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

Authors: Farah M. Al-Askar; Wael W. Mohammed; Sallieu K. Samura; M. El-Morshedy
Journal: Journal of Function Spaces
 (2022)

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## Abstract

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 significantly more useful and effective in understanding various important challenging physical phenomena. In addition, we use the MATLAB Package to generate 3D graphs for specific FSDSWE solutions in order to discuss the impact of fractional order and the Wiener process on the solutions of FSDSWEs.


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## Body

## 1. Introduction

Partial differential equations (PDEs) have grown in popularity because of their broad spectrum of applications in nonlinear science including engineering [1], civil engineering [2], quantum mechanics [3], soil mechanics [4], statistical mechanics [5], population ecology [6], economics [7], and biology [8, 9]. Therefore, finding exact solutions is critical for a better understanding of nonlinear phenomena. To acquire exact solutions to these equations, a variety of methods such as Darboux transformation [10], Hirota’s function [11], sine-cosine [12, 13], G′/G-expansion [14–16], perturbation [17, 18], Riccati-Bernoulli sub-ODE [19], exp−ϕς-expansion [20, 21], tanh-sech [22, 23], Jacobi elliptic function [24, 25], and Riccati equation method [26] have been used.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 [27–30] 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 ofφ:ℝ+⟶ℝ of order α is defined as
(1)Dyαφx=limε⟶0φy+εy1−α−φyε.The CFD satisfies(1)
Dyαaφy+bψy=aDxαφy+bDxαψy,a,b∈ℝ(2)
DyαC=0, C is a constant(3)
Dyαφ∘ψy=x1−αψ′yφψy(4)
Dyαxÿ=ÿyÿ−α,ÿ∈ℝ(5)
Dyαψy=y1−αdψ/dyOn 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.Therefore, we consider in this paper the following fractional-stochastic Drinfel’d–Sokolov–Wilson equations (FSDSWEs):(2)dΨ+γ1ΦDxαΦdt=σΨdβ,(3)dΦ+γ2DxxxαΦ+γ3ΨDxαΦ+γ4ΦDxαΨdt=σΦdβ,where γk for k=1,2,3,4 are nonzero parameters. Dα, for 0<α≤1, is CFD [29]. βt is a standard Wiener process (SWP), and σ is the noise strength.The Drinfel’d–Sokolov–Wilson equations (DSWEs) ((2) and (3)), with α=1 and σ=0, evolved from shallow water wave models initially given by Drinfel’d and Sokolov [31, 32] and later refined by Wilson [33]. Due to the importance of DSWEs, several authors have created analytical solutions for this system using a variety of methods, including exp-function method [34], truncated Painlevè method [35], F−expansion method [36], Bäcklund transformation of Riccati equation [37], homotopy analysis method [38], and tanh and extended tanh methods [39]. Furthermore, a few authors obtained exact solutions for fractional DSW using various methods such as Jacobi elliptical function method [40] and complete discrimination system for polynomial method [41], while the analytical fractional-stochastic solutions of FSDSWEs ((2) and (3)) have never been obtained before.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 Section2, 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.


## 2. Analytical Solutions of FSDSWEs

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


### 2.1. Wave Equation for FSDSWEs

Let us apply the following wave transformation(4)Ψx,t=ψμeσβt−1/2σ2t,Φx,t=φμeσβt−1/2σ2t,μ=1αxα+ωt,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
(5)dΨ=ωψ′dt+σψdβeσβt−1/2σ2t,dΦ=ωφ′dt+σφdβeσβt−1/2σ2t,DxαΦ=φ′eσβt−1/2σ2t,DxαΨ=ψ′eσβt−1/2σ2t,DxxxαΦ=φ′′′eσβt−1/2σ2t,we attain
(6)ωψ′+γ1φφ′eσβt−1/2σ2t=0,(7)ωφ′+γ2φ′′′+γ3ψφ′eσβt−1/2σ2t+γ4φψ′eσβt−1/2σ2t=0.Taking expectationE· for Equations (6) and (7), we get
(8)ωψ′+γ1φφ′e−1/2σ2tEeσβt=0,(9)ωφ′+γ2φ′′′+γ3ψφ′+γ4φψ′e−1/2σ2tEeσβt=0.Sinceβt is a normal distribution, then Eeσβt=eσ2/2t. Now, Equations (8) and (9) take the type
(10)ωψ′+γ1φφ′=0,(11)ωφ′+γ2φ′′′+γ3ψφ′+γ4φψ′=0.Integrating Equation (10) and putting the constants of integration equal zero, we get
(12)ψ=−γ1ωφ2+C,where C is the integral constant. Plugging Equation (12) into (11) and using Equation (10), we have
(13)γ2φ′′′−γ1γ32ω+γ1γ4ωφ2φ′+ω+Cγ3φ′=0.Integrating Equation (13), we obtain
(14)φ′′−ℓ1φ3+ℓ2φ=0,where
(15)ℓ1=γ1γ36γ2ω+γ1γ43γ2ω,ℓ2=ωγ2+Cγ3γ2.


### 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
(16)φμ=∑i=0Naiχi,where N is fixed by balancing the linear term of the highest order derivative φ′′ with nonlinear term φ3, ai, for i=1,2,⋯aN, are constants to be calculated and χ satisfies the first kind of elliptic equation
(17)χ′=12pχ4+qχ2+r,where p, q, andr are real parameters.We notice that Equation (17) has a variety of solutions depending on p, q, and r as follows (Table 1).Table 1
All possible solutions for Equation (17) for different values of p, q, and r.
Casepqrχμ12m2−1+m21snμ222m2−1−m21−m2dsμ322−m21−m2csμ4−2m22m2−11−m2cnμ5−22−m2m2−1dnμ6m22m2−2214snμ1±dnμ7m22m2−22m24snμ1±dnμ8−12m2+12−1−m224mcnμ±dnμ9m2−12m2+12m2−14dnμ1±snμ101−m221−m221−m24cnμ1±snμ111−m2221−m22214snμdn±cnμ12200cμ13010ceμsnμ=snμ,m,cnμ=cnμ,m,dnμ,m=dnμ,m are the Jacobi elliptic functions (JEFs) for 0<m<1. When m⟶1, the JEFs are converted into the hyperbolic functions shown below:
(18)cnμ⟶sechμ,snμ⟶tanhμ,csμ⟶cschμ,ds⟶cschμ,dnμ⟶sechμ.


### 2.3. Solutions of FSDSWEs

Now, let us determine the parameterN by balancing φ′′ with φ3 in Equation (14) as
(19)N+2=3N⟹N=1.Rewriting Equation (17) with N=1 as
(20)φ=a0+a1χ.Differentiating Equation (20) twice, we have, by using (17),
(21)φ′′=a1qχ+a1pχ3.Substituting Equations (20) and (21) into Equation (14), we obtain
(22)a1p−ℓ1a13χ3−3a0a12ℓ1χ2+a1q−3ℓ1a02a1+ℓ2a1χ−ℓ1a03−ℓ2a0=0.Putting each coefficient ofχk for k=0,1,2,3 equal zero, we get
(23)a1p−ℓ1a13=0,3a0a12ℓ1=0,a1q−3ℓ1a02a1+ℓ2a1=0,ℓ1a03−ℓ2a0=0.Solving these equations, we obtain(24)a0=0,a1=±pℓ1,q=−ℓ2.Hence, the solution of Equation (14) is
(25)φμ=±pℓ1χμ,for p/ℓ1>0. There are two sets depending only on p and ℓ1 as follows.First set: if p>0 and ℓ1>0, then the solutions φμ, from Table 1, of wave Equation (14) are as follows (Table 2).Table 2
All possible solutions for wave Equation (14) when p>0.
Casepqrχμφμ12m2−1+m21snμ±pℓ1snμ222m2−1−m21−m2dsμ±pℓ1dsμ322−m21−m2csμ±pℓ1csμ4m22m2−221/4 or m2/4snμ1±dnμ±pℓ1snμ1±dnμ51−m221−m221−m24cnμ1±snμ±pℓ1cnμ1±snμ61−m2221−m22214snμdn±cnμ±pℓ1snμdn±cnμ7200cμ±pℓ1cμIfm⟶1, then Table 2 degenerates to Table 3.Table 3
All possible solutions for wave Equation (14) when p>0 and m⟶1.
Casepqrχμφμ12−21tanhμ±pℓ1tanhμ2210sechμ±p/ℓ1sechμ3210cschμ±p/ℓ1cschμ412−1214tanhμ1±sechμ±pℓ1tanhμ1±sechμ5200cμ±pℓ1cμNow, using Table2 (or Table 3 when m⟶1) and Equations (25) and (12), we get the solutions of FSDSWEs ((2) and (3)), for p/ℓ1>0, as follows:
(26)Φx,t=φμeσβt−1/2σ2t,(27)Ψx,t=−γ1ωφ2μ+Ceσβt−1/2σ2t,where μ=xα/α+ωt.Second set: if p<0 and ℓ1<0, then the solutions φμ, from Table 1, of wave Equation (14) are as follows.Ifm⟶1, then Table 3 degenerates to Table 4.Table 4
All possible solutions for wave Equation (14) when p<0 and m⟶1.
Casepqrχμφμ1−210sechμ±pℓ1sechμ2−12202sechμ±2pℓ1sechμIn this case, using Table5 (or Table 4 when m⟶1), we can get the analytical solutions of FSDSWEs ((2) and (3)) as stated in Equations (26) and (27).Table 5
All possible solutions for wave Equation (14) when p<0.
Casepqrχμφμ1−2m22m2−11−m2cnμ±pℓ1cnμ2−22−m2m2−1dnμ±pℓ1dnμ3−12m2+12−1−m224mcnμ±dnμ±pℓ1mcnμ±dnμ4m2−12m2+12m2−14dnμ1±snμ±pℓ1dnμ1±snμ


## 2.1. Wave Equation for FSDSWEs

Let us apply the following wave transformation(4)Ψx,t=ψμeσβt−1/2σ2t,Φx,t=φμeσβt−1/2σ2t,μ=1αxα+ωt,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
(5)dΨ=ωψ′dt+σψdβeσβt−1/2σ2t,dΦ=ωφ′dt+σφdβeσβt−1/2σ2t,DxαΦ=φ′eσβt−1/2σ2t,DxαΨ=ψ′eσβt−1/2σ2t,DxxxαΦ=φ′′′eσβt−1/2σ2t,we attain
(6)ωψ′+γ1φφ′eσβt−1/2σ2t=0,(7)ωφ′+γ2φ′′′+γ3ψφ′eσβt−1/2σ2t+γ4φψ′eσβt−1/2σ2t=0.Taking expectationE· for Equations (6) and (7), we get
(8)ωψ′+γ1φφ′e−1/2σ2tEeσβt=0,(9)ωφ′+γ2φ′′′+γ3ψφ′+γ4φψ′e−1/2σ2tEeσβt=0.Sinceβt is a normal distribution, then Eeσβt=eσ2/2t. Now, Equations (8) and (9) take the type
(10)ωψ′+γ1φφ′=0,(11)ωφ′+γ2φ′′′+γ3ψφ′+γ4φψ′=0.Integrating Equation (10) and putting the constants of integration equal zero, we get
(12)ψ=−γ1ωφ2+C,where C is the integral constant. Plugging Equation (12) into (11) and using Equation (10), we have
(13)γ2φ′′′−γ1γ32ω+γ1γ4ωφ2φ′+ω+Cγ3φ′=0.Integrating Equation (13), we obtain
(14)φ′′−ℓ1φ3+ℓ2φ=0,where
(15)ℓ1=γ1γ36γ2ω+γ1γ43γ2ω,ℓ2=ωγ2+Cγ3γ2.


## 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
(16)φμ=∑i=0Naiχi,where N is fixed by balancing the linear term of the highest order derivative φ′′ with nonlinear term φ3, ai, for i=1,2,⋯aN, are constants to be calculated and χ satisfies the first kind of elliptic equation
(17)χ′=12pχ4+qχ2+r,where p, q, andr are real parameters.We notice that Equation (17) has a variety of solutions depending on p, q, and r as follows (Table 1).Table 1
All possible solutions for Equation (17) for different values of p, q, and r.
Casepqrχμ12m2−1+m21snμ222m2−1−m21−m2dsμ322−m21−m2csμ4−2m22m2−11−m2cnμ5−22−m2m2−1dnμ6m22m2−2214snμ1±dnμ7m22m2−22m24snμ1±dnμ8−12m2+12−1−m224mcnμ±dnμ9m2−12m2+12m2−14dnμ1±snμ101−m221−m221−m24cnμ1±snμ111−m2221−m22214snμdn±cnμ12200cμ13010ceμsnμ=snμ,m,cnμ=cnμ,m,dnμ,m=dnμ,m are the Jacobi elliptic functions (JEFs) for 0<m<1. When m⟶1, the JEFs are converted into the hyperbolic functions shown below:
(18)cnμ⟶sechμ,snμ⟶tanhμ,csμ⟶cschμ,ds⟶cschμ,dnμ⟶sechμ.


## 2.3. Solutions of FSDSWEs

Now, let us determine the parameterN by balancing φ′′ with φ3 in Equation (14) as
(19)N+2=3N⟹N=1.Rewriting Equation (17) with N=1 as
(20)φ=a0+a1χ.Differentiating Equation (20) twice, we have, by using (17),
(21)φ′′=a1qχ+a1pχ3.Substituting Equations (20) and (21) into Equation (14), we obtain
(22)a1p−ℓ1a13χ3−3a0a12ℓ1χ2+a1q−3ℓ1a02a1+ℓ2a1χ−ℓ1a03−ℓ2a0=0.Putting each coefficient ofχk for k=0,1,2,3 equal zero, we get
(23)a1p−ℓ1a13=0,3a0a12ℓ1=0,a1q−3ℓ1a02a1+ℓ2a1=0,ℓ1a03−ℓ2a0=0.Solving these equations, we obtain(24)a0=0,a1=±pℓ1,q=−ℓ2.Hence, the solution of Equation (14) is
(25)φμ=±pℓ1χμ,for p/ℓ1>0. There are two sets depending only on p and ℓ1 as follows.First set: if p>0 and ℓ1>0, then the solutions φμ, from Table 1, of wave Equation (14) are as follows (Table 2).Table 2
All possible solutions for wave Equation (14) when p>0.
Casepqrχμφμ12m2−1+m21snμ±pℓ1snμ222m2−1−m21−m2dsμ±pℓ1dsμ322−m21−m2csμ±pℓ1csμ4m22m2−221/4 or m2/4snμ1±dnμ±pℓ1snμ1±dnμ51−m221−m221−m24cnμ1±snμ±pℓ1cnμ1±snμ61−m2221−m22214snμdn±cnμ±pℓ1snμdn±cnμ7200cμ±pℓ1cμIfm⟶1, then Table 2 degenerates to Table 3.Table 3
All possible solutions for wave Equation (14) when p>0 and m⟶1.
Casepqrχμφμ12−21tanhμ±pℓ1tanhμ2210sechμ±p/ℓ1sechμ3210cschμ±p/ℓ1cschμ412−1214tanhμ1±sechμ±pℓ1tanhμ1±sechμ5200cμ±pℓ1cμNow, using Table2 (or Table 3 when m⟶1) and Equations (25) and (12), we get the solutions of FSDSWEs ((2) and (3)), for p/ℓ1>0, as follows:
(26)Φx,t=φμeσβt−1/2σ2t,(27)Ψx,t=−γ1ωφ2μ+Ceσβt−1/2σ2t,where μ=xα/α+ωt.Second set: if p<0 and ℓ1<0, then the solutions φμ, from Table 1, of wave Equation (14) are as follows.Ifm⟶1, then Table 3 degenerates to Table 4.Table 4
All possible solutions for wave Equation (14) when p<0 and m⟶1.
Casepqrχμφμ1−210sechμ±pℓ1sechμ2−12202sechμ±2pℓ1sechμIn this case, using Table5 (or Table 4 when m⟶1), we can get the analytical solutions of FSDSWEs ((2) and (3)) as stated in Equations (26) and (27).Table 5
All possible solutions for wave Equation (14) when p<0.
Casepqrχμφμ1−2m22m2−11−m2cnμ±pℓ1cnμ2−22−m2m2−1dnμ±pℓ1dnμ3−12m2+12−1−m224mcnμ±dnμ±pℓ1mcnμ±dnμ4m2−12m2+12m2−14dnμ1±snμ±pℓ1dnμ1±snμ


## 3. 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:
(28)Φx,t=pℓ1cnxαα+ωteσβt−1/2σ2t,(29)Ψx,t=−γ1pωℓ1cn2xαα+ωteσβt−1/2σ2t,with C=0,p=−2m2,γ1=γ2=1, γ3=γ4=3, p=−2, q=2−m2, and m=0.5. Then, ℓ1=−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.Figure 1
3D plot of Equations (28) and (29) with σ=0 and α=1.While in Figure2, if the noise is introduced and its strength σ is raised, the surface becomes substantially flatter as follows.Figure 2
3D plot of Equations (28) and (29) with σ=1,2 and α=1.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.Figure 3
3D plot of Equation (28) with σ=0 and different α.Figure 4
3D plot of Equation (29) with σ=0 and different α.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.


## 4. 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.



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