Novel Analytical Approach for the Space-Time Fractional (2+1)-Dimensional Breaking Soliton Equation via Mathematical Methods

: The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modiﬁed Riemann– Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modiﬁed F-expansion methods. We used the fractional complex transformation of the concern fractional differential equation to convert it for the solvable integer order differential equation. After the successful implementation of the presented methods, a comprehensive class of novel and broad-ranging exact and solitary travelling wave solutions were discovered, in terms of trigonometric, rational and hyperbolic functions. Hence, the present methods are reliable and efﬁcient for solving nonlinear fractional problems in mathematics physics.

Γ is defined as Definition 2. The Mittag-Leffler function via two parameters is explained in [53]: This function is used for utilization to scrutinize the fractional PDEs as the exponential function with integer order. For the fractional derivative, some postulates are as follows: First Postulate: where r ia a real number. Second Postulate: Third Postulate: Fourth Postulate: where η = G(x).

Description of the Proposed Methods
Let the NFPDEs be Let Substitute Equation (11) into Equation (10),

Modified F-Expansion Method
Let (12) have the following solution: Put Equation (15) with Equation (16) in Equation (12), solve the obtained algebraic system of equations for investigating the solution of Equation (10).

Applications
The space-time fractional breaking soliton equation: Put Equation (11) into Equation (1) (Figures 1-6), we have the following ODE form: Now we integrate the second equation, Equation (17), by taking the constant of integration as equal to zero, Putting Equation (18) into the first equation of (17), yields

Results and Discussion
The mathematical methods emphasize the wave solutions of Equation (1). In the derived solutions, parameters A 1 , A −1 , A 2 and A −2 received various specific values due to these exact solutions being converted into different solitary wave solutions in different forms, such as hyperbolic, trigonometric and rational functions (Figures 1-6). Currently several methods have been utilized to solve Equation (1) throughout the research literature [40][41][42][43][44][45][46][47][48]. Moreover, our investigated solutions are likely similar to other solutions in different research articles. Our solutions U 6 and V 6 in (35) and (36) are likely similar to the solutions U 41 in Equation (82) and U 42 in Equation (83), respectively, in [50]. Furthermore, our solutions U 141 and V 141 in (64) and (65) are similar in form to the solutions u 2 and v 2 in Equation (4.39), respectively, in [54]. Our solutions U 8 and V 9 in (41) and (42) are similar in form to the solutions u 23(η) and u 214(η) in Equation (50) and Equation (51), respectively, in [55]. The remainder of our derived solutions are novel and have not yet been reported in any research literature. Hence, our employed schemes are simple and useful for solving many other nonlinear problems in applied sciences.

Conclusions
In this study, we have proposed two novel techniques, namely, an improved form of simple equation and a modified form of F-expansion are utilized to construct exact solutions of the nonlinear fractional time space (2+1)-dimensional breaking soliton equation, via the properties of the modified Riemann-Liouville derivative. Multifarious transformation has been operated to renovate the fractional order differential equations. The constructed results have extensive potential to comprehend the interior configurations of the usual manifestations that arise in physics, mathematics and in other different fields. Hence, it is worth declaring that the execution of our techniques is extremely steady and well-organized for fractional differential equations.