New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.

• Exact Traveling Wave Solutions
• (2 + 1)-Dimensional Time-Fractional Zoomeron Equation
• The New Mapping Approach
• The New Extended Auxiliary Equation Approach

1. Introduction

Fractional partial differential equations (FPDEs) have a wide of applications in different fields, such as biology, physics, signal processing, fluid mechanics, and electromagnetic, and so on. In recent decades, many effective methods have been presented to obtain the exact traveling wave solutions of FPDEs, for example, ( G ′ / G ) expansion method [1] [2] [3], ( 1 / G ′ ) expansion method [4] [5] [6] [7], the exp ( − Φ ( ξ ) ) function method [8] [9], the F-expansion method [10] [11], sine-cosine method [12] [13] and others [14] [15] [16]. There are many important definitions of fractional derivative, such as Riemann-Liouville, Caputo, Atangana’ s-conformable and the conformable fractional derivative, etc. [17] [18] [19] [20] [21].

In this paper, we use the complex traveling wave transformation to deduce (2 + 1)-dimensional conformable time-fractional Zoomeron equation into ordinary differential equation. Furthermore, inspired by the reference [22], we introduce the new mapping approach and the new extended auxiliary equation approach [23] [24] [25] to investigate the exact solutions of (2 + 1)-dimensional time-fractional Zoomeron equation [20]:

∂ 2 α u ∂ t 2 α [ u x y u ] − ∂ 2 u ∂ x 2 [ u x y u ] + 2 ∂ α u ∂ t α [ u 2 ] x = 0 , 0 < α ≤ 1. (1)

when α = 1 , Equation (1) reduces to the (2 + 1)-dimensional Zoomeron equation [26]. Aksoy E. [27] obtained two types of exact analytical solutions including hyperbolic function solutions and trigonometric function solutions by using sub-equation and generalized Kudryashov methods in Equation (1). Hosseini K. [28] obtained several new wave form solutions of Equation (1) such as kink, singular kink, and periodic wave solutions using exp ( − Φ ( ε ) ) expansion approach and modified Kudryashov method. Akbulut A. [20] obtained analytical solutions of Equation (1) with auxiliary equation method. Based on the study of Akbulut A., Topsakal M. [21] obtained new exact solutions of Equation (1) by using the auxiliary equation method. These methods are effective in investigation of the solutions of Equation (1), the aim of this investigation is to establish more general solutions and some new solutions using the two methods mentioned above.

The organization of this paper is as follows: In Section 2, we introduce the conformable fractional derivative. In Section 3, we introduce the new mapping approach and the new extended auxiliary equation approach to investigate the solutions of (2 + 1)-dimensional time-fractional Zoomeron equation, and analyze the dynamic behaviors of the solutions in Section 4. Finally, we give some conclusions in Section 5.

2. The Conformable Fractional Derivative

In this section, we introduce the conformable fractional derivative [20] [21].

Definition 2.1. [20] Suppose a function f : [ 0, ∞ ) → R . Then, the conformable fractional derivative of f of order α , which is defined by

( T α f ) ( t ) = l i m ε → 0 f ( t + ε t 1 − α ) − f ( t ) ε (2)

for all t > 0 , 0 < α ≤ 1 .

Properties. [20] [21] Let α ∈ ( 0,1 ) and f , g be α -differentiable at a point t > 0 , then some properties of the conformable fractional derivative are by follows:

1) Linearity: T α ( a f + b g ) = a ( T α f ) + b ( T α g ) , for all a , b ∈ R .

2) Leibniz rule: T α ( f g ) = f   T α ( g ) + g T α ( f ) .

3) T α ( t p ) = p t p − α , for all p ∈ R .

4) T α ( λ ) = 0 , for all constant functions f ( t ) = λ .

5) T α ( f / g ) = g ( T α f ) − f ( T α g ) g 2 .

6) Additionally, if f is differentiable, then

T α ( f ) ( t ) = t 1 − α d f d t ( t ) .

Theorem 2.1 (Chain rule). [20] [21] Assume function f , g : [ 0, ∞ ) → R be α -differentiable, then the following rule is obtained

T α ( f ∘ g ) ( t ) = t 1 − α g ( t ) f ′ ( g ( t ) ) (3)

where 0 < α ≤ 1 .

Definition 2.2 (Conformable fractional integral). [21] Let 0 < α ≤ 1 and 0 ≤ a < b . A function f : [ a , b ] → R is α -ractional integrable on [ a , b ] if the integral

I α f ( x ) = ∫ a b     f ( x ) d α x = ∫ a b     f ( x ) x α − 1 d x (4)

exist and is finite.

Theorem 2.2. [29] Let f ∈ C [ a , b ] and 0 < α ≤ 1 . Then

d α d x α I α f ( x ) = f ( x ) . (5)

3. Description of the Methods

Suppose that a nonlinear fractional differential equation with the conformable time-fractional derivative:

H ( u , ∂ α u ∂ t α , ∂ u ∂ x , ∂ 2 α u ∂ t 2 α , ∂ 2 u ∂ x 2 , ⋯ ) = 0 (6)

where H is a polynomial of u ( x , t ) and its partial conformable derivatives including the highest order derivative and the nonlinear term.

We use the complex traveling wave transformation

u ( x , y , t ) = u ( ξ ) , ξ = k x + h y − l t α α (7)

where k , h , l are non-zero arbitrary constants. Equation (1) converts into a nonlinear ordinary differential equation:

P ( u , u ′ , u ″ , ⋯ ) = 0 (8)

where P is a polynomial of u ( x , t ) and its partial derivatives, ' = d d ξ .

3.1. The New Mapping Approach

We suppose the solution of Equation (8) as follow:

u ( ξ ) = ∑ i = 0 2 N     a i φ i ( ξ ) (9)

where a i ( i = 0 , 1 , ⋯ , 2 N ) are constants, the positive integer N can be determined by balancing the highest order derivative and the nonlinear term in Equation (8). φ ( ξ ) satisfies the following equation:

( φ ′ ) 2 ( ξ ) = r + p φ 2 + 1 2 q φ 4 + 1 3 s φ 6 (10)

where r , p , q and s are arbitrary constants, the solutions of Equation (10) given by reference [23] with s ≠ 0 .

3.2. The New Extended Auxiliary Equation Approach

We suppose the solution of Equation (8) as follow:

u ( ξ ) = ∑ i = 0 2 N     a i F i ( ξ ) (11)

where a i ( i = 0 , 1 , ⋯ , 2 N ) are constants and the positive integer N can be determined by balancing the highest order derivative and the nonlinear term in Equation (8). F ( ξ ) satisfies the following equation:

( F ′ ) 2 ( ξ ) = c 0 + c 2 F 2 ( ξ ) + c 4 F 4 ( ξ ) + c 6 F 6 ( ξ ) (12)

where c j ( j = 0 , 2 , 4 , 6 ) are constants and c 6 ≠ 0 . Equation (12) has the following solutions:

F ( ξ ) = 1 2 [ − c 4 c 6 ( 1 ± f i ( ξ ) ) ] 1 2 (13)

where the function f i ( ξ ) ( i = 1 , 2 , ⋯ , 12 ) is the Jacobi elliptic function s n ( ξ ) , c n ( ξ ) , d n ( ξ ) , while 0 < m < 1 is the modulus of the Jacobi elliptic functions. When m → 1 or m → 0 , the Jacobi elliptic function solutions degenerate to hyperbolic functions and trigonometric functions [24] [25].

4. Applications

We substitute Equation (7) into Equation (1), which deduce the nonlinear ordinary differential equation:

k h l 2 ( u ″ u ) ′ ​ ′ − k 3 h ( u ″ u ) ′ ​ ′ − 2 k l ( u 2 ) ′ ​ ′ = 0 (14)

We integrate Equation (14) twice, then we have

k h ( l 2 − κ 2 ) u ″ − 2 k l u 3 − β u = 0 (15)

where the prime denotes the derivative with respect to ξ , the second constant of integration is zero. Balancing the highest order derivative term u ″ and the highest order nonlinear term u 3 , we get N + 2 = 3 N , hence N = 1 .

4.1. Application of the New Mapping Approach

We assume that the solution of Equation (9) as follow:

u ( ξ ) = a 0 + a 1 φ ( ξ ) + a 2 φ 2 ( ξ ) (16)

where a 0 , a 1 , a 2 are constants.

Substituting Equation (16) and its derivatives and Equation (10) into Equation (15), yields a system of equations of φ i ( ξ ) , then setting the coefficients of φ i ( ξ ) ( i = 0 , 1 , 2 , ⋯ ) to zero, we can deduce the following set of algebraic polynomials with the respect a 0 , a 1 , a 2 :

φ 0 ( ξ ) : 2 r k h ( l 2 − k 2 ) a 2 − 2 k l a 0 3 − β a 0 = 0 φ 1 ( ξ ) : p k h ( l 2 − k 2 ) a 1 − 6 k l a 0 2 a 1 − β a 1 = 0 φ 2 ( ξ ) : 4 p k h ( l 2 − k 2 ) a 2 − 6 k l a 0 ( a 1 2 + a 0 a 2 ) − β a 2 = 0 φ 3 ( ξ ) : q k h ( l 2 − k 2 ) a 1 − 2 k l a 1 ( a 1 2 + 6 a 0 a 2 ) = 0

φ 4 ( ξ ) : 3 q k h ( l 2 − κ 2 ) a 2 − 6 k l a 2 ( a 1 2 + a 0 a 2 ) = 0 φ 5 ( ξ ) : s k h ( l 2 − κ 2 ) a 1 − 6 k l a 1 a 2 2 = 0 φ 6 ( ξ ) : 8 3 s k h ( l 2 − κ 2 ) a 2 − 2 k l a 2 3 = 0 (17)

Solving the above algebraic equations, we obtain the following two results:

Type 1. Substituting s = 3 q 2 16 p , r = 0 into Equation (17), we have

a 0 = ± − β 2 k l , a 1 = 0 , a 2 = ± q k h ( l 2 − k 2 ) β − β 2 k l , p = − β 2 k h ( l 2 − k 2 ) , q = q (18)

Substituting Equation (18) and the solutions in reference [23] into Equation (16), we get

u 1 ( ξ ) = ± − β 2 k l ( 2 + tanh ( ε − β 2 k h ( l 2 − k 2 ) ξ ) ) (19)

u 2 ( ξ ) = ± − β 2 k l ( 2 + coth ( ε − β 2 k h ( l 2 − k 2 ) ξ ) ) (20)

where ε = ± 1 , β > 0 , h > 0 , k < 0 .

Type 2. Substituting r = 0 into Equation (17), we have

a 0 = a 0 , a 1 = 0 , a 2 = q h ( l 2 − k 2 ) 2 l a 0 , p = l a 0 2 h ( l 2 − k 2 ) , q = q , s = 3 q 2 h ( l 2 − k 2 ) 16 l a 0 2 (21)

Substituting Equation (21) and the solutions in reference [23] into Equation (16), we have

u 3 ( ξ ) = a 0 [ 1 − sech 2 ( l a 0 2 h ( l 2 − k 2 ) ξ ) 1 − 1 4 ( 1 + ε tanh l a 0 2 h ( l 2 − k 2 ) ξ ) 2 ] (22)

u 4 ( ξ ) = a 0 [ 1 + csch 2 ( l a 0 2 h ( l 2 − k 2 ) ξ ) 1 − 1 4 ( 1 + ε coth l a 0 2 h ( l 2 − k 2 ) ξ ) 2 ] (23)

u 5 ( ξ ) = a 0 [ 1 − sech 2 ( l a 0 2 h ( l 2 − k 2 ) ξ ) 1 + ε tanh l a 0 2 h ( l 2 − k 2 ) ξ ] (24)

u 6 ( ξ ) = a 0 [ 1 + csch 2 ( l a 0 2 h ( l 2 − k 2 ) ξ ) 1 + ε coth l a 0 2 h ( l 2 − k 2 ) ξ ] (25)

where ε = ± 1 , l > 0 , h > 0 , ξ = k x + h y − l t α α .

4.2. Application of the New Extended Auxiliary Equation Approach

We assume the solution of Equation (11) as follow:

u ( ξ ) = a 0 + a 1 F ( ξ ) + a 2 F 2 ( ξ ) (26)

where a 0 , a 1 , a 2 are constants.

Substituting Equation (26) and its derivatives and Equation (12) into Equation (15), yields a system of equations of F i ( ξ ) , then setting the coefficients of F i ( ξ ) ( i = 0 , 1 , 2 , ⋯ ) to zero, we can deduce the following set of algebraic polynomials with the respect a 0 , a 1 , a 2 :

F 0 ( ξ ) : 2 c 0 k h ( l 2 − k 2 ) a 2 − 2 k l a 0 3 − β a 0 = 0 F 1 ( ξ ) : c 2 k h ( l 2 − k 2 ) a 1 − 6 k l a 0 2 a 1 − β a 1 = 0 F 2 ( ξ ) : 4 c 2 k h ( l 2 − k 2 ) a 2 − 6 k l a 0 ( a 1 2 + a 0 a 2 ) − β a 2 = 0 F 3 ( ξ ) : 2 c 4 k h ( l 2 − k 2 ) a 1 − 2 k l a 1 ( a 1 2 + 6 a 0 a 2 ) = 0

F 4 ( ξ ) : 6 c 4 k h ( l 2 − k 2 ) a 2 − 6 k l a 2 ( a 1 2 + a 0 a 2 ) = 0 F 5 ( ξ ) : 3 c 6 k h ( l 2 − k 2 ) a 1 − 6 k l a 1 a 2 2 = 0 F 6 ( ξ ) : 8 c 6 k h ( l 2 − k 2 ) a 2 − 2 k l a 2 3 = 0 (27)

Solving the above algebraic equations, we obtain the following results:

a 0 = a 0 , a 1 = 0 , a 2 = a 2 c 0 = 2 k l a 0 3 + β a 0 2 k h ( l 2 − k 2 ) a 2 , c 2 = 6 k l a 0 2 + β 4 k h ( l 2 − k 2 ) , c 4 = l a 0 a 2 h ( l 2 − k 2 ) , c 6 = l a 2 2 4 h ( l 2 − k 2 ) (28)

Substituting Equation (28) into (13), we have

F ( ξ ) = 1 2 [ − 4 a 0 a 2 ( 1 ± f i ( ξ ) ) ] 1 2 (29)

Substituting Equation (28) and (29) into (26), we get the solution

u ( ξ ) = ∓ a 0 f i ( ξ ) (30)

where f i ( ξ ) ( i = 1 , 2 , ⋯ , 12 ) given by reference [24]. Insetting them into Equation (30), we obtain the following Jacobi elliptic function solutions of Equation (1):

1) If c 0 = c 4 3 ( m 2 − 1 ) 32 c 6 2 m 2 ,   c 2 = c 4 2 ( 5 m 2 − 1 ) 16 c 6 m 2 ,   c 6 > 0 , then

u 1 ( ξ ) = ∓ a 0 s n ( l a 0 2 m 2 h ( l 2 − k 2 ) ξ , m ) (31)

u 2 ( ξ ) = ∓ a 0 m s n ( l a 0 2 m 2 h ( l 2 − k 2 ) ξ , m ) (32)

where ξ = k x + h y + m 2 β t α α k a 0 2 ( m 2 + 1 ) .

If m → 1 , then s n ( ξ ) → tanh ( ξ ) , we get the hyperbolic function solutions:

u ′ 1 ( ξ ) = ∓ a 0 tanh ( l a 0 2 h ( l 2 − k 2 ) ξ ) (33)

u ′ 2 ( ξ ) = ∓ a 0 coth ( l a 0 2 h ( l 2 − k 2 ) ξ ) (34)

where ξ = k x + h y + β t α 2 α k a 0 2 .

2) If c 0 = c 4 3 ( 1 − m 2 ) 32 c 6 2 ,   c 2 = c 4 2 ( 5 − m 2 ) 16 c 6 ,   c 6 > 0 , then

u 3 ( ξ ) = ∓ a 0 m s n ( l a 0 2 h ( l 2 − k 2 ) ξ , m ) (35)

u 4 ( ξ ) = ∓ a 0 s n ( l a 0 2 h ( l 2 − k 2 ) ξ , m ) (36)

where s n is elliptic sine, ξ = k x + h y + β t α α k a 0 2 ( m 2 + 1 ) .

If m → 1 , then we get the same solutions with Equation (33)-(34).

If m → 0 , then s n ( ξ ) → sin ( ξ ) , we get the periodic function solution:

u ′ 4 ( ξ ) = ∓ a 0 csc ( l a 0 2 h ( l 2 − k 2 ) ξ ) (37)

where ξ = k x + h y + β t α α κ a 0 2 .

3) If c 0 = c 4 3 32 c 6 2 m 2 ,   c 2 = c 4 2 ( 4 m 2 + 1 ) 16 c 6 m 2 ,   c 6 < 0 , then

u 5 ( ξ ) = ∓ a 0 c n ( − l a 0 2 m 2 h ( l 2 − k 2 ) ξ , m ) (38)

u 6 ( ξ ) = ∓ a 0 1 − m 2 s n ( − l a 0 2 m 2 h ( l 2 − k 2 ) ξ , m ) d n ( − l a 0 2 m 2 h ( l 2 − k 2 ) ξ , m ) (39)

where ξ = k x + h y − m 2 β t α α k a 0 2 ( 1 − 2 m 2 ) .

If m → 1 , then c n ( ξ ) → sech ( ξ ) , we get the hyperbolic function solution:

u ′ 5 ( ξ ) = ∓ a 0 sech ( − l a 0 2 h ( l 2 − k 2 ) ξ ) (40)

where ξ = k x + h y + β t α α k a 0 2 .

4) If c 0 = c 4 3 m 2 32 c 6 2 ( m 2 − 1 ) ,   c 2 = c 4 2 ( 5 m 2 − 4 ) 16 c 6 ( m 2 − 1 ) ,   c 6 < 0 , then

u 7 ( ξ ) = ∓ a 0 d n ( − l a 0 2 h ( l 2 − k 2 ) ( 1 − m 2 ) ξ , m ) 1 − m 2 (41)

u 8 ( ξ ) = ∓ a 0 d n ( − l a 0 2 h ( l 2 − k 2 ) ( 1 − m 2 ) ξ , m ) (42)

where ξ = k x + h y − ( m 2 − 1 ) β t α α k a 0 2 ( 2 − m 2 ) .

If m → 0 , then d n ( ξ ) → 1 , we get the solutions:

u ′ 7 ( ξ ) = u ′ 8 ( ξ ) = ∓ a 0 (43)

where ξ = k x + h y + β t α 2 α k a 0 2 .

5) If c 0 = c 4 3 32 c 6 2 ( 1 − m 2 ) ,   c 2 = c 4 2 ( 4 m 2 − 5 ) 16 c 6 ( m 2 − 1 ) ,   c 6 > 0 , then

u 9 ( ξ ) = ∓ a 0 c n ( l a 0 2 h ( l 2 − k 2 ) ( 1 − m 2 ) ξ , m ) (44)

u 10 ( ξ ) = ∓ a 0 d n ( l a 0 2 h ( l 2 − k 2 ) ( 1 − m 2 ) ξ , m ) 1 − m 2 s n ( l a 0 2 h ( l 2 − k 2 ) ( 1 − m 2 ) ξ , m ) (45)

where ξ = k x + h y − ( m 2 − 1 ) β t α α k a 0 2 ( 1 − 2 m 2 ) .

If m → 0 , then c n ( ξ ) → cos ( ξ ) , s n ( ξ ) → sin ( ξ ) , d n ( ξ ) → 1 , we get the periodic function solutions:

u ′ 9 ( ξ ) = ∓ a 0 sec ( l a 0 2 h ( l 2 − k 2 ) ) (46)

u ′ 10 ( ξ ) = ∓ a 0 csc ( l a 0 2 h ( l 2 − k 2 ) ) (47)

where ξ = k x + h y + β t α α k a 0 2 .

6) If c 0 = c 4 3 m 2 32 c 6 2 ,   c 2 = c 4 2 ( m 2 + 4 ) 16 c 6 ,   c 6 < 0 , then

u 11 ( ξ ) = ∓ a 0 d n ( − l a 0 2 h ( l 2 − k 2 ) ξ , m ) (48)

u 12 ( ξ ) = ∓ a 0 1 − m 2 d n ( − l a 0 2 h ( l 2 − k 2 ) ξ , m ) (49)

where ξ = k x + h y + β t α α k a 0 2 ( 2 − m 2 ) .

If m → 0 , then d n ( ξ ) → 1 , we have the same solutions with Equation (43).

If m → 1 , then d n ( ξ ) → sech ( ξ ) , we get the hyperbolic function solutions:

u ′ 11 ( ξ ) = ∓ a 0 sech ( − l a 0 2 h ( l 2 − k 2 ) ξ ) (50)

where ξ = k x + h y + β t α α k a 0 2 .

4.3. Dynamical Behaviors

In this section, we analyze the dynamic behaviors of the solutions in (2 + 1)-dimensional time-fractional Zoomeron equation.

Figure 1 and Figure 2 are the 3D and 2D graphs of the solutions (19) and (20), (22) and (23), where the solutions are kink and anti-kink soliton solutions within the interval − 10 ≤ x , y ≤ 10 with the values of parameters a 0 = t = ε = h = q = y = 1 , β = 8 , α = 1 2 , l = 2 , k = − 1 . And we only give graphs of the solutions with the parameter a 0 = t = ε = h = q = y = 1 , β = 8 , α = 1 2 , l = 2 , k = − 1 .

Figure 3 is the 3D and 2D graphs of the solution (31), where the solution is Jacobi elliptic function solution within the interval − 5 ≤ x , y ≤ 5 with the values of parameters a 0 = t = h = y = 1 , α = m = 1 2 , l = 2 , k = − 1 , β = 8 . While 2D graph of the solution (31) is in the interval − 10 ≤ x ≤ 10 .

Figure 4 is the 3D and 2D graphs of the solutions (46) and (47), where the solutions are the periodic function solutions within the interval − 5 ≤ x , y ≤ 5

with the values of parameters a 0 = t = h = y = 1 , α = 1 2 , l = 2 , k = − 1 , β = 8 . While 2D graphs of the solutions (46) and (47) are in the interval − 10 ≤ x ≤ 10 .

5. Conclusion

In conclusion, (2 + 1)-dimensional time-fractional Zoomeron equation has been investigated by the new mapping approach and the new extended auxiliary equation approach. Singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation have been obtained, where Jacobi elliptic function solutions are new solutions. When m → 1 or m → 0 , the Jacobi elliptic function solutions degenerate into the hyperbolic function solutions and the periodic function solutions. Consequently, it is obvious that the application of these two methods is effective to the time-fractional equations.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

1. Sahoo, S. and Ray, S.S. (2020) New Travelling Wave and Anti-Kink Wave Solutions of Space-Time Fractional (3 + 1)-Dimensional Jimbo-Miwa Equation. Chinese Journal of Physics, 67, 79-85. https://doi.org/10.1016/j.cjph.2020.04.016

2. Alam, M.N. and Li, X. (2019) Exact Traveling Wave Solutions to Higher Order Nonlinear Equations. Journal of Ocean Engineering and Science, 4, 276-288. https://doi.org/10.1016/j.joes.2019.05.003

3. Wang, M.L., Li, X.Z. and Zhang, J.L. (2008) The (G’/G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423. https://doi.org/10.1016/j.physleta.2007.07.051

4. Yokus, A., Durur, H., Ahmad, H., Thounthong, P. and Zhang, Y.-F. (2020) Construction of Exact Traveling Wave Solutions of the Bogoyavlenskii Equation by (G’/G, 1/G)-Expansion and (1/G’)-Expansion Techniques. Results in Physics, 19, Article ID: 103409. https://doi.org/10.1016/j.rinp.2020.103409

5. Yoku&#351;, A. and Durur, H. (2019) Complex Hyperbolic Traveling Wave Solutions of Kuramoto-Sivashinsky Equation Using (1/G’)-Expansion Method for Nonlinear Dynamic Theory. Balkesir &#220;niversitesi Fen Bilimleri Enstitüsü Dergisi, 21, 590-599. https://doi.org/10.25092/baunfbed.631193

6. Durur, H. and Yoku&#351;, A. (2019) (1/G')-A&#231;&#305;l&#305;m Metodunu Kullanarak Sawada-Kotera Denkleminin Hiperbolik Yürüyen Dalga &#199;&#246;zümleri. Afyon Kocatepe üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 19, 615-619. https://dergipark.org.tr/en/doi/10.35414/akufemubid.559048

7. Yokus, A., Durur, H., Ahmad, H. and Yao, S.-W. (2020) Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation. Mathematics, 8, Article No. 908. https://doi.org/10.3390/math8060908

8. Arshed, S. (2020) New Soliton Solutions to the Perturbed Nonlinear Schr&#246;dinger Equation by exp(&#934;(&#958;))-Expansion method. Optik, 220, Article ID: 165123. https://doi.org/10.1016/j.ijleo.2020.165123

9. Elboree, M.K. (2021) Soliton Molecules and exp(&#934;(&#958;))-Expansion Method for the New (3 + 1)-Dimensional Kadomtsev-Petviashvili (KP) Equation. Chinese Journal of Physics, 71, 623-633. https://doi.org/10.1016/j.cjph.2021.04.001

10. Yldrm, Y. (2021) Optical Solitons with Biswas-Arshed Equation by F-Expansion Method. Optik, 227, Article ID: 165788. https://doi.org/10.1016/j.ijleo.2020.165788

11. Yusuf, P. and Nail, T. (2021) Multiple Soliton Solutions for Nonlinear Differential Equations with a New Version of Extended F-Expansion Method. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 91, 495-501. https://doi.org/10.1007/s40010-020-00687-9

12. Ali Akbar, M., Akinyemi, L., Yao, S.-W., Jhangeer, A., Rezazadeh, H., et al. (2021) Soliton Solutions to the Boussinesq Equation through Sine-Gordon Method and Kudryashov Method. Results in Physics, 25, Article ID: 104228. https://doi.org/10.1016/j.rinp.2021.104228

13. Ali, K.K., Osman, M.S. and Abdel-Aty, M. (2020) New Optical Solitary Wave Solutions of Fokas-Lenells Equation in Optical Fiber via Sine-Gordon Expansion Method. Alexandria Engineering Journal, 59, 1191-1196. https://doi.org/10.1016/j.aej.2020.01.037

14. Hou, L., Niu, B., Li, X., Shen, H., Li, F. and Wang, Y. (2020) Analysis of Soil Fertility Quality and Heavy Metal Pollution in the Dawen River Basin, China. Open Journal of Soil Science, 10, 602-615. https://doi.org/10.4236/ojss.2020.1012029

15. Kome, G., Enang, R., Tabi, F. and Yerima, B. (2019) Influence of Clay Minerals on Some Soil Fertility Attributes: A Review. Open Journal of Soil Science, 9, 155-188. https://doi.org/10.4236/ojss.2019.99010

16. Barman, H.K., Roy, R., Mahmud, F., Akbar, M.A. and Osman, M.S. (2021) Harmonizing Wave Solutions to the Fokas-Lenells Model through the Generalized Kudryashov Method. Optik, 229, Article No. 166294. https://doi.org/10.1016/j.ijleo.2021.166294

17. Atangana, A. and Gómez-Aguilar, J.F. (2018) Numerical Approximation of Riemanndefinition of Fractional Derivative: From Riemannto Atangana. Numerical Methods for Partial Differential Equations, 34, 1502-1523. https://doi.org/10.1002/num.22195

18. Yépez-Martinez, H. and Gómez-Aguilar, J.F. (2021) Optical Solitons Solution of Resonance Nonlinear Schr&#246;dinger Type Equation with Atangana’s-Conformable Derivative Using Sub-Equation Method. Waves in Random and Complex Media, 31, 573-596. https://doi.org/10.1080/17455030.2019.1603413

19. Yépez-Martnez, H. and Gómez-Aguilar, J.F. (2019) Fractional Sub-Equation Method for Hirota-Satsuma-Coupled KdV Equation and Coupled mKdV Equation Using the Atangana’s Conformable Derivative. Waves in Random and Complex Media, 29, 678-693. https://doi.org/10.1080/17455030.2018.1464233

20. Akbulut, A. and Kaplan, M. (2018) Auxiliary Equation Method for Time-Fractional Differential Equations with Conformable Derivative. Computers Mathematics with Applications, 75, 876-882. https://doi.org/10.1016/j.camwa.2017.10.016

21. Topsakal, M. and Ta&#351;can, F. (2020) Exact Travelling Wave Solutions for Space-Time Fractional Klein-Gordon Equation and (2+1)-Dimensional Time-Fractional Zoomeron Equation via Auxiliary Equation Method. Applied Mathematics and Nonlinear Sciences, 5, 437-446. https://doi.org/10.2478/amns.2020.1.00041

22. Zayed, E.M.E., Nofal, T.A., Al-Nowehy, A.G. and Alngar, M.E.M. (2021) Optical Solitons and Other Solutions to the (2+1)-Dimensional Coupled System of NLSE by Two Integration Approaches. Optik, 232, Article ID: 166510. https://doi.org/10.1016/j.ijleo.2021.166510

23. Zayed, E.M.E. and Al-Nowehy, A.-G. (2017) Solitons and Other Exact Solutions for a Class of Nonlinear Schr&#246;dinger-Type Equations. Optik, 130, 1295-1311. https://doi.org/10.1016/j.ijleo.2016.11.115

24. Zayed, E.M.E. and Alurrfi, K.A.E. (2016) New Extended Auxiliary Equation Method and Its Applications to Nonlinear Schr&#246;dinger-Type Equations. Optik, 127, 9131-9151. https://doi.org/10.1016/j.ijleo.2016.05.100

25. Zayed, E.M.E. and Al-Nowehy, A.-G. (2017) New Extended Auxiliary Equation Method for Finding Many New Jacobi Elliptic Function Solutions of Three Nonlinear Schr&#246;dinger Equations. Waves in Random and Complex Media, 27, 420-439. https://doi.org/10.1080/17455030.2016.1259690

26. Manafian, J., Lakestani, M. and Bekir, A. (2016) Study of the Analytical Treatment of the (2+1)-Dimensional Zoomeron, the Duffing and the SRLW Equations via a New Analytical Approach. International Journal of Applied and Computational Mathematics, 2, 243-268. https://doi.org/10.1007/s40819-015-0058-2

27. Aksoy, E., &#199;evikel, A.C. and Bekir, A. (2016) Soliton Solutions of (2+1)-Dimensional Time-Fractional Zoomeron Equation. Optik, 127, 6933-6942. https://doi.org/10.1016/j.ijleo.2016.04.122

28. Hosseini, K., Korkmaz, A., Bekir, A., Samadani, F., Zabihi, A. and Topsakal, M. (2021) New Wave Form Solutions of Nonlinear Conformable Time-Fractional Zoomeron Equation in (2+1)-Dimensions. Waves in Random and Complex Media, 31, 228-238. https://doi.org/10.1080/17455030.2019.1579393

29. Lazo, M.J. and Torres, D.F.M. (2017) Variational Calculus with Conformable Fractional Derivatives. IEEE/CAA Journal of Automatica Sinica, 4, 340-352. https://doi.org/10.1109/JAS.2016.7510160