Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme

: In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, ﬁber optics, deep water-waves, plasma physics, mathematical physics, ﬂuid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are veriﬁed with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).


Introduction
Fractional Calculus (FC) is a generalization of classical calculus concerned with operations of integration and differentiation of non-integer (fractional) order. Since the 19th century, the theory of fractional calculus developed rapidly, mostly as a foundation for a number of applied disciplines, including fractional geometry, fractional differential equations (FDE) and fractional dynamics. The applications of FC are very wide nowadays. It is safe to say that almost no discipline of modern engineering, and science in general, remains untouched by the tools and techniques of fractional calculus. For example, wide and fruitful applications can be found in rheology, viscoelasticity, acoustics, optics, chemical and statistical physics, robotics, control theory, electrical and mechanical engineering, bio-engineering etc. In fact, one could argue that real world processes are fractional order systems in general. The main reason for the success of FC applications is that these new fractional-order models are often more accurate than integer-order ones, i.e., there are more degrees of freedom in the fractional order model than in the corresponding classical one. Fractional calculus is a field of mathematics study that grew out of the traditional definitions of calculus integral and derivative operators in much the same way fractional exponents are an outgrowth of exponents with integer values. A fractional equation (FE) is a differential equation that contains fractional derivatives or integrals. The awareness of the importance of this kind of equation has grown continually in the last decade. Many applications have become apparent: wave propagation in a complex or porous media [1], the fractional complex Ginzburg-Landau model [2], fractional order modified Duffing systems [3], the fractional order Boussinesq-Like equations occurring in physical sciences and engineering [4], symmetric regularized long-wave (SRLW) equations arising in long The main aim of this research is to discover new exact wave solutions to the truncated M-fractional complex three coupled Maccari's system with the help of the Sardar subequation method.
The motivation of this paper is to explain the effect of the M-fractional derivative on the solutions for the space-time fractional complex three coupled Maccari's system that are gained with the use of the Sardar sub-equation method. The significance of the M-fractional derivative is that it fulfills both properties of the integer and fractional order derivatives. This derivative generalizes the many fractional derivatives and satisfies important properties of the integer-order derivatives. Via our method, we can observe some elementary relationships between nonlinear fractional partial differential equations (NLFPDEs) and other simple nonlinear ordinary differential equations (NLODEs). It has been found that with the use of simple schemes and solvable ODEs, different types of exact-wave solutions for some complicated NLFPDEs can be easily obtained. The solutions attained are newer than the existing solutions in the literature.
This paper consists of different sections. In Section 2, we describe the main steps of our concerned method Sardar sub-equation. In Section 3, we explain our concerned model and its mathematical analysis. In Section 4, we apply the method to gain the new exact wave solutions for our concerned model. In Section 5, we explain the obtained solutions through 2-D, 3-D and contour graphs. In Section 6, we give the conclusion of our research work.

Description of Sardar Sub-Equation Method
Here, we explain the fundamental points of the Sardar sub-equation method [29]. Let us consider the non-linear fractional partial differential equation: Here, g = g(x, y, t) represents a wave function.
Applying the wave transformations gives as follow: where parameter λ represents the wave number, κ shows the frequency and µ denotes the speed of soliton. We obtain a non-linear ordinary differential equation (ODE), shown as: Consider that Equation (3) possesses the results in the given shape: where m is a natural number.
Here, ψ(ζ) fulfills the ODE, shown as: where σ and κ are constants. Using Equation (4) into Equation (3) with Equation (5) and collecting the coefficients of each power of ψ i . Putting the co-efficient of each power equal to 0, we gain a set of algebraic equations in the term b i , λ, µ. By solving the obtained system of equations, we obtain the values of the parameters.

The Governing Model and Its Mathematical Treatment
The three coupled non-linear Maccari's system explains how isolated waves are propagated in a finite region of space, in optical communications, hydrodynamics and plasma physics. Assume the M-fractional three coupled non-linear Maccari's system shown in [30]: where Here, E γ (.) represents the truncated Mittag-Leffler function of one parameter given in [31,32].
Integrating the fourth equation of system (23) yields Putting Equation (25) into the first three equations of the system (23), we obtain Taking V = θ 2 U and taking W = c U, we obtain

Solutions through Sardar Sub-Equation Method
By applying the homogeneous balance technique and balancing U and U 3 in Equation (27), we gain 3m = m + 2, So, Equation (4) changes into the given form for m = 1.
Putting Equation (30) into Equation (27) with Equation (5). By collecting co-efficients of every order of ψ(ζ) and taking them equal to 0, we obtain a set of algebraic equations. By solving the obtained set of equations using Mathematica software, we gain the following sets.

Graphical Representation of Solutions
In this section we present the graphical behavior of the solutions in the 3D, contour and 2D surfaces. The effect of the fractional order derivatives is also shown by the following graphs that are not given in [30]. The fractional order derivative gives more appropriate results than the integer order derivatives.

Conclusions
Overall, this paper contributes to our understanding of the truncated M-fractional complex three coupled Maccari's system and provides a useful technique for handling nonlinear partial differential equations. This paper describes the successful application of the Sardar sub-equation method to obtain new optical wave solutions for the truncated M-fractional complex three coupled Maccari's system. The obtained solutions are useful for future studies of the concerned model and provide insights into the behavior of optical waves in complex media. As far as we know, for the first time, we used this method for this model so that all the solutions are new and cannot be found in the literature to our best knowledge. The solutions were verified using Mathematica software and were also described graphically using 2-dimensional, 3-dimensional, and contour plots through Maple software. The obtained results in this work have some interesting applications in some plasma environments, such as new earth's ionoshere zone. The Sardar sub-equation method is shown to be a simple, fruitful and reliable technique for handling nonlinear partial differential equations. We will extend the proposed method for fractional models in a future work.