New exact solution for the (3+1) conformable space–time fractional modified Korteweg–de-Vries equations via Sine-Cosine Method

ABSTRACT In this research work, we established exact solution for the conformable space–time fractional (3 + 1) dimensional modified Korteweg de Vries equations (mKdV). A Sine- Cosine method is used for obtaining travelling wave solutions for these models with minimal algebra. We can conclude that the proposed scheme is reliable and efficient as its required minimal algebra without using sophisticated Mathematical tools (maple, Mathematica and others). The goal has been achieved with minimal computational cost and the present solutions obtained will serve as new solutions to the modified Korteweg–de-Vries (mKdV) equations.


Introduction
Nonlinear partial differential equations are known in fluid mechanics, plasma physics and nonlinear dispersive wave among others due to their vital roles and applications in many engineering and science application problems. However, a very good and well known example of the nonlinear partial differential equation describing shallow water waves is the Korteweg-de-Vries equation (KdV) given by The KdV equation also plays vital role in modelling blood pressure pulses and internal gravity waves in oceans among many other applications. Thus, efforts of investigating exact travelling wave solutions to problem (1) is of tremendous benefit since they help in understanding the physics behind. In addition, as higher dimensional models turn to be more realistic; some modifications to Eq. (1) have been demonstrated in literature among which the (3 + 1)-dimensional modified KdV equation given in Eq. (2) by Hereman [1,2] as Another recent (3 + 1)-dimensional modified KdV equations given by Wazwaz that read in [3] as; u t + 6u 2 u y + u xyz = 0, These equations play important role in threedimensional non-linear dispersion problems. However, this research aims to study this modified KdV equations given in Equations (1)-(3) with the introduction of the fractional order derivative in both the space and time variables using the new conformable fractional derivative [4] using sine and cosine method proposed by Wazwaz [5]. Also, it is important to note that several analytical methods have been used in this regard in treating such problems, read [6][7][8][9][10][11][12][13][14][15][16][17]. The definition of the conformable fractional derivative [4] read; let u: [0, ∞] → R, the α's order conformable derivative of u is defined by , t > 0, α (0, 1) (4) The paper is organized as follows: Section 2 devoted to properties of the conformable fractional derivative and method. Section 3 tackles the main problems and Section 4 is for conclusion

The properties of the conformable fractional derivative and methodology of solution
Some properties of the conformable fractional derivative are given using the following definition: Definition [6]: Let α ∈ (0, 1) and suppose u (t) and v (t) are α-differentiable at t > 0. Then Let's consider a nonlinear partial differential equation of the form which describes the dynamical wave solutionu(x, t). It is vital to summarize the main steps of the Sine-Cosine method given by Wazwaz in [5] as follows: Step 1: To find the travelling wave solution of equation (5), we introduce the wave variable Step 2: Based on this, we use the following changes: and so on for other derivatives. Using equation (8) changes the PDE equation (5) to an ODE where u ξ denotes du dξ .
Step 3: We then integrate the ordinary differential equation (9) as many times as possible, and setting the constant of integration to be zero.
Step 4: The solution may be set in the form u(x, t) = sin β (μξ ) (10) or in the form where , µ, and β are parameters to be determined.

Application
In this section, we the application of Sine-Cosine method for the exact solutions of (3 + 1)-dimensional conformable space-time fractional modified Kortewegde-Vries (mKdV) equations.

The first conformable space-time fractional mKdV equation
We consider the first (3 + 1) -dimensional conformable space time fractional mKdV equation given by Applying the following wave transformation We have; on integrating equation (23) we have; Having use of sine-cosine method, we are to consider U(x, y, z, t) = U(ξ ), Therefore, by substituting U and U'' in equation (24) respectively, we have; The equation (28) is satisfied if the following system of algebraic equations holds: −d = abcμ 2 β 2 , Solving this system, we obtained The same results are obtained if we use the cosine method, in view of the above equation, we obtain the solutions For d > 0, the following solutions