The conformable space-time fractional mKdV equations and their exact solutions

Asim Zafar; Aly R. Seadawy

doi:10.1016/j.jksus.2019.09.003

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Original article

31 (

); 1478-1484

doi:

10.1016/j.jksus.2019.09.003

The conformable space-time fractional mKdV equations and their exact solutions

Asim Zafar^{^a}, Aly R. Seadawy^{^b,^c,⁎}

a Department of Mathematics, CUI, Vehari Campus, Pakistan

b Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

c Mathematics Department, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

⁎Corresponding author at: Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia. aly742001@yahoo.com (Aly R. Seadawy)

Received: 2018-10-21, Accepted: 2019-9-16, Published: 2019-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

In this study, we are interested in exploring the conformable space-time fractional mKdV equations via hyperbolic function approach. A traveling wave transformation and the conformable derivative are used to convert the nonlinear fractional differential equation into a nonlinear ordinary differential equation. Then, the resulting equation is elucidated by utilizing the hyperbolic function approach through Mathematica. A variety of soliton type solutions including, hyperbolic and trigonometric functions, is formulated and the graphical representation for these solutions is given by using MATLAB.

Keywords

Space-time-fractional mKdV equations

Conformable derivatives

Hyperbolic function approach

Exact soliton solutions

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1 Introduction

The Korteweg-deVries (KdV), Boussinesq, Kadomtsev-Petviashvili, and Whitham-Broer-Kaup (WBK) equations are the well-known completely integrable models that describe the propagation of shallow waDepartment of Mathematicster (Wadati, 1973; Ohkuma and Wadati, 1983; Ablowitz and Segur, 1981 ). The adynamic of shallow water waves in various places like sea beaches are governed bt the KdV and Boussinesq Equations (Korteweg and de Vries, 1895; Yan and Zhang, 2001). The KdV equation has an impact in modeling blood pressure pulses (Schamel, 1973; Mousavian et al., 2011). Furthermore, Wazwaz (Wazwaz, 2017) introduced the nonlinear modified KdV equations of (3 + 1)-dimension and investigate their exact soliton and kink solutions. In particular, Nuruddeen (Nuruddeen, 2018) has explored the soliton solutions for the conformable space-time fractional modified KdV equations of (3 + 1)-dimension.

There are various mathematical approaches to solve PDEs with nonlinear characteristics or fractional derivatives. Some of the commonly used approaches are: The ansatz (Hosseini et al., 2018), modified simple equation (Jawad et al., 2010), the extended trial equation (Mahmood et al., 2014), the first integral (Eslami, 2016), $(\frac{G^{'}}{G})$ -expansion (Younis and Zafar, 2014), sine-Gordon expansion (Hosseini et al., 2017). Furthermore, some other excellent works like generalized Kudryashov method (Pandir et al., 2012), a modified form of Kudryashov and functional variable methods (Ayati et al., 2017) have been done by different researchers. In Bekir et al. (2014), Fan (2000) and Zhang (2007), the auxiliary equation, the extended tanh-function, the improved $\tan (\frac{ϕ (η)}{2})$ -expansion methods and the exp function approach have been explored for discrete and fractional order PDEs as well. Many more in several theoretical works about solitons and their applications (Seadawy, 2017; Khater et al., 2000; Khater et al., 2000; Khater et al., 2006; Khater et al., 2006; Helal and Seadawy, 2009; Helal and Seadawy, 2011; Seadawy, 2011; Seadawy, 2012; Helal and Seadawy, 2012 ). Moreover, the $\exp_{a}$ function method has been utilized to explore the PDEs in (Ali and Hassan, 2010; Zafar, 2018; Seadawy and El-Rashidy, 2016; Ul-Haq Tariq and Seadawy, 2017; Seadawy, 2017; Seadawy, 2017; Seadawy and Manafian, 2018; Selima et al., 2016 ). In particular, an efficient approach namely the hyperbolic function approach has been studied in Xie et al. (2001), Bai (2001), Hosseini et al. (2018) and Seadawy et al. (2018) to procure exact solutions. One can observe that aforementioned approach produces the soliton type and periodic function solutions of nonlinear evolution equations. The study of fractional calculus has been carried out in Samko et al. (1993) and Chung (2015).

This paper aims to explore the conformable space-time fractional modified KdV equations of (3 + 1)-dimension for exact soliton type solutions via hyperbolic function approach using conformable derivative and the traveling wave transformation. The scheme of this paper is as follows: A brief description of the conformable derivative and the hyperbolic function approach is given in Section 2. Section 3 and 4 illustrate how to utilize this approach for producing new solutions with their graphs. The latter part summarizes results of the current study.

2 Conformable fractional derivative approach

We recall the conformable derivative with some of its properties (Khalil et al., 2014).

Defination 1 Suppose $h : R_{> 0} \to R$ be a function. Then, for all $t > 0$ , $D_{t}^{α} (p (t)) = \lim_{ε \to 0} \frac{p (t + ε t^{1 - α}) - p (t)}{ε}$ is known as $α, 0 < α ⩽ 1$ order conformable fractional derivative of p. The followings are some useful properties:

$D_{t}^{α} (ap + bg) = {aD}_{t}^{α} (p) + {bD}_{t}^{α} (g)$ , for all $a, b \in R$ .

$D_{t}^{α} (pg) = {pD}_{t}^{α} (g) + {gD}_{t}^{α} (p)$

Let $p : R_{> 0} \to R$ be an $α$ -differentiable function, g be a differentiable function defined in the range of p. $D_{t}^{α} (p \circ g (t)) = t^{1 - α} g^{'} (t) p^{'} (g (t)) .$ On the top of that, the following rules hold.

$D_{t}^{α} (t^{h}) = {ht}^{h - α}$ , for all $h \in R$ .

$D_{t}^{α} (δ) = 0$ , where $δ$ is constant.

$D_{t}^{α} (p / g) = \frac{{gD}_{t}^{α} (p) - {pD}_{t}^{α} (g)}{g^{2}}$ .

Conjointly, if p is differentiable, then $D_{t}^{α} (p (t)) = t^{1 - α} \frac{dp (t)}{dt}$ .

2.1

2.1 The hyperbolic function approach

The present subsection provides a transitory explanation of hyperbolic function approach in engendering new exact solutions to nonlinear conformable space-time fractional modified KdV equations. For this purpose, suppose that we have a nonlinear conformable space-time FDE that can be presented in the form

(1)

F (u, D_{t}^{γ} u, D_{x}^{γ} u, D_{tt}^{2 γ} u, D_{xx}^{2 γ} u, \dots) = 0

The FDE (1) can be changed into the following nonlinear ODE of integer order

(2)

P (U, U^{'}, U^{″}, \dots,) = 0

with the use of following wave variable

(3)

u (x, t) = U (η), η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - l \frac{t^{γ}}{γ},

where

k, r, s

and l are nonzero arbitrary constants.

Let us try to search a non-trivial solution to the Eq. (2) in the following form (Xie et al., 2001; Bai, 2001; Hosseini et al., 2018; Seadawy et al., 2018 )

(4)

U (η) = A_{0} + \sum_{i = 1}^{N} \sinh^{i - 1} (ρ) [B_{i} \sinh (ρ) + A_{i} \cosh (ρ)]

where

ρ

is some specific functions. By calculating the positive integer N, setting the Eq. (4) in Eq. (2), and comparing coefficients, we will find a set of nonlinear equations whose solution, finally provides explicit exact solutions of the Eq. (1). It is worth mentioning that using the separation of variables techniques on

\frac{d ρ}{d η} = \sinh (ρ)

, we find

\sinh (ρ) = \pm csch (η), \cosh (ρ) = - \coth (η)

and

\sinh (ρ) = \pm ı sech (η), \cosh (ρ) = - \tanh (η)

. Accordingly, the solution (4) can be rewritten as

U (η) = A_{0} + \sum_{i = 1}^{N} {(\pm csch)}^{i - 1} (η)) [\pm B_{i} csch (η) - A_{i} \coth (η)],

and

U (η) = A_{0} + \sum_{i = 1}^{N} {(\pm ı sech)}^{i - 1} (η) [\pm ı B_{i} sech (η) - A_{i} \tanh (η)] .

Likewise, it is obvious that from

\frac{dw}{d η} = \cosh (ρ)

, we find

\sinh (ρ) = - \cot (η), \cosh (ρ) = \pm \csc (η)

and

\sinh (ρ) = \tan (η), \cosh (ρ) = \pm \sec (η)

. Accordingly, the solution (4) can be rewritten as

U (η) = A_{0} + \sum_{i = 1}^{N} {(- \cot)}^{i - 1} (η) [- B_{i} \cot (η) \pm A_{i} \csc (η)],

and

U (η) = A_{0} + \sum_{i = 1}^{N} (\tan^{i - 1}) (η) [B_{i} \tan (η) \pm A_{i} \sec (η)] .

On similar lines one can see Xian-Lin and Jia-Shi (2008), the extended sinh-Gordon equation expansion method based on the sinh-Gordon equation, was proposed to construct hyperbolic, trigonometric and Jacobi elliptic function solutions of nonlinear evolution equations.

3 The conformable fractional (3 + 1)-dimensional mKdV equation-I

Firstly, we consider the following space-time fractional mKdV equation (Nuruddeen, 2018):

(5)

D_{t}^{γ} u + 6 D_{x}^{γ} u^{3} + u_{xyz}^{3 γ} = 0, 0 < γ ⩽ 1 .

Using the transformation (3), and integrating once w.r.t.

η

with zero constant of integration, we get

(6)

- lU + {krsU}^{″} + 6 {kU}^{3} = 0 .

The balance between

U^{″}

and

U^{3}

gives

N = 1

, then the nontrivial solution (4) reduces to:

(7)

U (η) = B_{1} \sinh (ρ) + A_{1} \cosh (ρ) + A_{0} .

Case-I: $\frac{d ρ}{d η} = \sinh (ρ)$ By inserting the above solution in reduced equation Eq. (6) and equating the coefficients of each hyperbolic function to zero, we procure a set of nonlinear algebraic equations and its solution yields the following new exact solutions:

(8)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{rs}}{2 \sqrt{3}}, l = - \frac{1}{2} krs \\ u_{1, 2} (η) = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}} \tanh (\frac{η}{2}) \end{matrix}

(9)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}}, B_{1} = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}}, l = - \frac{1}{2} krs \\ u_{3, 4} (η) = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}} \coth (\frac{η}{2}) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} + \frac{1}{2} krs \frac{t^{γ}}{γ}

(10)

\begin{matrix} A_{0} = 0, A_{1} = 0, B_{1} = \mp \frac{i \sqrt{rs}}{\sqrt{3}}, l = krs \\ u_{5, 6} (η) = \mp \frac{i \sqrt{rs}}{\sqrt{3}} csch (η) \end{matrix}

(11)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{rs}}{\sqrt{3}}, B_{1} = 0, l = - 2 krs \\ u_{7, 8} (η) = \mp \frac{i \sqrt{rs}}{\sqrt{3}} \coth (η) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - l \frac{t^{γ}}{γ}

Case-2: $\frac{d ρ}{d η} = \cosh (ρ)$ and $N = 1$ , (7), gives a set of nonlinear algebraic equations and its solution yields the following exact solutions:

(12)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}}, B_{1} = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}}, l = \frac{krs}{2} \\ u_{1, 2} (η) = \pm \frac{i \sqrt{rs}}{2 \sqrt{3}} \tan (\frac{η}{2}) \end{matrix}

(13)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{rs}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{rs}}{2 \sqrt{3}}, l = \frac{krs}{2} \\ u_{3, 4} (η) = \pm \frac{i \sqrt{rs}}{2 \sqrt{3}} \cot (\frac{η}{2}) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - \frac{krs}{2} \frac{t^{γ}}{γ}

(14)

\begin{matrix} A_{0} = 0, A_{1} = 0, B_{1} = \mp \frac{i \sqrt{rs}}{\sqrt{3}}, l = 2 krs \\ u_{5, 6} (η) = \mp \frac{i \sqrt{rs}}{\sqrt{3}} \cot (η) \end{matrix}

(15)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{rs}}{\sqrt{3}}, B_{1} = 0, l = - krs \\ u_{7, 8} (η) = \pm \frac{i \sqrt{rs}}{\sqrt{3}} \csc (η) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - l \frac{t^{γ}}{γ}

3.1

3.1 The conformable fractional (3 + 1)-dimensional mKdV equation-II

The equation-II can be read as (Nuruddeen, 2018)

(16)

D_{t}^{γ} u + 6 D_{y}^{γ} u^{3} + u_{xyz}^{3 γ} = 0, 0 < γ ⩽ 1 .

Using the transformation (3), and integrating once w.r.t.

η

, we get

(17)

- lU + {krsU}^{″} + 6 {rU}^{3} = 0 .

Case-I: $\frac{d ρ}{d η} = \sinh (ρ)$ Through balancing the terms $U^{″}$ and $U^{3}$ , we select $N = 1$ . Then, by inserting (7) in (17), and equating coefficients of hyperbolic functions to zero in the resulting equation, a nonlinear algebraic set of equations is obtained and its solution yields the following exact solutions to (16).

(18)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{ks}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{ks}}{2 \sqrt{3}}, l = - \frac{1}{2} krs \\ u_{1, 2} (η) = \mp \frac{i \sqrt{ks}}{2 \sqrt{3}} \tanh (\frac{η}{2}) \end{matrix}

(19)

\begin{matrix} A_{0} = 0, A_{1} = \pm \frac{i \sqrt{ks}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{ks}}{2 \sqrt{3}}, l = - \frac{1}{2} krs \\ u_{3, 4} (η) = \pm \frac{i \sqrt{ks}}{2 \sqrt{3}} \coth (\frac{η}{2}) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} + \frac{1}{2} krs \frac{t^{γ}}{γ}

(20)

\begin{matrix} A_{0} = 0, A_{1} = 0, B_{1} = \mp \frac{i \sqrt{ks}}{\sqrt{3}}, l = krs \\ u_{5, 6} (η) = \mp \frac{i \sqrt{ks}}{\sqrt{3}} csch (η) \end{matrix}

(21)

A_{0} = 0, A_{1} = \mp \frac{i \sqrt{ks}}{\sqrt{3}}, B_{1} = 0, l = - 2 krs

(22)

u_{7, 8} (η) = \mp \frac{i \sqrt{ks}}{\sqrt{3}} \coth (η)

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - l \frac{t^{γ}}{γ}

Case-2: By using $\frac{d ρ}{d η} = \cosh (ρ)$ and $N = 1$ in (4), (7) yields a set of nonlinear equations and its solution gives the following exact solutions:

(23)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{ks}}{2 \sqrt{3}}, B_{1} = \mp \frac{i \sqrt{ks}}{2 \sqrt{3}}, l = \frac{krs}{2} \\ u_{1, 2} (η) = \pm \frac{i \sqrt{ks}}{2 \sqrt{3}} \tan (\frac{η}{2}) \end{matrix}

(24)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{ks}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{ks}}{2 \sqrt{3}}, l = \frac{krs}{2} \\ u_{3, 4} (η) = \mp \frac{i \sqrt{ks}}{2 \sqrt{3}} \cot (\frac{η}{2}) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - \frac{krs}{2} \frac{t^{γ}}{γ}

(25)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{ks}}{\sqrt{3}}, B_{1} = 0, l = - krs \\ u_{5, 6} (η) = \pm \frac{i \sqrt{ks}}{\sqrt{3}} \csc (η) \end{matrix}

(26)

\begin{matrix} A_{0} = 0, A_{1} = 0, B_{1} = \pm \frac{i \sqrt{ks}}{\sqrt{3}}, l = 2 krs \\ u_{7, 8} (η) = \pm \frac{i \sqrt{ks}}{\sqrt{3}} \cot (η) \end{matrix}

where

η = k \frac{x^{γ}}{γ} + r \frac{y^{γ}}{γ} + s \frac{z^{γ}}{γ} - l \frac{t^{γ}}{γ}

3.2

3.2 The conformable fractional $(3 + 1)$ -dimensional mKdV equation-III

In this section, the following conformable space–time fractional mKdV equation (Nuruddeen, 2018), is going to be considered for solutions.

(27)

D_{t}^{γ} u + 6 D_{z}^{γ} u^{3} + u_{xyz}^{3 γ} = 0, 0 < γ ⩽ 1 .

Using the transformation (3), and integrating once w.r.t.

η

with zero constant of integration, we get

(28)

- lU + {krsU}^{″} + 6 {sU}^{3} = 0 .

Case-I: $\frac{d ρ}{d η} = \sinh (ρ)$ Through balancing the terms $U^{″}$ and $U^{3}$ we select $N = 1$ , the non-trivial solution (7) and its derivatives in (28) produce a set of nonlinear algebraic equations. Then the solution of this set yields the following exact solutions for (27).

(29)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{kr}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}}, l = - \frac{1}{2} krs \\ u_{1, 2} (η) = \mp \frac{i \sqrt{kr}}{2 \sqrt{3}} \tanh (\frac{η}{2}) \end{matrix}

(30)

\begin{matrix} A_{0} = 0, A_{1} = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}}, l = - \frac{1}{2} krs \\ u_{3, 4} (η) = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}} \coth (\frac{η}{2}) \end{matrix}

(31)

\begin{matrix} A_{0} = 0, A_{1} = 0, B_{1} = \mp \frac{i \sqrt{kr}}{\sqrt{3}}, l = krs \\ u_{5, 6} (η) = \mp \frac{i \sqrt{kr}}{\sqrt{3}} csch (η) \end{matrix}

(32)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{kr}}{\sqrt{3}}, B_{1} = 0, l = - 2 krs \\ u_{7, 8} (η) = \mp \frac{i \sqrt{kr}}{\sqrt{3}} \coth (η) \end{matrix}

Case-2: $\frac{d ρ}{d η} = \cosh (ρ)$ and $N = 1$ in (4), yield a set of nonlinear equations and its solution yields the following exact solutions:

(33)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{kr}}{2 \sqrt{3}}, B_{1} = \mp \frac{i \sqrt{kr}}{2 \sqrt{3}}, l = \frac{krs}{2} \\ u_{1, 2} (η) = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}} \tan (\frac{η}{2}) \end{matrix}

(34)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{kr}}{2 \sqrt{3}}, B_{1} = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}}, l = \frac{krs}{2} \\ u (η) = \pm \frac{i \sqrt{kr}}{2 \sqrt{3}} \cot (\frac{η}{2}) \end{matrix}

(35)

\begin{matrix} A_{0} = 0, A_{1} = 0, B_{1} = \mp \frac{i \sqrt{kr}}{\sqrt{3}}, l = 2 krs \\ u (η) = \mp \frac{i \sqrt{kr}}{\sqrt{3}} \cot (η) \end{matrix}

(36)

\begin{matrix} A_{0} = 0, A_{1} = \mp \frac{i \sqrt{kr}}{\sqrt{3}}, B_{1} = 0, l = - krs \\ u (η) = \pm \frac{i \sqrt{kr}}{\sqrt{3}} \csc (η) \end{matrix}

3.3

3.3 Graphical representation of the solutions for Eq. (5)

The obtained solutions of Eq. (5) are graphed here for different $γ$ -values corresponding to $k = 1 = r$ , and $s = 3$ .

Case-I: The Figs. 1 and 2 reveal the two solutions given in Eq. (8) of Eq. (5) for $γ = 0.25, 0.5, 0.75$ and 1 respectively.

Fig. 1

Solitary wave profile of

u_{1, 2}

appears in Eq. (8).

Fig. 2

Solitary wave profile of

u_{3, 4}

appears in Eq. (8).

Case-II: The Figs. 3 and 4 represent two solutions given in Eq. (12) of Eq. (5) for $γ = 0.25, 0.5, 0.75$ and 1 respectively.

Fig. 3

Solitary wave profile of

u_{1, 2}

appears in Eq. (12).

Fig. 4

Solitary wave profile of

u_{3, 4}

appears in Eq. (12).

4 Conclusion

We investigated the exact soliton type solutions, different from those reported in Nuruddeen (2018), for the space-time fractional mKdV equations via hyperbolic function approach. For this purpose, we utilized a wave transformation and the conformable derivative to alter the nonlinear differential equation of fractional order into some nonlinear ordinary differential equation. Then, some real and complex valued solutions are calculated for three types of space-time fractional mKdV equations with the aid of soft computation in MATHEMATICA. Additionally, the graphs of some solutions indicates the well-preserved shape and hight of initial wave prorogations.

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Pandir Y., Gurefe Y., Misirli E., . A new approach to kudryashov’s method for solving some nonlinear physical models. Int. J. Phys. Sci.. 2012;7:2860-2866.
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Samko G., Kilbas A.A., Marichev, . Fractional Integrals and Derivatives: Theory and Applications. Yverdon: Gordon and Breach; 1993. Gordon and Breach, Yverdon
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Seadawy A.R., . New exact solutions for the KDV equation with higher order nonlinearity by using the variational method. Comput. Math. Appl.. 2011;62:3741-3755.
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Seadawy A.R., . Exact solutions of a two-dimensional nonlinear schrodinger equation. Appl. Math. Lett.. 2012;25:687-691.
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Seadawy Aly, . Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its a solitary wave solutions via mathematical methods. Eur. Phys. J. Plus. 2017;132:518.
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Seadawy A.R., El-Rashidy K., . Rayleigh-Taylor instability of the cylindrical flow with mass and heat transfer. Pramana – J. Phys.. 2016;87:20.
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Seadawy Aly R., Kumar D., Hosseini K., Samadani F., . The system of equations for the ion sound and langmuir waves and its new exact solutions. Results Phys.. 2018;9:1631-1634.
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Selima Ehab S., Seadawy Aly R., Yao Xiaohua, . The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability. Eur. Phys. J. Plus. 2016;131:425.
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Ul-Haq Tariq Kalim, Seadawy Aly, . Bistable Bright-Dark solitary wave solutions of the (3+1)-Dimensional Breaking Soliton, Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equations and their applications. Results Phys.. 2017;7:1143-1149.
[Google Scholar]
Wadati Miki, . The modified korteweg-de vries equation. J. Phys. Soc. Jpn.. 1973;34(5):1289-1296.
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Xian-Lin Y.A.N.G., Jia-Shi T.A.N.G., . Travelting solutions for konopelchenko-dubrovsky equation using an extended sinh-gordon equation expansion method. Commun. Theor. Phys.. 2008;50(5):1047-1051.
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Yan Zhenya, Zhang Hongqing, . New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water. Phys. Lett. A. 2001;285(5):355-362.
[Google Scholar]
Younis M., Zafar Asim, . Exact solutions to nonlinear differential equations of fractional order via $(\frac{G^{'}}{G})$ -expansion method. Appl. Math.. 2014;5:1-6.
[Google Scholar]
Zafar Asim, . Rational exponential solutions of conformable space-time fractional equal-width equations. Nonlinear Eng. 2018
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Zhang S., . Application of exp-function to a KDV equation with variable-coefficients. Phys. Lett. A. 2007;365:448-453.
[Google Scholar]

Introduction

Conformable fractional derivative approach

The hyperbolic function approach

The conformable fractional (3 + 1)-dimensional mKdV equation-I

The conformable fractional (3 + 1)-dimensional mKdV equation-II
The conformable fractional ( 3 + 1 ) -dimensional mKdV equation-III
Graphical representation of the solutions for Eq. (5)

Conclusion

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

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[Google Scholar]

[38] Ul-Haq Tariq Kalim, Seadawy Aly, . Bistable Bright-Dark solitary wave solutions of the (3+1)-Dimensional Breaking Soliton, Boussinesq equation with dual dispersion and modified Korteweg-de Vries-Kadomtsev-Petviashvili equations and their applications. Results Phys.. 2017;7:1143-1149.
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[Google Scholar]

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[Google Scholar]

[41] Xian-Lin Y.A.N.G., Jia-Shi T.A.N.G., . Travelting solutions for konopelchenko-dubrovsky equation using an extended sinh-gordon equation expansion method. Commun. Theor. Phys.. 2008;50(5):1047-1051.
[Google Scholar]

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[Google Scholar]

[43] Yan Zhenya, Zhang Hongqing, . New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water. Phys. Lett. A. 2001;285(5):355-362.
[Google Scholar]

[44] Younis M., Zafar Asim, . Exact solutions to nonlinear differential equations of fractional order via $(\frac{G^{'}}{G})$ -expansion method. Appl. Math.. 2014;5:1-6.
[Google Scholar]

[45] Zafar Asim, . Rational exponential solutions of conformable space-time fractional equal-width equations. Nonlinear Eng. 2018
[Google Scholar]

[46] Zhang S., . Application of exp-function to a KDV equation with variable-coefficients. Phys. Lett. A. 2007;365:448-453.
[Google Scholar]

The conformable space-time fractional mKdV equations and their exact solutions

Abstract

Keywords

Space-time-fractional mKdV equations

Conformable derivatives

Hyperbolic function approach

Exact soliton solutions

1 Introduction

2 Conformable fractional derivative approach

2.1 The hyperbolic function approach

3 The conformable fractional (3 + 1)-dimensional mKdV equation-I

3.1 The conformable fractional (3 + 1)-dimensional mKdV equation-II

3.2 The conformable fractional ( 3 + 1 ) -dimensional mKdV equation-III

3.3 Graphical representation of the solutions for Eq. (5)

4 Conclusion

References

Suggested read for related articles:

3 The conformable fractional (3 + 1)-dimensional mKdV equation-I

3.1 The conformable fractional (3 + 1)-dimensional mKdV equation-II

3.2 The conformable fractional $(3 + 1)$ -dimensional mKdV equation-III