Eq. (9) in Eq. (8) and equating all the coefficients to zero, we will achieve an algebraic set in the form
Applying the symbolic computation package, results in
Thus, the exact solutions to the fractional form of EW equation in the conformable sense can be constructed as follows
For λ2 − 4μ > 0 and μ ≠ 0
For λ2 − 4μ > 0, μ = 0, and λ ≠ 0
For λ2 − 4μ < 0 and μ ≠ 0
Consequently, the exact solutions to the fractional form of EW equation in the conformable sense can be established as follows
For λ2 − 4μ > 0 and μ ≠ 0
For λ2 − 4μ > 0, μ = 0, and λ ≠ 0
For λ2 − 4μ < 0 and μ ≠ 0
4.2 The conformable space-time fractional modified EW equation
By using the transformation
Eq. (2) can be converted to an ODE as follows
Integrating (10) once with respect to ϵ, yields
where integrating constant is assumed to be zero.
4.2.1 Applying the exp(−ϕ(ϵ))-expansion method
By balancing f3 and f″ in Eq. (11), we take N =1. Consequently, Eq. (11) has the following formal solution
By inserting Eq. (12) along with its necessary derivative in Eq. (11) and equating all the coefficients to zero, we will derive an algebraic set as
Using the symbolic computation package, we will find
Thus, the exact solutions to the fractional form of modified EW equation in the conformable sense can be constructed as follows
For λ2 − 4μ > 0 and μ ≠ 0
For λ2 − 4μ > 0, μ = 0, and λ ≠ 0
For λ2 − 4μ < 0 and μ ≠ 0
5 Conclusion
In this investigation, the fractional forms of EW and modified EW equations in the conformable sense were studied, successfully. First, by adopting a traveling wave transformation, the governing space-time fractional models were converted to the nonlinear ordinary differential equations; and then, the resulting NLODEs were solved using an effective method called the exp(−ϕ(ϵ))-expansion method. As a consequence, a variety of exact solutions to the fractional forms of EW and modified EW equations were formally extracted; confirming the competence of the scheme.
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