New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation ()
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,
expansion method [1] [2] [3],
expansion method [4] [5] [6] [7], the
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]:
(1)
when
, 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
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
. Then, the conformable fractional derivative of f of order
, which is defined by
(2)
for all
.
Properties. [20] [21] Let
and
be
-differentiable at a point
, then some properties of the conformable fractional derivative are by follows:
1) Linearity:
, for all
.
2) Leibniz rule:
.
3)
, for all
.
4)
, for all constant functions
.
5)
.
6) Additionally, if f is differentiable, then
.
Theorem 2.1 (Chain rule). [20] [21] Assume function
be
-differentiable, then the following rule is obtained
(3)
where
.
Definition 2.2 (Conformable fractional integral). [21] Let
and
. A function
is
-ractional integrable on
if the integral
(4)
exist and is finite.
Theorem 2.2. [29] Let
and
. Then
(5)
3. Description of the Methods
Suppose that a nonlinear fractional differential equation with the conformable time-fractional derivative:
(6)
where H is a polynomial of
and its partial conformable derivatives including the highest order derivative and the nonlinear term.
We use the complex traveling wave transformation
(7)
where
are non-zero arbitrary constants. Equation (1) converts into a nonlinear ordinary differential equation:
(8)
where P is a polynomial of
and its partial derivatives,
.
3.1. The New Mapping Approach
We suppose the solution of Equation (8) as follow:
(9)
where
are constants, the positive integer N can be determined by balancing the highest order derivative and the nonlinear term in Equation (8).
satisfies the following equation:
(10)
where
and s are arbitrary constants, the solutions of Equation (10) given by reference [23] with
.
3.2. The New Extended Auxiliary Equation Approach
We suppose the solution of Equation (8) as follow:
(11)
where
are constants and the positive integer N can be determined by balancing the highest order derivative and the nonlinear term in Equation (8).
satisfies the following equation:
(12)
where
are constants and
. Equation (12) has the following solutions:
(13)
where the function
is the Jacobi elliptic function
,
,
, while
is the modulus of the Jacobi elliptic functions. When
or
, 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:
(14)
We integrate Equation (14) twice, then we have
(15)
where the prime denotes the derivative with respect to
, the second constant of integration is zero. Balancing the highest order derivative term
and the highest order nonlinear term
, we get
, hence
.
4.1. Application of the New Mapping Approach
We assume that the solution of Equation (9) as follow:
(16)
where
are constants.
Substituting Equation (16) and its derivatives and Equation (10) into Equation (15), yields a system of equations of
, then setting the coefficients of
to zero, we can deduce the following set of algebraic polynomials with the respect
:
(17)
Solving the above algebraic equations, we obtain the following two results:
Type 1. Substituting
into Equation (17), we have
(18)
Substituting Equation (18) and the solutions in reference [23] into Equation (16), we get
(19)
(20)
where
.
Type 2. Substituting
into Equation (17), we have
(21)
Substituting Equation (21) and the solutions in reference [23] into Equation (16), we have
(22)
(23)
(24)
(25)
where
.
4.2. Application of the New Extended Auxiliary Equation Approach
We assume the solution of Equation (11) as follow:
(26)
where
are constants.
Substituting Equation (26) and its derivatives and Equation (12) into Equation (15), yields a system of equations of
, then setting the coefficients of
to zero, we can deduce the following set of algebraic polynomials with the respect
:
(27)
Solving the above algebraic equations, we obtain the following results:
(28)
Substituting Equation (28) into (13), we have
(29)
Substituting Equation (28) and (29) into (26), we get the solution
(30)
where
given by reference [24]. Insetting them into Equation (30), we obtain the following Jacobi elliptic function solutions of Equation (1):
1) If
, then
(31)
(32)
where
.
If
, then
, we get the hyperbolic function solutions:
(33)
(34)
where
.
2) If
, then
(35)
(36)
where
is elliptic sine,
.
If
, then we get the same solutions with Equation (33)-(34).
If
, then
, we get the periodic function solution:
(37)
where
.
3) If
, then
(38)
(39)
where
.
If
, then
, we get the hyperbolic function solution:
(40)
where
.
4) If
, then
(41)
(42)
where
.
If
, then
, we get the solutions:
(43)
where
.
5) If
, then
(44)
(45)
where
.
If
, then
,
,
, we get the periodic function solutions:
(46)
(47)
where
.
6) If
, then
(48)
(49)
where
.
If
, then
, we have the same solutions with Equation (43).
If
, then
, we get the hyperbolic function solutions:
(50)
where
.
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
with the values of parameters
,
,
,
,
. And we only give graphs of the solutions with the parameter
,
,
,
,
.
Figure 3 is the 3D and 2D graphs of the solution (31), where the solution is Jacobi elliptic function solution within the interval
with the values of parameters
,
,
,
,
. While 2D graph of the solution (31) is in the interval
.
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
with the values of parameters
,
,
,
,
. While 2D graphs of the solutions (46) and (47) are in the interval
.
Figure 2. (a), (b) are the 3D and 2D graphs of the solutions (22); (c), (d) are the 3D and 2D graphs of the solutions (23) with the values of parameters
,
,
,
,
. (a) 3D graph; (b) 2D graph; (c) 3D graph; (d) 2D graph.
Figure 3. The 3D and 2D graphs of the solution (31) with the values of parameters
,
,
,
,
. (a) 3D graph; (b) 2D graph.
Figure 4. (a), (b) are the 3D and 2D graphs of the solutions (46); (c), (d) are the 3D and 2D graphs of the solutions (47) with the values of parameters
,
,
,
,
. (a) 3D graph; (b) 2D graph; (c) 3D graph; (d) 2D graph.
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
or
, 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.