Various Exact Solutions for the Conformable Time-Fractional Generalized Fitzhugh–Nagumo Equation with Time-Dependent Coefficients

In this paper, the subequation method and the sine-cosine method are improved to give a set of traveling wave solutions for the time-fractional generalized Fitzhugh–Nagumo equation with time-dependent coefficients involving the conformable fractional derivative. Various structures of solutions such as the hyperbolic function solutions, the trigonometric function solutions, and the rational solutions are constructed. These solutions may be useful to describe several physical applications. The results show that these methods are shown to be affective and easy to apply for this type of nonlinear fractional partial differential equations (NFPDEs) with time-dependent coefficients.


Introduction
Fractional calculus is considered as a generalization of classical concepts of integration and differentiation. e first appearance of this idea was in 1695. en, its importance began to increase due to its many applications in several fields such as biology, plasma physics, solid state physic, engineering, economy, finance, liquid crystals, electrical network, numerical analysis, dynamical systems, and control systems [1][2][3][4][5][6][7][8][9].
ere are many properties achieved in the classical derivative, but they are not satisfied with the definitions that are mentioned above. e most important of these properties are product rule, quotient rule, chain rule, Rolle theorem, and mean value theorem.
In 2014, Khalil et al. [14] introduced a new definition of fractional derivative, which is called the conformable fractional derivative. Unlike other definitions, this new definition satisfies the properties mentioned above [15].
A number of researchers presented analytical solutions to a large number of NFPDEs with constant coefficients by using the conformable fractional derivative, such as Al-Shawba et al. [16] solved the KdV-ZK equation with timefractional derivative using the ((G ' /G), (1/G))-expansion method. e space-time-fractional modified equal-width equation was solved by Zafar et al. [17] using a hyperbolic function method. A time-fractional biological population model was solved by Zhang and Zhang [18] using a fractional subequation method. Also, Çenesiz and Kurt [19] presented the solution of the space-fractional telegraph equation by introducing the conformable fractional complex transform.
Nowadays, the equations with time-dependent coefficients are more important than equations with constant coefficients as they describe cases that are more general. In 2020, Injrou [20] modified the subequation method to obtain a set of exact solutions for the space-time-fractional Zeldovich equation with time-dependent coefficients. e Fitzhugh-Nagumo equation system has been derived by both Fitzhugh [21] and Nagumo et al. [22]. It is a simplified form of the Hodgkin-Huxley Model because it is too difficult to be solved analytically.
Recently, many researchers have been interested in the time-fractional Fitzhugh-Nagumo equation with different applications in the areas of neurophysiology, logistical population growth, flame spread, catalytic chemical reaction, and nuclear reactor theory where it combines diffusion and nonlinearity which are controlled by the term (1) In 2012, Merdan [23] obtained analytical solutions to the time-fractional Fitzhugh-Nagumo equation (1) by a new application of fractional variational iteration method. At the same year, Pandir and Tandogan [24] presented analytical solution for equation (1) using the modified trial equation method. Ahmet Bekir et al. [25], in 2016, applied (G ′ /G)-expansion method to obtain exact solutions for equation (1). While in 2017, Bekir et al. [26] solved equation (1) by using the exp-function method. Taşbozan and Kurt [27], in 2020, introduced new exact solutions for equation (1) using the Sine-Gordon expansion method.
In this paper, we utilize the improved subequation method mentioned in [20] and improve the sine-cosine method to solve the time-fractional generalized Fitzhugh-Nagumo equation with time-dependent coefficients and linear dispersion term where β(t), c(t), and δ(t) are arbitrary real-valued function of t, μ is a constant, and u(x, t) is the unknown function depending on the temporal variable t and the spatial variable (2) will be reduced to the standard fractional Fitzhugh-Nagumo equation. Equation (2) has never solved in the case where the coefficients are time-dependent in literature before. e advantages of these methods are that they are more general because they are used in the predicted solution as time-dependent coefficients instead of using constant coefficients and they are effective and easy to apply to NFPDEs with time-dependent coefficients. However, we have noticed that some fractional differential equations are solved with time-dependent coefficients in a variety of methods but with constant coefficients in the predicted solution, for example [28,29]. On the other hand, these methods require fewer calculations than other methods like the exp-function method or the ((G ' /G), (1/G))-expansion method. [14].

Definition of Conformable Fractional Derivative
Given a function f: [0, ∞) ⟶ R, the conformable fractional derivative of f of order α is defined by . Furthermore, if α � 1, the definition is equivalent to the classical definition of the first-order derivative of the function. [14]. e conformable fractional integral of f (function continuous) of order α, on interval [0, t], is defined by f function continuous: [30]. Some important properties of the conformable fractional derivative and the conformable fractional integral are as follows.

Properties of Conformable Fractional Derivative
Let α ∈ (0, 1] and f, g be α-differentiable at a point t > 0. en, there are the following properties Property 1. (differential of a constant).
2 International Journal of Differential Equations Property 6. [31]. Let f: [0, ∞) ⟶ R, be a function such that f is differentiable and also α-differentiable. Let g be a function define in the range of f and also differentiable, then

Description of the Improved Subequation Method.
In this section, we summarize the main steps of the improved subequation method for finding exact solutions of (NFPDEs). Suppose an NFPDE where D α t u is the conformable fractional derivative of u, u � u(x, t) is an unknown function, and F is a polynomial in u and its conformable time-fractional partial derivatives and space partial derivatives. e main steps of the improved subequation method are presented in [18,20,32] as follows: Step 1: suppose that By substituting (15), NFPDEs turn to the ordinary differential equation (ODE): where P is a polynomial in U and its derivatives, Step 2: suppose that the solution of equation (15) can be expressed in the following form: where a i (t)(i � − m, . . . , m) is the function of t to be determined later, and φ � φ(ξ) satisfies the following Riccati equation: where σ is a constant. e following solutions of Riccati equation (19) are given by [33] where ω is a constant.
Step 3: determine the positive integer m by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in equation (17), then substituting (18) with equation (19) into equation (17), then using the properties of the conformable fractional derivative (5)-(14), then collecting all terms with the same order of φ(ξ), and then setting each coefficient of φ i (ξ) to zero, one can get an overdetermined system of nonlinear differential equations for a i (t) and ξ.
Step 4: assume that a i (t) and ξ can be obtained by solving the overdetermined system of Step 3, then substituting these results and the solutions, one can obtain the exact solutions of equation (15) immediately.

Description of the Improved Sine-Cosine Method.
In this section, the main steps of the sine-cosine method [34,35] are given as follows: Step 1: suppose that where a 0 (t), . . . , a n (t) and b 1 (t), . . . , b n (t) are all functions of t to be determined later; equation (15) reduces to ODE where ϕ ′ � dϕ/dξ. Now, taking the target equation as considering the following: sin9 � sechξ and cos9 � ∓tanhξ (24) or sin9 � ∓tanhξ and Step 2: equating the highest order nonlinear term and highest order linear partial derivative in equation (22), it yields the value of n.
Step 3: substituting (21) and (23) into equation (15), and collecting all the terms with the same power of sin9, cos9, sin9 cos9, sin 2 9, . . ., then setting the coefficients to be zero, one can obtain an overdetermined system of nonlinear differential equations for Step 4: assuming that b i (t), a i (t), and ξ can be obtained by solving the overdetermined system of Step 3, then substituting these results and the solutions, the exact solutions of equation (15) can be obtained immediately.

Applications of the Improved Subequation Method.
Using the generalized traveling wave transformation, and Property 2.1.3 and eorem 2.1.4, equation (2) can be reduced to the following nonlinear ordinary differential equation: where . We suppose that equation (27) has a solution in the form of (18). Balancing the highest order derivative term U ' ′ and with nonlinear term U 3 in equation (27), one can get m + 2 � 3m⇒m � 1. So, we have Substituting (28) into equation (2), collecting all the terms with the same power of φ i (i � − 3, − 2, . . . , 2, 3), then setting the coefficients of φ i (i � − 3, − 2, . . . , 2, 3) to be zero, one can obtain the overdetermined system of nonlinear differential equations for a − 1 (t), a 0 (t), a 1 (t), and τ(t) as follows: 4 International Journal of Differential Equations is a continuous funcation. Solving this system by Maple software, and for simplicity, we introduce the notation.
Finally, one can obtain these four cases: Case 1.
If we assume that a − 1 (t) � 0, a 0 (t) � Ψ(t), It can be obtained that When σ < 0, the following hyperbolic function solution of equation (2) is obtained as follows: When σ > 0, we obtain the following trigonometric function solution of equation (2), as follows: When σ � 0, substituting σ � 0 into the abovementioned system and then solving the obtained system, one can find a 1 (t) will not change but τ(t) is changed and becomes equal to When σ � 0, we have the following rational solution of equation (2), as follows: Case 2.
If we assume that a 1 (t) � 0, a 0 (t) � Ψ(t), it can be obtained that We construct the following hyperbolic function solution of equation (2), where σ < 0 is as follows: We construct the following hyperbolic function solution of equation (2), where σ > 0 is as follows: When σ � 0, we have the following solution of equation (2), as follows: Case 3.
If we assume that a 0 (t) � c 3 Ψ(t), a 1 (t) � c 4 Ψ(t), it can be obtained that International Journal of Differential Equations where When σ < 0, we obtain the following hyperbolic function solution of equation (2), as follows: When σ > 0, we obtain the following trigonometric function solution of equation (2), as follows: When σ � 0, we have the following rational solution of equation (2), as follows: (55) 6 International Journal of Differential Equations When σ < 0, we obtain the following hyperbolic function solution of equation (2), as follows: When σ > 0, we obtain the following trigonometric function solution of equation (2), as follows: When σ � 0, we have the following rational solution of equation (2), as follows: where c 1 , c 2 , c 3 , c 4 , c 5 , and k are arbitrary constants.
In Figure 1, we compare the space-time graph of solution (38) with the space-time graph of solution (39) at x ∈ [− 10, 10], t ∈ [0.0001, 0.01]. In (a), we find the kinkshape wave soliton solution (38) in the x t plane, while (b) describes the propagation of the anti-kink-shape wave soliton solution in the x t plane, with the same time-dependent coefficients where α � 0.5.
In Figure 2. (a) clarifies the propagation of the kinkshape wave soliton solution in the x t plane. e wave moves with the same shape and amplitude over its the track in the x t domain, while (b) describes the propagation of the spikeshape wave soliton solution in the x t plane, with the same time-dependent coefficients where α � 0.3.

Applications of the Improved Sine-Cosine Method.
Using the generalized traveling wave transformation, and Property 2.1.3 and eorem 2.1.4, equation (2) can be reduced to the following nonlinear ordinary differential equation: Balancing the highest order derivative term ϕ '' with nonlinear term ϕ 3 , we can get n + 2 � 3n⇒n � 1. So, we have Substituting (64) and (23) into equation (2) and collecting all the terms with the same power ofsin i 9 cos j 9 for i � 0, 1 and j � 0, 1, 2, 3, then setting the coefficients to be zero, we can obtain overdetermined system of nonlinear differential equations for b 1 (t), a 0 (t), a 1 (t), and τ(t) as follows: International Journal of Differential Equations sin 0 9 cos 0 9:    International Journal of Differential Equations Solving this system by Maple, for simplicity, we introduce the notation is a continuous function.
Finally, one finds these four cases as follows: Case 1.
If we assume that b 1 (t) � 0, a 0 (t) � Ψ(t), it can be obtained that We substitute what mentioned before into (64) and using (24) then, we obtain the hyperbolic functions solution of equation (2), as follows: If we assume that a 0 (t) � 0, a 1 (t) � Ψ(t), it can be obtained that where We substitute the solutions of the Case 2 into (64) and using (24), and then we give the hyperbolic functions solution of equation (2), as follows: Case 3.

International Journal of Differential Equations
We substitute the solutions of the Case 4 into (64) and using (24), and then we give the hyperbolic functions solution of equation (2), as follows: where c 6 , c 7 , c 8 , c 9 , and k are arbitrary constants.

Conclusions
In this work, we have obtained three kinds of exact solutions that include the trigonometric function solutions, the hyperbolic function solutions, and the rational solutions which are successfully established by using the subequation method. Also, a kind of exact analytical solutions which are the hyperbolic function solutions are found by using the sine-cosine method and conformable fractional derivative of the time-fractional generalized Fitzhugh-Nagumo equation with time-dependent coefficients. We have found that the subequation method gives more general solutions than the solutions that are given by the sine-cosine method. Moreover, it is observed that the solutions obtained in this research may be important to describe certain nonlinear phenomena in mathematical physics, engineering, and biology. Remarkably, these solutions and the proposed traveling wave transformation have not been reported in other literature. Also, some 3D graphical representations are offered for the obtained results with the different time-dependent coefficients and different values of α. We can conclude that the used method is a very powerful, convenient, and efficient technique and it can be used for many other partial fractional nonlinear differential equations.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.