The Exact Solutions for Fractional-Stochastic Drinfel ’ d – Sokolov – Wilson Equations Using a Conformable Operator

The fractional-stochastic Drinfel ’ d – Sokolov – Wilson equations (FSDSWEs) perturbed by the multiplicative Wiener process are studied. The mapping method is used to obtain rational, hyperbolic, and elliptic stochastic solutions for FSDSWEs. Due to the importance of FSDSWEs in describing the propagation of shallow water waves, the derived solutions are signi ﬁ cantly more useful and e ﬀ ective in understanding various important challenging physical phenomena. In addition, we use the MATLAB Package to generate 3D graphs for speci ﬁ c FSDSWE solutions in order to discuss the impact of fractional order and the Wiener process on the solutions of FSDSWEs.

Recently, fractional derivatives are used to characterize a wide range of physical phenomena in mathematical biology, engineering disciplines, electromagnetic theory, signal processing, and other scientific research. These new fractional-order models are better than the previously used integer-order models because fractional-order derivatives and integrals allow for the modeling of distinct substances' memory and hereditary capabilities.
The conformable fractional derivative (CFD) helps us to develop an idea of how physical phenomena act. The CFD is very useful for modelling a variety of physical issues since differential equations with CFD are simpler to solve numerically than those with Caputo fractional derivative or the Riemann-Liouville. Currently, authors are focusing on fractional calculus and creating new operators such the Caputo Fabrizio, Caputo, Riemann Liouville, and Atangana Baleanu derivatives. The conformable fractional operator [27][28][29][30] eliminates some of the restrictions of current fractional operators and provides standard calculus properties such as the derivative of the quotient of two functions, the product of two functions, Rolle's theorem, the chain rule, and the mean value theorem. Here, we use CFD stated in [29]. Therefore, let us state the definition of CFD and its properties as follows [29]: The CFD of φ : ℝ + ⟶ ℝ of order α is defined as The CFD satisfies On the other hand, in the practically physical system, random perturbations emerge from a variety of natural sources. They cannot be avoided, because noise can cause statistical properties and significant phenomena. Consequently, stochastic differential equations emerged and they started to play a major role in modeling phenomena in oceanography, physics, biology, chemistry, atmosphere, fluid mechanics, and other fields.
Our aim of this paper is to attain a wide range of solutions including rational, hyperbolic, and elliptic functions for FSDSWEs ( (2) and (3)) by using the mapping method. This is the first study to obtain exact solutions to FSDSWEs with combination of a stochastic term and fractional derivative. Also, we utilize MATLAB to generate 3D diagrams for a number of the FSDSWEs ( (2) and (3)) developed in this study to demonstrate how the SWP affects these solutions. This paper will be formatted as follows. In Section 2, the mapping method is used to generate analytic solutions for FSDSWEs ( (2) and (3)). In Section 3, we investigate the effect of the SWP and fractional order on the derived solutions. Section 4 presents the paper's conclusion.

Analytical Solutions of FSDSWEs
First, let us derive the wave equation of FSDSWEs as follows.

Wave Equation for
FSDSWEs. Let us apply the following wave transformation to attain the wave equation of FSDSWEs ( (2) and (3)), where ψ and φ are real deterministic functions and ω is a constant. Putting Equation (4) into Equations (2) and (3) and using we attain Taking expectation Eð·Þ for Equations (6) and (7), we get

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Since βðtÞ is a normal distribution, then Eðe σβðtÞ Þ = e ðσ 2 /2Þt : Now, Equations (8) and (9) take the type Integrating Equation (10) and putting the constants of integration equal zero, we get where C is the integral constant. Plugging Equation (12) into (11) and using Equation (10), we have Integrating Equation (13), we obtain where 2.2. The Mapping Method Description. Here, let us describe the mapping method stated in [42]. Assuming the solutions of Equation (14) have the form where N is fixed by balancing the linear term of the highest order derivative φ ′ ′ with nonlinear term φ 3 , a i , for i = 1, 2, ⋯a N , are constants to be calculated and χ satisfies the first kind of elliptic equation where p, q, and r are real parameters. We notice that Equation (17) has a variety of solutions depending on p, q, and r as follows (Table 1). snðμÞ = snðμ, mÞ, cnðμÞ = cnðμ, mÞ, dnðμ, mÞ = dnðμ, mÞ are the Jacobi elliptic functions (JEFs) for 0 < m < 1: When m ⟶ 1, the JEFs are converted into the hyperbolic functions shown below: 2.3. Solutions of FSDSWEs. Now, let us determine the parameter N by balancing φ′′ with φ 3 in Equation (14) as Rewriting Equation (17) with N = 1 as Differentiating Equation (20) twice, we have, by using (17), Substituting Equations (20) and (21) into Equation (14), we obtain Putting each coefficient of χ k for k = 0, 1, 2, 3 equal zero, we get Solving these equations, we obtain

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Hence, the solution of Equation (14) is for p/ℓ 1 > 0: There are two sets depending only on p and ℓ 1 as follows.

The Impact of Noise and Fractional Order on the Solutions
The impact of the noise and fractional order on the acquired solutions of FSDSWEs ( (2) and (3)) is addressed. MATLAB tools are used to generate graphs for the following solutions: with C = 0, p = −2m 2 , γ 1 = γ 2 = 1, γ 3 = γ 4 = 3, p = −2, q = 2 − m 2 , and m = 0:5: Then, ℓ 1 = −6/7 and ω = 7/4: Firstly the impact of noise: in the absence of the noise, the surface is periodic (not flat) as we see in Figure 1.
While in Figure 2, if the noise is introduced and its strength σ is raised, the surface becomes substantially flatter as follows.
Secondly the impact of fractional order: in Figures 3  and 4, if σ = 0, we can see that the surface expands when α is increasing.
From the previous simulations, we may examine the nature of the solution as a double-periodic wave in physical form. We may conclude that it is critical to incorporate some fluctuation when modelling any phenomenon since the ignored terms may have an influence on the solutions.

Conclusions
In this paper, we considered the fractional-stochastic Drinfel'd-Sokolov-Wilson equations. This equation is well known in mathematical physics, population dynamics, surface physics, plasma physics, and applied sciences. The analytical solutions to FSDSWEs ( (2) and (3)) were successfully attained by utilizing the mapping method. Due to the importance of FSDSWEs, these established solutions are significantly more useful and effective in understanding a variety of critical physical processes. In addition, we utilized the MATLAB software to demonstrate how multiplicative noise and fractional order affected the solutions of FSDSWEs. We may employ additive noise to address the FSDSWEs ( (2) and (3)) in future study.

Data Availability
All data are available in this paper.