# Exact solutions of \((3+1)\)-dimensional fractional mKdV equations in conformable form via \(\exp (-\phi (\tau ))\) expansion method

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## Abstract

In this study, three conformable \((3+1)\)-dimensional fractional mKdV equations are explored via \(\exp (-\phi (\tau ))\) expansion method. A traveling wave transformation along with conformable derivative is used to transformed the nonlinear fractional differential equation into an ordinary differential equation. Then, the implementation of \(\exp (-\phi (\tau ))\) expansion method gives a variety of exact solutions of space-time fractional mKdV equations.

## Keywords

Exact solutions Fractional derivatives Soliton solutions Fractional mKdV equations## 1 Introduction

Among all above approaches, the \(\exp (-\phi (\tau ))\) technique has achieved substantial consideration due to its competency in inaugurating the exact solutions of nonlinear differential equations, see for instance, [41, 42, 43, 44]. In fractional calculus, many definitions of fractional derivatives, Like Hilfer, Riemann–Liouville, Caputo form and so on, have been introduced in the literature but the well known product, quotient and the chain rules were the setbacks of one definition or another [45, 46, 47, 48, 49]. Therefore the most fascinating definition of fractional derivative with some of its properties are given in [50].

This paper aims to explore the conformable space-time fractional modified KdV equations of \((3+ 1)\)-dimensional for exact soliton type solutions via the \(\exp (-\phi (\tau ))\) 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 \(\exp (-\phi (\tau ))\) expansion approach is given in Sect. 2. Section 3, illustrate how to utilize this approach for producing new solutions with their graphs. The last parts summarized results and discussion of the current study.

## 2 Conformable fractional derivative approach

We recall the conformable derivative with some of its properties [50].

### Definition 1

is known as \(\alpha , ~~0 <\alpha \le 1\) order conformable fractional derivative of *p*. The followings are some useful properties:

\(D^{\alpha }_t (a~p+b~g)=a D^{\alpha }_t(p)+b D^{\alpha }_t (g)\), for all \(a,~ b \in {\mathbb {R}}\)

\( D^{\alpha }_t(p~g)=p~D^{\alpha }_t(g)+g~D^{\alpha }_t(p)\).

*g*be a differentiable function defined in the range of

*p*.

\(D^{\alpha }_t(t^{h})=h~t^{h-\alpha }\), for all \(h\in {\mathbb {R}}\)

\(D^{\alpha }_t(\delta )=0\), where \(\delta \) is constant.

\(D^{\alpha }_t(p/g)=\frac{g D^{\alpha }_t(p)-p D^{\alpha }_t(g)}{g^{2}}\).

Conjointly, if *p* is differentiable, then \(D^{\alpha }_t(p(t))=t^{1-\alpha }\frac{d p(t)}{dt}\).

### 2.1 Demarcation of the \(\exp (-\phi (\tau ))\) method

*N*is calculated using the homogeneous balance principle (HBP) and \(\phi (\tau )\) is a function that satisfies a first-order equation as

**Case 1**: If \(\lambda _{1}^{2}-4\mu _{1}>0\) and \(\mu _{1} \ne 0\), then

**Case 2**: If \(\lambda _{1}^{2}-4\mu _{1}>0\) , \(\mu _{1}=0\) and \(\lambda _{1} \ne 0\), then

**Case 3**: If \(\lambda _{1}^{2}-4\mu _{1}<0\) and \(\mu _{1} \ne 0\), then

**Case 4**: If \(\lambda _{1}^{2}-4\mu _{1}=0\) , \(\mu _{1} \ne 0\) and \(\lambda _{1} \ne 0\), then

**Case 5**If \(\lambda _{1}^{2}-4\mu _{1}=0\) , \(\mu _{1}=0\) and \(\lambda _{1}=0\), then

## 3 Execution of the method

Firstly, we consider the space-time fractional mKdV equation (1).

### 3.1 Exact solutions of \((3+1)\)-dimensional conformable space-time fractional Eq. (1)

### 3.2 Exact solutions of \((3+1)\)-dimensional conformable space-time fractional Eq. (2)

### 3.3 Exact solutions of \((3+1)\)-dimensional conformable space-time fractional Eq. (3)

## 4 Results and discussion

Furthermore, for suitable parametric choices, we plotted three dimensional graphics of some solutions of the fractional mKDV equations for Figs. 1, 2 and 3. The obtained solutions are periodic wave, solitary wave and traveling wave solutions. It is more advantageous than other methods because different, various and more solutions are obtained with our methods. Note that our solutions are new and more extensive than the given ones in [13, 14]. When the parameters are given special values, the optical solitary waves are derived from the travelling waves.

## 5 Conclusion

In this study, three conformable fractional \((3+1)\)-dimensional mKdV equations have been explored via \(\exp (-\phi (\tau ))\) expansion method. A traveling wave transformation along with conformable derivative has used to transformed the nonlinear fractional differential equation into an ordinary differential equation. We plot some sketches for some of the analytical and exact solutions to express more physical properties of this model. Then, the implementation of \(\exp (-\phi (\tau ))\) expansion method procured a variety of exact solutions of aforementioned fractional mKdV equations. This method and the mathematical tool can be used to derive a localized wave solutions for different nonlinear models in engineering and mathematical physics.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors involved in this manuscript declare that they have no conflict of interest.

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