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

Authors: Sami Injrou; Ramez Karroum; Nadia Deeb
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Fractional calculus is considered as a generalization of classical concepts of integration and differentiation. The first appearance of this idea was in 1695. Then, 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 [1–9].Many mathematicians have proposed different kinds for the definition of the fractional derivative. The most popular of these are Riemann–Liouville, Caputo, Grunwald–Letnikov Erdelyi, Riesz, Hadamard, Marchaud, and Kober [10–13].There are many properties achieved in the classical derivative, but they are not satisfied with the definitions that are mentioned above. The 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 time-fractional derivative using the G'/G,1/G-expansion method. The 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.The 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 termu1−uμ−u:(1)Dtαu−uxx+u1−uμ−u=0;t>0,0<α≤1,x∈ℝ.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(2)Dtαu+βtux−γtuxx+δtu1−uμ−u=0;t>0,0<α≤1,x∈ℝ,where βt, γt, and δt are arbitrary real-valued function of t,μ is a constant, and ux,t is the unknown function depending on the temporal variable t and the spatial variable x. For βt=0, γt=δt=1, equation (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.The 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.


## 2. Preliminaries

### 2.1. Conformable Fractional Calculus

#### 2.1.1. Definition of Conformable Fractional Derivative [14]

Given a functionf:0,∞⟶ℝ, the conformable fractional derivative of f of order α is defined by(3)Dtαft=limξ⟶0ft+ξt1−α−ftξ,for t∈0,∞, α∈0,1. If f is α-differentiable in some interval 0,a, a>0, and limt⟶0+Dtαft exists, then Dtαf0=limt⟶0+Dtαft.Furthermore, ifα=1, the definition is equivalent to the classical definition of the first-order derivative of the function.


#### 2.1.2. Definition of Conformable Fractional Integral [14]

The conformable fractional integral off (function continuous) of order α, on interval 0,t, is defined by f function continuous:(4)Itαft=∫0txα−1fxdx.


#### 2.1.3. Properties of Conformable Fractional Derivative [30]

Some important properties of the conformable fractional derivative and the conformable fractional integral are as follows.Letα∈0,1 and f,g be α-differentiable at a point t>0. Then, there are the following propertiesProperty 1.
(differential of a constant).(5)Dtαc=0,wherecis a constant.Property 2.
(the linearity property).(6)Dtαaft+bgt=aDtαft+bDtαgt;a,bare constants.Property 3.
(product rule).(7)Dtαf.gt=fDtαgt+gDtαft.Property 4.
(quotient rule).(8)Dtαfg=gDtαft−fDtαgtg2t.Property 5.
(chain rule).(9)Dtαfοgt=f′gtDtαgt.Property 6.
(10)Dtαtp=ptp−α;p∈ℝ.Property7.
Letα∈0,1 and f is be α-differentiable at a point t>0. If f is differentiable, then(11)Dtαft=t1−αdfdt.Property 8.
Letα∈0,1 and f is any continuous function in a domain of Itα, for t>a, we have(12)DtαItαft=ft,(13)ItαDtαft=ft−f0;on the interval0,t.


#### 2.1.4. Theorem [31]

Letf: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(14)Dtαfοgt=t1−αg′tf′gt.


### 2.2. Description of Two Methods

#### 2.2.1. 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(15)Fu,Dtαu,ux,Dt2α,uxx,…=0;0<α≤1,where Dtαu is the conformable fractional derivative of u, u=ux,t is an unknown function, and F is a polynomial inu and its conformable time-fractional partial derivatives and space partial derivatives.The main steps of the improved subequation method are presented in [18, 20, 32] as follows:Step 1: suppose that(16)ux,t=Uξ;ξ=kx+Itατt.By substituting (15), NFPDEs turn to the ordinary differential equation (ODE):(17)PU,U′,U'′,…=0,whereP is a polynomial in U and its derivatives, U′=dU/dξ.Step 2: suppose that the solution of equation (15) can be expressed in the following form:(18)Uξ=∑i=−mmaitφiξ,whereaiti=−m,…,m is the function of t to be determined later, and φ=φξ satisfies the following Riccati equation:(19)φ'=σ+φ2ξ,whereσ is a constant. The following solutions of Riccati equation (19) are given by [33](20)φξ=−−σtanh−σξ,σ<0,−−σcoth−σξ,σ<0,σtanσξ,σ>0,−σcotσξ,σ>0,−1ξ+ω,σ=0,whereωis a constant.Step 3: determine the positive integerm 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 φξ, and then setting each coefficient of φiξ to zero, one can get an overdetermined system of nonlinear differential equations for ait and ξ.Step 4: assume thatait 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.


#### 2.2.2. 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(21)ux,t=ϕξ;ξ=kx+Itατt,ϕξ=∑i=1nsini−1ϱbitsinϱ+aitcosϱ+a0t,wherea0t,…,antandb1t,…,bnt are all functions of t to be determined later; equation (15) reduces to ODE(22)Pϕ,ϕ′,ϕ′′,…=0,whereϕ′=dϕ/dξ.Now, taking the target equation as(23)dϱdξ=sinϱ,ordϱdξ=cosϱ,considering the following:(24)sinϱ=sechξandcosϱ=∓tanhξor(25)sinϱ=∓tanhξandcosϱ=−sechξStep 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 sinϱ,cosϱ,sinϱcosϱ,sin2ϱ,…, then setting the coefficients to be zero, one can obtain an overdetermined system of nonlinear differential equations for b0t,bit,ait and τt for i=1,…,n.Step 4: assuming thatbit,ait, 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.


## 2.1. Conformable Fractional Calculus

### 2.1.1. Definition of Conformable Fractional Derivative [14]

Given a functionf:0,∞⟶ℝ, the conformable fractional derivative of f of order α is defined by(3)Dtαft=limξ⟶0ft+ξt1−α−ftξ,for t∈0,∞, α∈0,1. If f is α-differentiable in some interval 0,a, a>0, and limt⟶0+Dtαft exists, then Dtαf0=limt⟶0+Dtαft.Furthermore, ifα=1, the definition is equivalent to the classical definition of the first-order derivative of the function.


### 2.1.2. Definition of Conformable Fractional Integral [14]

The conformable fractional integral off (function continuous) of order α, on interval 0,t, is defined by f function continuous:(4)Itαft=∫0txα−1fxdx.


### 2.1.3. Properties of Conformable Fractional Derivative [30]

Some important properties of the conformable fractional derivative and the conformable fractional integral are as follows.Letα∈0,1 and f,g be α-differentiable at a point t>0. Then, there are the following propertiesProperty 1.
(differential of a constant).(5)Dtαc=0,wherecis a constant.Property 2.
(the linearity property).(6)Dtαaft+bgt=aDtαft+bDtαgt;a,bare constants.Property 3.
(product rule).(7)Dtαf.gt=fDtαgt+gDtαft.Property 4.
(quotient rule).(8)Dtαfg=gDtαft−fDtαgtg2t.Property 5.
(chain rule).(9)Dtαfοgt=f′gtDtαgt.Property 6.
(10)Dtαtp=ptp−α;p∈ℝ.Property7.
Letα∈0,1 and f is be α-differentiable at a point t>0. If f is differentiable, then(11)Dtαft=t1−αdfdt.Property 8.
Letα∈0,1 and f is any continuous function in a domain of Itα, for t>a, we have(12)DtαItαft=ft,(13)ItαDtαft=ft−f0;on the interval0,t.


### 2.1.4. Theorem [31]

Letf: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(14)Dtαfοgt=t1−αg′tf′gt.


## 2.1.1. Definition of Conformable Fractional Derivative [14]

Given a functionf:0,∞⟶ℝ, the conformable fractional derivative of f of order α is defined by(3)Dtαft=limξ⟶0ft+ξt1−α−ftξ,for t∈0,∞, α∈0,1. If f is α-differentiable in some interval 0,a, a>0, and limt⟶0+Dtαft exists, then Dtαf0=limt⟶0+Dtαft.Furthermore, ifα=1, the definition is equivalent to the classical definition of the first-order derivative of the function.


## 2.1.2. Definition of Conformable Fractional Integral [14]

The conformable fractional integral off (function continuous) of order α, on interval 0,t, is defined by f function continuous:(4)Itαft=∫0txα−1fxdx.


## 2.1.3. Properties of Conformable Fractional Derivative [30]

Some important properties of the conformable fractional derivative and the conformable fractional integral are as follows.Letα∈0,1 and f,g be α-differentiable at a point t>0. Then, there are the following propertiesProperty 1.
(differential of a constant).(5)Dtαc=0,wherecis a constant.Property 2.
(the linearity property).(6)Dtαaft+bgt=aDtαft+bDtαgt;a,bare constants.Property 3.
(product rule).(7)Dtαf.gt=fDtαgt+gDtαft.Property 4.
(quotient rule).(8)Dtαfg=gDtαft−fDtαgtg2t.Property 5.
(chain rule).(9)Dtαfοgt=f′gtDtαgt.Property 6.
(10)Dtαtp=ptp−α;p∈ℝ.Property7.
Letα∈0,1 and f is be α-differentiable at a point t>0. If f is differentiable, then(11)Dtαft=t1−αdfdt.Property 8.
Letα∈0,1 and f is any continuous function in a domain of Itα, for t>a, we have(12)DtαItαft=ft,(13)ItαDtαft=ft−f0;on the interval0,t.


## 2.1.4. Theorem [31]

Letf: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(14)Dtαfοgt=t1−αg′tf′gt.


## 2.2. Description of Two Methods

### 2.2.1. 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(15)Fu,Dtαu,ux,Dt2α,uxx,…=0;0<α≤1,where Dtαu is the conformable fractional derivative of u, u=ux,t is an unknown function, and F is a polynomial inu and its conformable time-fractional partial derivatives and space partial derivatives.The main steps of the improved subequation method are presented in [18, 20, 32] as follows:Step 1: suppose that(16)ux,t=Uξ;ξ=kx+Itατt.By substituting (15), NFPDEs turn to the ordinary differential equation (ODE):(17)PU,U′,U'′,…=0,whereP is a polynomial in U and its derivatives, U′=dU/dξ.Step 2: suppose that the solution of equation (15) can be expressed in the following form:(18)Uξ=∑i=−mmaitφiξ,whereaiti=−m,…,m is the function of t to be determined later, and φ=φξ satisfies the following Riccati equation:(19)φ'=σ+φ2ξ,whereσ is a constant. The following solutions of Riccati equation (19) are given by [33](20)φξ=−−σtanh−σξ,σ<0,−−σcoth−σξ,σ<0,σtanσξ,σ>0,−σcotσξ,σ>0,−1ξ+ω,σ=0,whereωis a constant.Step 3: determine the positive integerm 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 φξ, and then setting each coefficient of φiξ to zero, one can get an overdetermined system of nonlinear differential equations for ait and ξ.Step 4: assume thatait 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.


### 2.2.2. 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(21)ux,t=ϕξ;ξ=kx+Itατt,ϕξ=∑i=1nsini−1ϱbitsinϱ+aitcosϱ+a0t,wherea0t,…,antandb1t,…,bnt are all functions of t to be determined later; equation (15) reduces to ODE(22)Pϕ,ϕ′,ϕ′′,…=0,whereϕ′=dϕ/dξ.Now, taking the target equation as(23)dϱdξ=sinϱ,ordϱdξ=cosϱ,considering the following:(24)sinϱ=sechξandcosϱ=∓tanhξor(25)sinϱ=∓tanhξandcosϱ=−sechξStep 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 sinϱ,cosϱ,sinϱcosϱ,sin2ϱ,…, then setting the coefficients to be zero, one can obtain an overdetermined system of nonlinear differential equations for b0t,bit,ait and τt for i=1,…,n.Step 4: assuming thatbit,ait, 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.


## 2.2.1. 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(15)Fu,Dtαu,ux,Dt2α,uxx,…=0;0<α≤1,where Dtαu is the conformable fractional derivative of u, u=ux,t is an unknown function, and F is a polynomial inu and its conformable time-fractional partial derivatives and space partial derivatives.The main steps of the improved subequation method are presented in [18, 20, 32] as follows:Step 1: suppose that(16)ux,t=Uξ;ξ=kx+Itατt.By substituting (15), NFPDEs turn to the ordinary differential equation (ODE):(17)PU,U′,U'′,…=0,whereP is a polynomial in U and its derivatives, U′=dU/dξ.Step 2: suppose that the solution of equation (15) can be expressed in the following form:(18)Uξ=∑i=−mmaitφiξ,whereaiti=−m,…,m is the function of t to be determined later, and φ=φξ satisfies the following Riccati equation:(19)φ'=σ+φ2ξ,whereσ is a constant. The following solutions of Riccati equation (19) are given by [33](20)φξ=−−σtanh−σξ,σ<0,−−σcoth−σξ,σ<0,σtanσξ,σ>0,−σcotσξ,σ>0,−1ξ+ω,σ=0,whereωis a constant.Step 3: determine the positive integerm 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 φξ, and then setting each coefficient of φiξ to zero, one can get an overdetermined system of nonlinear differential equations for ait and ξ.Step 4: assume thatait 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.


## 2.2.2. 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(21)ux,t=ϕξ;ξ=kx+Itατt,ϕξ=∑i=1nsini−1ϱbitsinϱ+aitcosϱ+a0t,wherea0t,…,antandb1t,…,bnt are all functions of t to be determined later; equation (15) reduces to ODE(22)Pϕ,ϕ′,ϕ′′,…=0,whereϕ′=dϕ/dξ.Now, taking the target equation as(23)dϱdξ=sinϱ,ordϱdξ=cosϱ,considering the following:(24)sinϱ=sechξandcosϱ=∓tanhξor(25)sinϱ=∓tanhξandcosϱ=−sechξStep 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 sinϱ,cosϱ,sinϱcosϱ,sin2ϱ,…, then setting the coefficients to be zero, one can obtain an overdetermined system of nonlinear differential equations for b0t,bit,ait and τt for i=1,…,n.Step 4: assuming thatbit,ait, 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.


## 3. Applications of Two Methods

### 3.1. Applications of the Improved Subequation Method

Using the generalized traveling wave transformation,(26)ux,t=Uξ,ξ=kx+Itατt,and Property 2.1.3 and Theorem 2.1.4, equation (2) can be reduced to the following nonlinear ordinary differential equation:(27)τt+kβtU′−γtk2U'′+δtU1−Uμ−U=0,where U′=dU/dξ and U'′=d2U/dξ2. We suppose that equation (27) has a solution in the form of (18). Balancing the highest order derivative term U'′ and with nonlinear term U3 in equation (27), one can get m+2=3m⇒m=1. So, we have(28)Uξ=∑i=−11aitφiξ=a0t+a1tφξ+a−1tφ−1ξ.Substituting (28) into equation (2), collecting all the terms with the same power ofφii=−3,−2,…,2,3, then setting the coefficients ofφii=−3,−2,…,2,3 to be zero, one can obtain the overdetermined system of nonlinear differential equations for a−1t,a0t,a1t, and τt as follows:(29)φ−3:−2k2σ2γta−1t+δta−13t=0,(30)φ−2:−kσβta−1t+3δta0ta−12t−στta−1t−1+μδta−12t=0,(31)φ−1:t1−αda−1tdt−21+μδta0ta−1t+3δta−1ta02t+3δta1ta−12t+μδta−1t−2k2σγta−1t=0,(32)φ0:t1−αda0tdt+kσβta1t+6δta0ta−1ta1t−21+μδta1ta−1t−βtka−1t+τtσa1t−τta−1t−1+μδta02t+δta03t+μδta0t=0,(33)φ:t1−αda1tdt−21+μδta0ta1t+3δta02ta1t+3δta−1ta12t+μδta1t−2k2σγta1t=0,(34)φ2:τta1t+kβta1t+3δta0ta12t−1+μδta12t=0,(35)φ3:−2k2γta1t+δta13t=0.Ψt=exp−∫tα−1ftdt,whereft 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 thata−1t=0, a0t=Ψt, It can be obtained that(36)a1t=c1exp∫tα−121+μδtΨt−3δtΨ2t−μδt+2k2σγtdt,(37)τt=−1σa1tβtσka1t−ftΨt+δt−1+2μΨ2t+Ψ3t+μΨt.Whenσ<0, the following hyperbolic function solution of equation (2) is obtained as follows:(38)ux,t=Ψt−−σa1ttanh−σkx+Itατt,(39)ux,t=Ψt−−σa1tcoth−σkx+Itατt.Whenσ>0, we obtain the following trigonometric function solution of equation (2), as follows:(40)ux,t=Ψt+σa1ttanσkx+Itατt,(41)ux,t=Ψt−σa1tcotσkx+Itατt.Whenσ=0, substituting σ=0 into the abovementioned system and then solving the obtained system, one can find a1t will not change but τt is changed and becomes equal to τ1t=−βtk+−3Ψt+1+μa1tδt.Whenσ=0, we have the following rational solution of equation (2), as follows:(42)ux,t=Ψt−1kx+Itατ1t+ωa1t.Case 2.If we assume thata1t=0, a0t=Ψt, it can be obtained that(43)a−1t=c2exp∫​tα−12δtΨt−3δtΨ2t−μδt+2k2σγt+2μδtΨtdt,τt=−1a−1tβtka1t+ftΨt+δt−Ψ3t+1+μΨ2t−μΨt.We construct the following hyperbolic function solution of equation (2), where σ<0 is as follows:(44)ux,t=Ψt−a−1t−σtanh−σkx+Itατt,(45)ux,t=Ψt−a−1t−σcoth−σkx+Itατt.We construct the following hyperbolic function solution of equation (2), where σ>0 is as follows:(46)ux,t=Ψt+a−1tσtanσkx+Itατt,(47)ux,t=Ψt−a−1tσcotσkx+Itατt.Whenσ=0, we have the following solution of equation (2), as follows:(48)ux,t=Ψt−kx+Itατt+ωa−1t.Case 3.If we assume thata0t=c3Ψt, a1t=c4Ψt, it can be obtained that(49)a−1t=exp∫​tα−1ϵtdtc5+3∫exp∫ϵttα−1dtδtc4Ψttα−1dt,where(50)ϵt=2c3δtΨt−3c32δtΨ2t−μδt+2k2σγt+2c3μδtΨt,τt=1σc4Ψt−a−1t−kσβtc4Ψt−6δtc3Ψ2tc4a−1t+δt1+μ2c4Ψta−1t+c32Ψ2t+βtka−1t+c3Ψtft−δtc33Ψ3t+μc3Ψt.Whenσ<0, we obtain the following hyperbolic function solution of equation (2), as follows:(51)ux,t=c3Ψt−c4−σΨttanh−σkx+Itατt−a−1t−σtanh−σkx+Itατt,(52)ux,t=c3Ψt−c4−σΨtcoth−σkx+Itατt−a−1t−σcoth−σkx+Itατt.Whenσ>0, we obtain the following trigonometric function solution of equation (2), as follows:(53)ux,t=c3Ψt+c4σΨttanσkx+Itατt+a−1tσtanσkx+Itατt,(54)ux,t=c3Ψt−c4σΨtcotσkx+Itατt−a−1tσcotσkx+Itατt.Whenσ=0, we have the following rational solution of equation (2), as follows:(55)ux,t=c3Ψt−1kx+Itατt+ωc4Ψt−kx+Itατt+ωa−1t.Case 4.If we assume thata0t=c3Ψt, a1t=c4Ψt, it is obtained that(56)τt=−kβt−3c4δtc3Ψ2t+μ+1δtc4Ψt,a−1t=δtΨ2t−3c42σc3+c33+δt1+μΨtc42σ−c32−c3ft+c3μδt3c4δt−3c3Ψt+μ+1.Whenσ<0, we obtain the following hyperbolic function solution of equation (2), as follows: (57)ux,t=c3Ψt−c4−σΨttanh−σkx+Itατt−a−1t−σtanh−σkx+Itατt,(58)ux,t=c3Ψt−c4−σΨtcoth−σkx+Itατt−a−1t−σcoth−σkx+Itατt.Whenσ>0, we obtain the following trigonometric function solution of equation (2), as follows:(59)ux,t=c3Ψt+c4σΨttanσkx+Itατt+a−1tσtanσkx+Itατt,(60)ux,t=c3Ψt−c4σΨtcotσkx+Itατt−a−1tσcotσkx+Itατt.Whenσ=0, we have the following rational solution of equation (2), as follows:(61)ux,t=c3Ψt−1kx+Itατt+ωc4Ψt−kx+Itατt+ωa−1t,wherec1, c2, c3, c4, c5, and k are arbitrary constants.In Figure1, 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 kink-shape 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.Figure 1
The 3D graph whenα=0.5,ft=t0.5,σ=−1,c1=k=μ=1,δt=et,γt=2−3e−t/2−et/2+t0.5/2,andβt=−t0.5. (a) of solution (38) and (b) of solution (39).
(a)(b)In Figure2. (a) clarifies the propagation of the kink-shape wave soliton solution in the x_t plane. The wave moves with the same shape and amplitude over its the track in the x_t domain, while (b) describes the propagation of the spike-shape wave soliton solution in the x_t plane, with the same time-dependent coefficients where α=0.3.Figure 2
The 3D graph whenα=0.3,ft=t0.7,σ=−1,c1=k=μ=1,δt=et,γt=2−3/2e−t−1/2et−1/2t0.7,andβt=−3e−2t. (a) of solution (38) and (b) of solution (39).
(a)(b)


### 3.2. Applications of the Improved Sine-Cosine Method

Using the generalized traveling wave transformation,(62)ux,t=ϕξ,ξ=kx+Itατt,and Property 2.1.3 and Theorem 2.1.4, equation (2) can be reduced to the following nonlinear ordinary differential equation:(63)τt+kβtϕ′−γtk2ϕ″+δtμϕ−ϕ2−μϕ2+ϕ3=0.Balancing the highest order derivative termϕ'' with nonlinear term ϕ3, we can get n+2=3n⇒n=1. So, we have(64)ϕξ=b1tsinϱ+a1tcosϱ+a0t.Substituting (64) and (23) into equation (2) and collecting all the terms with the same power ofsiniϱcosjϱ 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 b1t,a0t, a1t, and τt as follows:(65)sinϱ:t1−αdb1tdt+k2γtb1t+μδtb1t+3δta02tb1t−21+μδta0tb1t+δtb13t=0,(66)sinϱcosϱ:τtb1t+kβtb1t−21+μδta1tb1t+6δta0tb1ta1t=0,(67)cosϱ:t1−αda1tdt+2k2γta1t+3δta1ta02t+μδta1t+3δta1tb12t−21+μδta0ta1t=0,(68)sin0ϱcos0ϱ:t1−αda0tdt−kβta1t+3δtb12ta0t−a1tτt−1+μδtb12t−1+μδta02t+δta03t+μδta0t=0,(69)sinϱcos2ϱ:−2k2γtb1t−δtb13t+3δta12tb1t=0,(70)cos2ϱ:τta1t+kβta1t−3δta0tb12t+3δta0ta12t−1+μδta12t+1+μδtb12t=0,(71)cos3ϱ:−2k2γta1t+δta13t−3δta1tb12t=0.Solving this system by Maple, for simplicity, we introduce the notationΨt=exp−∫tα−1ftdt, where ft is a continuous function.Finally, one finds these four cases as follows:Case 1.If we assume thatb1t=0, a0t=Ψt, it can be obtained that(72)a1t=c6exp∫tα−1−2k2γt+2δtΨt−3δtΨ2t−μδt+2μδtΨtdt,τt=1+μ−3Ψtδtc6exp∫tα−1−2k2γt+2δtΨt−3δtΨ2t−μδt+2μδtΨtdt−kβt.We substitute what mentioned before into (64) and using (24) then, we obtain the hyperbolic functions solution of equation (2), as follows:(73)u1x,t=±a1ttanhkx+Itατt+Ψt.Case 2.If we assume thata0t=0, a1t=Ψt, it can be obtained that(74)τt=−kβt+2μ+1δtΨt,(75)b1t=±1ω1+c7ω1t+c7exp−2tαk2γtα−2μω2t,where(76)ω1t=∫​2tα−1δtexp2∫tα−1k2γt+μδtdtdt,ω2t=∫tα−1δtdt.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:(77)u2x,t=±Ψttanhkx+Itατt+b1tsechkx+Itατt.Case 3.If we assume thata0t=c8Ψt, a1t=c9Ψt, it is obtained that(78)b1t=±1δt−2tαk2γt+3c92δtΨ2t,τt=−6c8c9δtΨ2t−kβt+2c9μ+1δtΨt.We substitute the solutions of the Case 3 into (64) and using (24), and then we give the hyperbolic functions solution of equation (2), as follows:(79)u3x,t=±c9Ψttanhkx+Itατt+b1tsechkx+Itατt+c8Ψt.Case 4.If we assume thata0t=c8Ψt, a1t=c9Ψt, it is obtained that(80)b1t=0,τt=−6c8c9δtΨ2t−kβt+2c9μ+1δtΨt.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:(81)u4x,t=±c9tanhkx+Itατt+c8Ψt,wherec6, c7, c8, c9, and k are arbitrary constants.In Figure3, it is clearly shown that (a) the space-time graph of solution (73) when α=0.5,ft=t0.5,c6=k=μ=1,δt=et,βt=e−t,andγt=1/24−3e−t−et+t0.5 clarify the propagation of the kink-shape wave soliton solution and (b) the space-time graph of solution (79) when α=0.5,ft=t0.5,c8=c9=k=μ=1,δt=e−2t,βt=4e−3t,andγt=t−0.5, represents the propagation of the anti-kink-shape wave soliton solution.Figure 3
(a) The 3D graph of solution (73) when α=0.5. (b) The 3D graph of solution (79) when α=0.5.
(a)(b)


## 3.1. Applications of the Improved Subequation Method

Using the generalized traveling wave transformation,(26)ux,t=Uξ,ξ=kx+Itατt,and Property 2.1.3 and Theorem 2.1.4, equation (2) can be reduced to the following nonlinear ordinary differential equation:(27)τt+kβtU′−γtk2U'′+δtU1−Uμ−U=0,where U′=dU/dξ and U'′=d2U/dξ2. We suppose that equation (27) has a solution in the form of (18). Balancing the highest order derivative term U'′ and with nonlinear term U3 in equation (27), one can get m+2=3m⇒m=1. So, we have(28)Uξ=∑i=−11aitφiξ=a0t+a1tφξ+a−1tφ−1ξ.Substituting (28) into equation (2), collecting all the terms with the same power ofφii=−3,−2,…,2,3, then setting the coefficients ofφii=−3,−2,…,2,3 to be zero, one can obtain the overdetermined system of nonlinear differential equations for a−1t,a0t,a1t, and τt as follows:(29)φ−3:−2k2σ2γta−1t+δta−13t=0,(30)φ−2:−kσβta−1t+3δta0ta−12t−στta−1t−1+μδta−12t=0,(31)φ−1:t1−αda−1tdt−21+μδta0ta−1t+3δta−1ta02t+3δta1ta−12t+μδta−1t−2k2σγta−1t=0,(32)φ0:t1−αda0tdt+kσβta1t+6δta0ta−1ta1t−21+μδta1ta−1t−βtka−1t+τtσa1t−τta−1t−1+μδta02t+δta03t+μδta0t=0,(33)φ:t1−αda1tdt−21+μδta0ta1t+3δta02ta1t+3δta−1ta12t+μδta1t−2k2σγta1t=0,(34)φ2:τta1t+kβta1t+3δta0ta12t−1+μδta12t=0,(35)φ3:−2k2γta1t+δta13t=0.Ψt=exp−∫tα−1ftdt,whereft 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 thata−1t=0, a0t=Ψt, It can be obtained that(36)a1t=c1exp∫tα−121+μδtΨt−3δtΨ2t−μδt+2k2σγtdt,(37)τt=−1σa1tβtσka1t−ftΨt+δt−1+2μΨ2t+Ψ3t+μΨt.Whenσ<0, the following hyperbolic function solution of equation (2) is obtained as follows:(38)ux,t=Ψt−−σa1ttanh−σkx+Itατt,(39)ux,t=Ψt−−σa1tcoth−σkx+Itατt.Whenσ>0, we obtain the following trigonometric function solution of equation (2), as follows:(40)ux,t=Ψt+σa1ttanσkx+Itατt,(41)ux,t=Ψt−σa1tcotσkx+Itατt.Whenσ=0, substituting σ=0 into the abovementioned system and then solving the obtained system, one can find a1t will not change but τt is changed and becomes equal to τ1t=−βtk+−3Ψt+1+μa1tδt.Whenσ=0, we have the following rational solution of equation (2), as follows:(42)ux,t=Ψt−1kx+Itατ1t+ωa1t.Case 2.If we assume thata1t=0, a0t=Ψt, it can be obtained that(43)a−1t=c2exp∫​tα−12δtΨt−3δtΨ2t−μδt+2k2σγt+2μδtΨtdt,τt=−1a−1tβtka1t+ftΨt+δt−Ψ3t+1+μΨ2t−μΨt.We construct the following hyperbolic function solution of equation (2), where σ<0 is as follows:(44)ux,t=Ψt−a−1t−σtanh−σkx+Itατt,(45)ux,t=Ψt−a−1t−σcoth−σkx+Itατt.We construct the following hyperbolic function solution of equation (2), where σ>0 is as follows:(46)ux,t=Ψt+a−1tσtanσkx+Itατt,(47)ux,t=Ψt−a−1tσcotσkx+Itατt.Whenσ=0, we have the following solution of equation (2), as follows:(48)ux,t=Ψt−kx+Itατt+ωa−1t.Case 3.If we assume thata0t=c3Ψt, a1t=c4Ψt, it can be obtained that(49)a−1t=exp∫​tα−1ϵtdtc5+3∫exp∫ϵttα−1dtδtc4Ψttα−1dt,where(50)ϵt=2c3δtΨt−3c32δtΨ2t−μδt+2k2σγt+2c3μδtΨt,τt=1σc4Ψt−a−1t−kσβtc4Ψt−6δtc3Ψ2tc4a−1t+δt1+μ2c4Ψta−1t+c32Ψ2t+βtka−1t+c3Ψtft−δtc33Ψ3t+μc3Ψt.Whenσ<0, we obtain the following hyperbolic function solution of equation (2), as follows:(51)ux,t=c3Ψt−c4−σΨttanh−σkx+Itατt−a−1t−σtanh−σkx+Itατt,(52)ux,t=c3Ψt−c4−σΨtcoth−σkx+Itατt−a−1t−σcoth−σkx+Itατt.Whenσ>0, we obtain the following trigonometric function solution of equation (2), as follows:(53)ux,t=c3Ψt+c4σΨttanσkx+Itατt+a−1tσtanσkx+Itατt,(54)ux,t=c3Ψt−c4σΨtcotσkx+Itατt−a−1tσcotσkx+Itατt.Whenσ=0, we have the following rational solution of equation (2), as follows:(55)ux,t=c3Ψt−1kx+Itατt+ωc4Ψt−kx+Itατt+ωa−1t.Case 4.If we assume thata0t=c3Ψt, a1t=c4Ψt, it is obtained that(56)τt=−kβt−3c4δtc3Ψ2t+μ+1δtc4Ψt,a−1t=δtΨ2t−3c42σc3+c33+δt1+μΨtc42σ−c32−c3ft+c3μδt3c4δt−3c3Ψt+μ+1.Whenσ<0, we obtain the following hyperbolic function solution of equation (2), as follows: (57)ux,t=c3Ψt−c4−σΨttanh−σkx+Itατt−a−1t−σtanh−σkx+Itατt,(58)ux,t=c3Ψt−c4−σΨtcoth−σkx+Itατt−a−1t−σcoth−σkx+Itατt.Whenσ>0, we obtain the following trigonometric function solution of equation (2), as follows:(59)ux,t=c3Ψt+c4σΨttanσkx+Itατt+a−1tσtanσkx+Itατt,(60)ux,t=c3Ψt−c4σΨtcotσkx+Itατt−a−1tσcotσkx+Itατt.Whenσ=0, we have the following rational solution of equation (2), as follows:(61)ux,t=c3Ψt−1kx+Itατt+ωc4Ψt−kx+Itατt+ωa−1t,wherec1, c2, c3, c4, c5, and k are arbitrary constants.In Figure1, 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 kink-shape 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.Figure 1
The 3D graph whenα=0.5,ft=t0.5,σ=−1,c1=k=μ=1,δt=et,γt=2−3e−t/2−et/2+t0.5/2,andβt=−t0.5. (a) of solution (38) and (b) of solution (39).
(a)(b)In Figure2. (a) clarifies the propagation of the kink-shape wave soliton solution in the x_t plane. The wave moves with the same shape and amplitude over its the track in the x_t domain, while (b) describes the propagation of the spike-shape wave soliton solution in the x_t plane, with the same time-dependent coefficients where α=0.3.Figure 2
The 3D graph whenα=0.3,ft=t0.7,σ=−1,c1=k=μ=1,δt=et,γt=2−3/2e−t−1/2et−1/2t0.7,andβt=−3e−2t. (a) of solution (38) and (b) of solution (39).
(a)(b)


## 3.2. Applications of the Improved Sine-Cosine Method

Using the generalized traveling wave transformation,(62)ux,t=ϕξ,ξ=kx+Itατt,and Property 2.1.3 and Theorem 2.1.4, equation (2) can be reduced to the following nonlinear ordinary differential equation:(63)τt+kβtϕ′−γtk2ϕ″+δtμϕ−ϕ2−μϕ2+ϕ3=0.Balancing the highest order derivative termϕ'' with nonlinear term ϕ3, we can get n+2=3n⇒n=1. So, we have(64)ϕξ=b1tsinϱ+a1tcosϱ+a0t.Substituting (64) and (23) into equation (2) and collecting all the terms with the same power ofsiniϱcosjϱ 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 b1t,a0t, a1t, and τt as follows:(65)sinϱ:t1−αdb1tdt+k2γtb1t+μδtb1t+3δta02tb1t−21+μδta0tb1t+δtb13t=0,(66)sinϱcosϱ:τtb1t+kβtb1t−21+μδta1tb1t+6δta0tb1ta1t=0,(67)cosϱ:t1−αda1tdt+2k2γta1t+3δta1ta02t+μδta1t+3δta1tb12t−21+μδta0ta1t=0,(68)sin0ϱcos0ϱ:t1−αda0tdt−kβta1t+3δtb12ta0t−a1tτt−1+μδtb12t−1+μδta02t+δta03t+μδta0t=0,(69)sinϱcos2ϱ:−2k2γtb1t−δtb13t+3δta12tb1t=0,(70)cos2ϱ:τta1t+kβta1t−3δta0tb12t+3δta0ta12t−1+μδta12t+1+μδtb12t=0,(71)cos3ϱ:−2k2γta1t+δta13t−3δta1tb12t=0.Solving this system by Maple, for simplicity, we introduce the notationΨt=exp−∫tα−1ftdt, where ft is a continuous function.Finally, one finds these four cases as follows:Case 1.If we assume thatb1t=0, a0t=Ψt, it can be obtained that(72)a1t=c6exp∫tα−1−2k2γt+2δtΨt−3δtΨ2t−μδt+2μδtΨtdt,τt=1+μ−3Ψtδtc6exp∫tα−1−2k2γt+2δtΨt−3δtΨ2t−μδt+2μδtΨtdt−kβt.We substitute what mentioned before into (64) and using (24) then, we obtain the hyperbolic functions solution of equation (2), as follows:(73)u1x,t=±a1ttanhkx+Itατt+Ψt.Case 2.If we assume thata0t=0, a1t=Ψt, it can be obtained that(74)τt=−kβt+2μ+1δtΨt,(75)b1t=±1ω1+c7ω1t+c7exp−2tαk2γtα−2μω2t,where(76)ω1t=∫​2tα−1δtexp2∫tα−1k2γt+μδtdtdt,ω2t=∫tα−1δtdt.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:(77)u2x,t=±Ψttanhkx+Itατt+b1tsechkx+Itατt.Case 3.If we assume thata0t=c8Ψt, a1t=c9Ψt, it is obtained that(78)b1t=±1δt−2tαk2γt+3c92δtΨ2t,τt=−6c8c9δtΨ2t−kβt+2c9μ+1δtΨt.We substitute the solutions of the Case 3 into (64) and using (24), and then we give the hyperbolic functions solution of equation (2), as follows:(79)u3x,t=±c9Ψttanhkx+Itατt+b1tsechkx+Itατt+c8Ψt.Case 4.If we assume thata0t=c8Ψt, a1t=c9Ψt, it is obtained that(80)b1t=0,τt=−6c8c9δtΨ2t−kβt+2c9μ+1δtΨt.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:(81)u4x,t=±c9tanhkx+Itατt+c8Ψt,wherec6, c7, c8, c9, and k are arbitrary constants.In Figure3, it is clearly shown that (a) the space-time graph of solution (73) when α=0.5,ft=t0.5,c6=k=μ=1,δt=et,βt=e−t,andγt=1/24−3e−t−et+t0.5 clarify the propagation of the kink-shape wave soliton solution and (b) the space-time graph of solution (79) when α=0.5,ft=t0.5,c8=c9=k=μ=1,δt=e−2t,βt=4e−3t,andγt=t−0.5, represents the propagation of the anti-kink-shape wave soliton solution.Figure 3
(a) The 3D graph of solution (73) when α=0.5. (b) The 3D graph of solution (79) when α=0.5.
(a)(b)


## 4. 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 ofα. 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.



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