# Solitary Wave Solutions of Conformable Time Fractional Equations Using Modified Simplest Equation Method

Authors: Waseem Razzaq; Mustafa Habib; Muhammad Nadeem; Asim Zafar; Ilyas Khan; Patrick Kandege Mwanakatwea
Journal: Complexity
 (2022)

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## Abstract

This study presents a modified simplest equation method (MSEM) to investigate some real and exact solutions of conformable time fractional Benjamin-Bona-Mahony (BBM) equation and Chan-Hilliard (CH) equation. We use traveling wave transformation to obtain the results in the form of series solution. Some calculations are performed through Mathematica software to analyze the accuracy of this approach. Graphical representations are reported for more significant results at different fractional-order which demonstrates that this approach is very simple, adequate, and legitimate.


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## Body

## 1. Introduction

Various applications of applied mathematics in the form of partial differential equations (PDEs) play important role with derivative and integral of arbitrary orders. Fractional differential equations (FDEs) are used for modeling in various fields of sciences such as physics, chemistry, mathematics, biology, engineering, fluid dynamics, and other physical phenomena [1–4]. So, they have gained much attention of many scholar. The exact/analytical solutions of these PDEs are ideal. Therefore, a lot of scholars have paid much attentions to acquire the solutions of these kinds of FDEs with the help of various numerical and analytical methods. Mainly, a wave transformation is used to transform the FDEs into a nonlinear ordinary differential equation, which leads for further exact solution. A lot of significant approaches have been generated and applied to search exact solutions in different research articles including Lie symmetry analysis method [5–8], the differential transform method [9], the iteration method [10, 11], the (G’/G)-expansion method [12], homotopy analysis method [13], Adomian decomposition method [14], homotopy perturbation method [15], the functional variable method [16], the first integral method [1], modified Kudryashov method [17], G′/G,1/G-expansion [18], and improved G′/G-expansion method [19]. Very recently, Wang presented a new idea on the fractal theory of nonlinear system to find the solitary wave solutions which show the promising results [20–22]. A full detail of conformable time fractional equations has been explained in [23, 24].In this work, we will go over the conformable time fractional BBM and nonlinear space-time fractional CH equation. Many inquirer quaere the exact solutions of these equations by applying different approaches in the past [25–27]. Some recent developments in time fractional equations developments can be studied through He’s fractional derivative and the two-scale fractal derivative [28, 29]. We discussed some of their investigations here. They used fractional exp-function method to construct the exact solutions for the conformable time fractional Benjamin-Bona-Mahony equation and conformable time fractional Chan-Hilliard equation. The scheme of this paper is as follows: in Section 2, we will briefly discuss some basic properties of conformable fractional derivative. In Section 3, we present the description of simplest equation method. Two examples of conformable time fractional BBM and CH have been tested in Section 4 to verify this approach and conclusion is discussed in Section 5.


## 2. Conformable Fractional Derivative

In this part, we shortly explain the definition and properties of conformable fractional derivatives [30–33].Definition: letf:0,∞⟶ℝ be a function. Then, the conformable fractional derivative of f for order δ is defined as(1)Tδft=limε⟶0ft+εt1−δ−ftε,∀t>0,δ∈0,1.Assuming the conformable fractional derivative onf regarding order δ, then we can say f is δ-differentiable.Properties: letf and g be δ-differentiable at a value t>0, then the conformable fractional derivative has some useful properties as follows:(1)
Tδaf+bg=aTδf+bTδg, for all a,b∈ℝ(2)
Tδfg=fTδg+gTδf(3)
Tδtμ=μtμ−δ, for all μ∈ℝ(4)
Tδλ=0, where λ is constant(5)
Tδf/g=gTδf−fTδg/g2(6)
Iff is differentiable, then Tδft=t1−δdft/dtThis definition also satisfies the chain rule as follows.Letf,g:0,∞⟶ℝ be an δ-differentiable function and δ∈0,1. Then, the following rule is defined.(2)Tδf°gt=t1−δg′tf′gt.


## 3. Description of the Modified Simplest Equation Method (MSEM)

Consider the following nonlinear PDE in the form time fractional(3)Fu,Dtδϑ,ϑxi,ϑxi,xj,ϑxi,xj,xk,…=0,where Dtδ is known as conformable fractional derivative and δ∈0,1. (3) has n+1 independent variables x=x1,x2,…,xn and dependent variable ϑ=ϑx,t, where(4)ϑxi=∂ϑ∂xi,ϑxi,xj=∂2ϑ∂xi∂xj,ϑxi,xj,xk=∂3ϑ∂xi∂xj∂xk,…We will execute MSEM to provide particular results of (3). Over here, we summarize the fundamental points of MSEM.Step 1.
Employing the fractional traveling wave transform,(5)ϑx,t=Vℑ,ℑ=∑i=1nkixδδ−ltδδ,where k, l are nonzero constants; we can rewrite (3) as the following nonlinear ODE:(6)NV,lVℑ,kVℑℑ,…=0,where V=dV/dℑ,Vℑℑ=dV2/dℑ2,…Step 2.
Consider the solution of equation (6) in the following form:(7)Vℑ=∑i=1maiϕiℑ,where aii=1,2,…,m are constants and am≠0, whereas ϕℑ conforms to some ordinary differential equations. Now, consider the Riccati equations such as(8)ϕ′ℑ=ϕ2ℑ+ϱ,where ϱ is an arbitrary constant and prime represents the differentiation according to ℑ. We can dissolve the solution of (8) in the following possible ways:
Ifϱ<0, then(9)ϕℑ=−−ϱtanh−ϱℑ,(10)ϕℑ=−−ϱcoth−ϱℑ,(11)ϕℑ=−ϱ−tanh2−ϱℑ±isech2−ϱℑ,(12)ϕℑ=−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.
Ifϱ>0, then(13)ϕℑ=ϱtanϱℑ,(14)ϕℑ=−ϱcotϱℑ,(15)ϕℑ=ϱtan2ϱℑ±sec2ϱℑ,(16)ϕℑ=ϱ−cot2ϱℑ±csc2ϱℑ,(17)ϕℑ=ϱ2tanϱ2ℑ−cotϱ2ℑ,
Ifϱ=0, then(18)ϕℑ=−1ℑ.Step 3.
After substituting equations (7) and (8) into (6), and setting all the coefficients of ϕi to zero, we can get the system of algebraic equations in terms of ai,k,l, which leads to a very simple solution.Step 4.
The obtained values ofai,k,l from Step 2 can be used in equation (5) to achieve the traveling wave solutions of equation (3).
Remarks. The results in equation (9) or (10) are referred to as the solitary wave solutions and the results in equation (13) or (14) are considered to be periodic function solutions, whereas the result in equation (18) is termed as rational function solution.


## 4. Applications of MSEM

In this section, MSEM is applied to compute the solitary wave solutions of conformable time fractional differential BBM and CH equation.


### 4.1. Example 1

Consider the conformable time fractional differential BBM equation:(19)Dtδϑ−νϑ2ϑx+ϑx+ϑxxx=0,where δ is a parameter representing the order of the fractional time derivative. When δ=1, equation (19) is called classical Benjamin-Bona-Mahony equation. We used a new proposed simplest equation method and applied it on the conformable time fractional differential Benjamin-Bona-Mahony equation for obtaining the exact solutions.By using the transformation,(20)ϑx,t=Vℑ,ℑ=kx−ltδδ.Equation (19) can be reduced to the following nonlinear ODE:(21)k−lV′−νkV′V2+k3V‴=0.BalancingV‴andV′V2 in equation (22), then we drive m=1.(22)Vℑ=a1ϕℑ+a0.Putting equation (22) into (21) and using some mathematical operations, we obtained an algebraic system. Solving this system for different value of a0,a1,k,l results in the following.Case 1.
(23)a0=0,a1=−6kν,l=2k3ϱ+k,where ϱ<0. Therefore, we obtain the following solitary wave solution for the equation:(24)ϑx,t=−6k−−ϱtanh−ϱkx−k−2k3ϱtδ/δν,(25)ϑx,t=−6k−−ϱcoth−ϱkx−k−2k3ϱtδ/δν,(26)ϑx,t=−6kν−ϱ−tanh2−ϱℑ±isech2−ϱℑ,(27)ϑx,t=−6kν−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ,where ϱ>0. Therefore, we obtain the following periodic function solution for the equation:(28)ϑx,t=−6kϱtanϱkx−2k3ϱ+ktδ/δν,(29)ϑx,t=−6k−ϱcotϱkx−2k3ϱ+ktδ/δν,(30)ϑx,t=−6kνϱtan2ϱℑ±sec2ϱℑ,(31)ϑx,t=−6kνϱ−cot2ϱℑ±csc2ϱℑ,(32)ϑx,t=−6kνϱ2tanϱ2ℑ−cotϱ2ℑ.
Ifϱ=0,(33)ϑx,t=−6kν−1ℑ.Case 2.
(34)a0=0,a1=6kν,l=2k3ϱ+k,where ϱ<0. Therefore, we obtain the following solitary wave solution for the equation:(35)ϑx,t=6k−−ϱtanh−ϱkx−k−2k3ϱtδ/δν,ϑx,t=6k−−ϱcoth−ϱkx−k−2k3ϱtδ/δν,ϑx,t=6kν−ϱ−tanh2−ϱℑ±isech2−ϱℑ,ϑx,t=6kν−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.
Thus, we can obtain the following periodic function solution for the equation:(36)ϑx,t=6kϱtanϱkx−2k3ϱ+ktδ/δν,ϑx,t=6k−ϱcotϱkx−2k3ϱ+ktδ/δν,ϑx,t=6kνϱtan2ϱℑ±sec2ϱℑ,ϑx,t=6kνϱ−cot2ϱℑ±csc2ϱℑ,ϑx,t=6kνϱ2tanϱ2ℑ−cotϱ2ℑ.
Ifϱ=0,(37)ϑx,t=6kν−1ℑ.


### 4.2. Example 2

Let us consider the conformable time fractional CH equation(38)Dtδϑ−3ϑ2−1ϑxx−6ϑϑx2−ϑx+ϑxxxx=0,wheret>0,0<δ≤1.Applying the transformation,(39)ϑx,t=Vℑ,ℑ=kx−ltδδ.Equation (38) is reduced to the following nonlinear ODE:(40)−k+lV′−6k2V′2V−k23V2−1V″+k4V″″=0.Through balancing the termsV‴′ and V′2V, we select m=1, then the nontrivial solution (40) reduces to(41)Vℑ=a1ϕℑ+a0.By setting the above solution in (40) and equating factors of each power of ϕi, we get nonlinear algebraic system, solving this system for different value of a0,a1,k,l. Case 1:(42)a0=0,a1=iϱ,k=±i2ϱ,l=±i2ϱ.Ifϱ<0,(43)ϑx,t=iϱ−−ϱtanh−ϱℑ,(44)ϑx,t=iϱ−−ϱcoth−ϱℑ,(45)ϑx,t=iϱ−ϱ−tanh2−ϱℑ±isech2−ϱℑ,(46)ϑx,t=iϱ−ϱ−coth2−ϱℑ±csch2−ϱℑ,(47)ϑx,t=iϱ−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.Ifϱ>0,(48)ϑx,t=iϱϱtanϱℑ,(49)ϑx,t=iϱ−ϱcotϱℑ,(50)ϑx,t=iϱϱtan2ϱℑ±sec2ϱℑ,(51)ϑx,t=iϱϱ−cot2ϱℑ±csc2ϱℑ,(52)ϑx,t=iϱϱ2tanϱ2ℑ−cotϱ2ℑ.Case 3.
(53)a0=0,a1=−iϱ,k=±i2ϱ,l=±i2ϱ.
Ifϱ<0,(54)ϑx,t=−iϱ−−ϱtanh−ϱℑ,ϑx,t=−iϱ−−ϱcoth−ϱℑ,ϑx,t=−iϱ−ϱ−tanh2−ϱℑ±isech2−ϱℑ,ϑx,t=−iϱ−ϱ−coth2−ϱℑ±csch2−ϱℑ,ϑx,t=−iϱ−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.
Ifϱ>0,(55)ϑx,t=−iϱϱtanϱℑ,ϑx,t=−iϱ−ϱcotϱℑ,ϑx,t=−iϱϱtan2ϱℑ±sec2ϱℑ,ϑx,t=−iϱϱ−cot2ϱℑ±csc2ϱℑ,ϑx,t=−iϱϱ2tanϱ2ℑ−cotϱ2ℑ.


## 4.1. Example 1

Consider the conformable time fractional differential BBM equation:(19)Dtδϑ−νϑ2ϑx+ϑx+ϑxxx=0,where δ is a parameter representing the order of the fractional time derivative. When δ=1, equation (19) is called classical Benjamin-Bona-Mahony equation. We used a new proposed simplest equation method and applied it on the conformable time fractional differential Benjamin-Bona-Mahony equation for obtaining the exact solutions.By using the transformation,(20)ϑx,t=Vℑ,ℑ=kx−ltδδ.Equation (19) can be reduced to the following nonlinear ODE:(21)k−lV′−νkV′V2+k3V‴=0.BalancingV‴andV′V2 in equation (22), then we drive m=1.(22)Vℑ=a1ϕℑ+a0.Putting equation (22) into (21) and using some mathematical operations, we obtained an algebraic system. Solving this system for different value of a0,a1,k,l results in the following.Case 1.
(23)a0=0,a1=−6kν,l=2k3ϱ+k,where ϱ<0. Therefore, we obtain the following solitary wave solution for the equation:(24)ϑx,t=−6k−−ϱtanh−ϱkx−k−2k3ϱtδ/δν,(25)ϑx,t=−6k−−ϱcoth−ϱkx−k−2k3ϱtδ/δν,(26)ϑx,t=−6kν−ϱ−tanh2−ϱℑ±isech2−ϱℑ,(27)ϑx,t=−6kν−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ,where ϱ>0. Therefore, we obtain the following periodic function solution for the equation:(28)ϑx,t=−6kϱtanϱkx−2k3ϱ+ktδ/δν,(29)ϑx,t=−6k−ϱcotϱkx−2k3ϱ+ktδ/δν,(30)ϑx,t=−6kνϱtan2ϱℑ±sec2ϱℑ,(31)ϑx,t=−6kνϱ−cot2ϱℑ±csc2ϱℑ,(32)ϑx,t=−6kνϱ2tanϱ2ℑ−cotϱ2ℑ.
Ifϱ=0,(33)ϑx,t=−6kν−1ℑ.Case 2.
(34)a0=0,a1=6kν,l=2k3ϱ+k,where ϱ<0. Therefore, we obtain the following solitary wave solution for the equation:(35)ϑx,t=6k−−ϱtanh−ϱkx−k−2k3ϱtδ/δν,ϑx,t=6k−−ϱcoth−ϱkx−k−2k3ϱtδ/δν,ϑx,t=6kν−ϱ−tanh2−ϱℑ±isech2−ϱℑ,ϑx,t=6kν−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.
Thus, we can obtain the following periodic function solution for the equation:(36)ϑx,t=6kϱtanϱkx−2k3ϱ+ktδ/δν,ϑx,t=6k−ϱcotϱkx−2k3ϱ+ktδ/δν,ϑx,t=6kνϱtan2ϱℑ±sec2ϱℑ,ϑx,t=6kνϱ−cot2ϱℑ±csc2ϱℑ,ϑx,t=6kνϱ2tanϱ2ℑ−cotϱ2ℑ.
Ifϱ=0,(37)ϑx,t=6kν−1ℑ.


## 4.2. Example 2

Let us consider the conformable time fractional CH equation(38)Dtδϑ−3ϑ2−1ϑxx−6ϑϑx2−ϑx+ϑxxxx=0,wheret>0,0<δ≤1.Applying the transformation,(39)ϑx,t=Vℑ,ℑ=kx−ltδδ.Equation (38) is reduced to the following nonlinear ODE:(40)−k+lV′−6k2V′2V−k23V2−1V″+k4V″″=0.Through balancing the termsV‴′ and V′2V, we select m=1, then the nontrivial solution (40) reduces to(41)Vℑ=a1ϕℑ+a0.By setting the above solution in (40) and equating factors of each power of ϕi, we get nonlinear algebraic system, solving this system for different value of a0,a1,k,l. Case 1:(42)a0=0,a1=iϱ,k=±i2ϱ,l=±i2ϱ.Ifϱ<0,(43)ϑx,t=iϱ−−ϱtanh−ϱℑ,(44)ϑx,t=iϱ−−ϱcoth−ϱℑ,(45)ϑx,t=iϱ−ϱ−tanh2−ϱℑ±isech2−ϱℑ,(46)ϑx,t=iϱ−ϱ−coth2−ϱℑ±csch2−ϱℑ,(47)ϑx,t=iϱ−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.Ifϱ>0,(48)ϑx,t=iϱϱtanϱℑ,(49)ϑx,t=iϱ−ϱcotϱℑ,(50)ϑx,t=iϱϱtan2ϱℑ±sec2ϱℑ,(51)ϑx,t=iϱϱ−cot2ϱℑ±csc2ϱℑ,(52)ϑx,t=iϱϱ2tanϱ2ℑ−cotϱ2ℑ.Case 3.
(53)a0=0,a1=−iϱ,k=±i2ϱ,l=±i2ϱ.
Ifϱ<0,(54)ϑx,t=−iϱ−−ϱtanh−ϱℑ,ϑx,t=−iϱ−−ϱcoth−ϱℑ,ϑx,t=−iϱ−ϱ−tanh2−ϱℑ±isech2−ϱℑ,ϑx,t=−iϱ−ϱ−coth2−ϱℑ±csch2−ϱℑ,ϑx,t=−iϱ−−ϱ2tanh−ϱ2ℑ+coth−ϱ2ℑ.
Ifϱ>0,(55)ϑx,t=−iϱϱtanϱℑ,ϑx,t=−iϱ−ϱcotϱℑ,ϑx,t=−iϱϱtan2ϱℑ±sec2ϱℑ,ϑx,t=−iϱϱ−cot2ϱℑ±csc2ϱℑ,ϑx,t=−iϱϱ2tanϱ2ℑ−cotϱ2ℑ.


## 5. Conclusion

By using simplest equation method, we have succeeded to find out the solitary wave and periodic and rational solutions of conformable nonlinear space-time fractional differential BBM and CH equations. These solutions have also been demonstrated through graph. Figures1–3 show the solitary wave profile of ϑx,t for conformable time fractional differential BBM equation, whereas Figures 4 and 5 represent the solitary wave profile of ϑx,t for conformable time fractional differential CH equation. All the results are obtained with the help of traveling wave transformation. At the end, it is concluded that the simplest equation method suggests an innovative and powerful mathematical aid to deal with nonlinear partial differential equations of a wide class of nonlinear fractional-order of mathematical models in science and engineering. The performance of this method can also be used for fractal theory and microgravity space in our future research.Figure 1
Solitary wave profile ofϑx,t appears in equation (24).
(a)(b)(c)(d)(e)Figure 2
Periodic wave profile ofϑx,t appears in equation (28).
(a)(b)(c)(d)(e)Figure 3
Wave profile ofϑx,t appears in equation (33).
(a)(b)(c)(d)(e)Figure 4
Solitary wave profile ofϑx,t appears in equation (43).
(a)(b)(c)(d)(e)Figure 5
Solitary wave profile ofϑx,t appears in equation (48).
(a)(b)(c)(d)(e)



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