# Fractional-Stochastic Solutions for the Generalized (2+1)-Dimensional Nonlinear Conformable Fractional Schrödinger System Forced by Multiplicative Brownian Motion

Authors: Sahar Albosaily; Wael W. Mohammed; Ekram E. Ali; R. Sidaoui; E. S. Aly; M. El-Morshedy
Journal: Journal of Function Spaces
 (2022)

---

## Abstract

In this paper, the (2+1)-dimensional nonlinear conformable fractional stochastic Schrödinger system (NCFSSS) generated by the multiplicative Brownian motion is treated. To get new rational, trigonometric, hyperbolic, and elliptic stochastic solutions, we use two different methods: the sine-cosine and the Jacobi elliptic function methods. Moreover, we use the MATLAB tools to plot our figures to introduce a variety of 2D and 3D graphs to highlight the effect of the multiplicative noise on the exact solutions of the NCFSSS. Finally, we illustrate that the multiplicative Brownian motion stabilizes the solutions of NCFSSS a round zero.


---

## Body

## 1. Introduction

Stochastic partial differential equations (SPDEs) can be used to represent a wide range of complicated nonlinear physical processes. These kinds of equations appear in a variety of areas including physics, finance, climate dynamics, chemistry, biology, geophysical, engineering, and other fields [1–3].On the other side, fractional partial differential equations (FPDEs) have gotten a lot of interest because they may illustrate the fundamental components underlying real-world issues. They have been seen in a number of physical phenomena, such as viscoelastic materials with relaxation and creeping functions, the motion of a heavy meager surface in a Newtonian fluid, and relapse subordinate dissipative occupancy of components. As a result, FPDEs are employed in a range of fields, including predicting, describing, and modeling the mechanisms engaged in finance, polymeric materials, a kinematic model of neutron points, engineering, electrical circuits, solid-state physics, optical fibers, chemical kinematics, biogenetics, plasma physics, physics of condensed matter, meteorology, electromagnetic, elasticity, and oceanic spectacles [4–9].The exact solutions of PDEs are important in nonlinear science. As a result, various analytical techniques, such as tanh-sech [10, 11], Darboux transformation [12], sine-cosine [13, 14], extended simple equation [15], extended sinh-Gordon equation expansion [16], F-expansion [17], Kudryashov technique [18], generalized Kudryashov [19–21], exp−ϕς-expansion [22], G′/G-expansion [23–25], Hirota’s function [26], perturbation [5, 27], the Jacobi elliptic function [28, 29], and Riccati-Bernoulli sub-ODE [30], have been created to deal with these types of equations.To reach a better level of qualitative agreement, the following (2+1)-dimensional nonlinear conformable fractional stochastic Schrödinger system (NCFSSS) is addressed:
(1)idu+γ1Dxyαu+γ2uvdt+iσudB=0,(2)γ3Dxαv+γ4Dyαu2=0,where v∈ℝ while u∈ℂ. Dα is the conformable derivative (CD) [31], and γi are arbitrary constants for i=1,..,4. Bt is a Brownian motion (BM), and udB is multiplicative noise in the Itô sense.The NCFSSS ((1) and (2)) is crucial in atomic physics, and the functions v and u have diverse physical meanings in various disciplines of physics such as plasma physics [32] and fluid dynamics [33]. In the hydrodynamic context, v is the induced mean flow, and u is the envelope of the wave packet [33], while, in the context of water waves, v is the velocity potential of the mean flow interacting with the surface waves and u is the amplitude of a surface wave packet [34]. The multiplicative noise iσudB plays an important role in the theory of measurements continuous in time in open quantum systems. For more physical interpretations, we refer to [35, 36] and the references therein.Recently, many authors have established exact solutions of NCFSSS ((1) and (2)) with σ=0 and α=1 by employing various techniques, such as applied Kudryashov approach [37], direct approach [38], and the extended modified auxiliary equation [39]. Moreover, Bilal and Ahmad [40] applied three methods such as generalized Kudryashov, modified direct algebraic, and G′/G2-expansion function to attain diverse forms of optical solutions of the NCFSSS ((1) and (2)) with σ=0, while the exact solutions of the NCFSSS ((1) and (2)) has not yet been studied.The motivations of this work are to obtain the exact fractional stochastic solutions of NCFSSS ((1) and (2)). This is the first investigation to acquire the exact solutions of NCFSSS ((1) and (2)) in the presence of stochastic term and fractional-space derivatives. To accomplish a wide variety of solutions, such as trigonometric, hyperbolic, elliptic, and rational functions, we apply two different methods such as the Jacobi elliptic function and the sine-cosine methods. Also, we study the effect of BM on the obtained solutions of NCFSSS ((1) and (2)) by using MATLAB to create 3D and 2D diagrams for some of the obtained solutions here.The document is laid out as follows: we define and state some features of the CD and BM in Section2. We employ an appropriate wave transformation in Section 3 to derive the wave equation of NCFSSS ((1) and (2)). While in Section 4, we utilize two methods to create the analytic solutions of the NCFSSS ((1) and (2)). In Section 5, the influence of the BM on the obtained solutions is investigated. The conclusion of the document is displayed last.


## 2. Preliminaries

Here, we define and state some features of the CD and BM.Definition 1 (see [31]).
Letϕ:0,∞⟶ℝ, then the CD of ϕ of order α∈0,1 is defined as
(3)Dxαϕx=limκ⟶0ϕx+κx1−α−ϕxκ.Theorem 2.
Letϕ,H:0,∞⟶ℝ be differentiable and also α be differentiable functions, then
(4)Dxαϕ∘Hx=x1−αH′xϕ′Hx.
Let us state some properties of the CD. Ifa and b are constant, then
(1)
Dxαaϕx+bHx=aDxαϕx+bDxαHx(2)
Dxαa=0(3)
Dxαxb=bxb−α(4)
DxαHx=x1−αdH/dxIn next definition, we define Brownian motionBt.Definition 3.
Stochastic processBtt≥0 is said a Brownian motion if it satisfies:
(1)
B0=0(2)
Bt is continuous function of t≥0(3)
Bt2−Bt1 is independent for t1<t2(4)
Bt2−Bt1 has a normal distribution ℵ0,t2−t1


## 3. Wave Equation for NCFSSS

The next wave transformation is used to get the wave equation of the NCFSSS ((1) and (2)):
(5)ux,y,t=φζeiℏ−σBt−1/2σ2t,vx,y,t=ψζe−σBt−1/2σ2t,with
(6)ζ=ζ1αxα+ζ2αyα−ζ3t,h=h1αxα+h2αyα−h3t,where φ and ψ are deterministic functions and ζk and ℏk for k=1,2,3, are nonzero constants. Plugging Equation (5) into Equations (1) and (2) and using
(7)du=−ζ3φ′+iℏ3φdt−σφdBeiℏ−σBt−1/2σ2t,Dxyαu=ζ1ζ2φ′′+iℏ2ζ1+ℏ1ζ2φ′−ℏ1ℏ2φeiℏ−σBt−1/2σ2t,Dyαu2=2ζ2φφ′e−σBt−1/2σ2t,Dxαv=ζ1ψ′e−σBt−1/2σ2t,we get for imaginary part
(8)ζ3=ℏ2ζ1+ℏ1ζ2,and for real part
(9)ζ1ζ2φ′′−ℏ3+γ1ℏ1ℏ2φ+γ2φψe−σBt−1/2σ2t=0,(10)γ3ζ1ψ′+2γ4ζ2φφ′e−σBt−1/2σ2t=0.Taking expectationE· on both sides for Equations (9) and (10), we have
(11)ζ1ζ2φ′′−ℏ3+γ1ℏ1ℏ2φ+γ2φψe−1/2σ2tEe−σBt=0,(12)γ3ζ1ψ′+2γ4ζ2φφ′e−1/2σ2tEe−σBt=0.SinceBt is standard Gaussian process, hence Ee−σBt=eσ2/2t. Now Equations (11) and (12) have the form
(13)ζ1ζ2φ′′−ℏ3+γ1ℏ1ℏ2φ+γ2φψ=0,(14)γ3ζ1ψ′+2γ4ζ2φφ′=0.Integrating Equation (14) once and setting the integral constant equal zero yields
(15)ψ=−γ4ζ2γ3ζ1φ2.Plugging Equation (15) into Equation (13), we get the following wave equation
(16)φ′′−Λ1φ3−Λ2φ=0,where
(17)Λ1=γ2γ4γ3ζ12,Λ2=ℏ3+γ1ℏ1ℏ2ζ1ζ2.


## 4. The Exact Solutions of the NCFSSS

To find the exact solutions of Equation (16), we use two different methods such as sine-cosine [13, 14] and the Jacobi elliptic function [29] methods. As a result, we are able to obtain the exact solutions of the NCFSSS ((1) and (2)).


### 4.1. Sine-Cosine Method

Assume the solutionφ of Equation (16) has the form
(18)φζ=AYn,where
(19)Y=cosBζorY=sinBζ.Setting Equation (18) into Equation (16) we get
(20)−AB2−n2Yn+nn−1Yn−2−Λ1A3Y3n−Λ2AYn=0,rewriting the above equation
(21)Λ2A−AB2n2Yn+nn−1AB2Yn−2+Λ1A3Y3n=0.Equalizing the term ofY in Equation (21), we attain
(22)n−2=3n⇒n=−1.Substituting Equation (22) into Equation (21)
(23)Λ2A−AB2Y−1+Λ1A3+2AB2Y−3=0.Equating each coefficient ofY−3 and Y−1 to zero, we have
(24)Λ2A−AB2=0,Λ1A3+2AB2=0.By solving these equations, we get(25)B=Λ2,A=−2Λ2Λ1.Hence, the solution of Equation (16) is
(26)φζ=AsecBζorφζ=AcscBζ.There are many cases depending on the sign ofΛ1 and Λ2.Case 1.
IfΛ2>0 and Λ1<0, then the exact solutions of the NCFSSS ((1) and (2)) are
(27)ux,y,t=−2Λ2Λ1secΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sec2Λ2ζe−σBt−1/2σ2t,or
(28)ux,y,t=−2Λ2Λ1cscΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1csc2Λ2ζe−σBt−1/2σ2t.Case 2.
IfΛ2<0 and Λ1<0, then the exact solutions of the NCFSSS ((1) and (2)) are
(29)ux,y,t=i2Λ2Λ1sech−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sech2−Λ2ζe−σBt−1/2σ2t,or
(30)ux,y,t=2Λ2Λ1csch−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=−2Λ2γ4ζ2Λ1γ3ζ1csch2−Λ2ζe−σBt−1/2σ2t.Case 3.
IfΛ2<0 and Λ1>0, then the exact solutions of the NCFSSS ((1) and (2)) are
(31)ux,y,t=−2Λ2Λ1sech−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sech2−Λ2ζe−σBt−1/2σ2t,or
(32)ux,y,t=−i−2Λ2Λ1csch−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=−2Λ2γ4ζ2Λ1γ3ζ1csch2−Λ2ζe−σBt−1/2σ2t.Case 4.
IfΛ2>0 and Λ1>0, then the exact solution of the NCFSSS ((1) and (2)) are
(33)ux,y,t=i2Λ2Λ1secΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sec2Λ2ζe−σBt−1/2σ2t,or
(34)ux,y,t=i2Λ2Λ1cscΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1csc2Λ2ζe−σBt−1/2σ2t,where Λ1 and Λ2 are defined in (17) and ζ=ζ1/αxα+ζ2/αyα−ζ3t and h=h1/αxα+h2/αyα+h3t.


### 4.2. The Jacobi Elliptic Function Method

We suppose that the solution to Equation (16) has the type
(35)φζ=a+bsnρζ,where a,b, and ρ are undefined constants and snρζ=snρζ,m is the Jacobi elliptic sine function (Latin: sinus amplitudinis) for 0<m<1. Differentiate Equation (35) two times, we have
(36)φ′′ζ=−m2+1bρ2snρζ+2m2bρ2sn3ρζ.Plugging Equations (35) and (36) into Equation (16), we attain
(37)2m2bρ2−Λ1b3sn3ρζ−3Λ1ab2sn2ρζ−m2+1bρ2+3Λ1a2b+Λ2bsnρζ−Λ1a3+aΛ2=0.Putting each coefficient ofsnρζn to be zero for n=0,1,2,3, we have
(38)Λ1a3+aΛ2=0,m2+1bρ2+3Λ1a2b+Λ2b=0,3Λ1ab2sn2=0,2m2bρ2−Λ1b3=0.When we solve the previous equations, we get(39)a=0,b=±−2m2Λ2m2+1Λ1,ρ2=−Λ2m2+1.As a result, using (35), the solution of Equation (16) is
(40)φζ=±−2m2Λ2m2+1Λ1sn−Λ2m2+1ζ.Therefore, the exact solution of the NCFSSS ((1) and (2)) is
(41)ux,y,t=±−2m2Λ2m2+1Λ1sn−Λ2m2+1ζeiℏ−σBt−1/2σ2t,(42)vx,y,t=γ4ζ2m2Λ2m2+1γ3ζ1Λ1sn2−Λ2m2+1ζe−σBt−1/2σ2t,for Λ2<0 and Λ1>0. If m⟶1, then the solutions (41) and (42) turn to:
(43)ux,y,t=±−Λ2Λ1tanh−Λ22ζeiℏ−σBt−1/2σ2t,vx,y,t=γ4ζ2Λ22γ3ζ1Λ1tanh2−Λ22ζe−σBt−1/2σ2t.In the same way, we can substitutesn in (35) with cnξ=cnξ,m (wherecnis the elliptic cosine (Latin: cosinus amplitudinis)) and dnξ,m=dnξ,m (where dn is the delta amplitude (Latin: delta amplitudinis)) to derive the solutions of Equation (16) as follows:
(44)φζ=±−2m2Λ22m2−1Λ1cn−Λ22m2−1ζ,φζ=±2m2Λ22−m2Λ1dn−Λ22−m2ζ.Therefore, the solutions of the NCFSSS ((1) and (2)) are as follows:
(45)ux,y,t=±−2m2Λ22m2−1Λ1cn−Λ22m2−1ζeiℏ−σBt−1/2σ2t,(46)vx,y,t=2γ4ζ2m2Λ2γ3ζ1Λ12m2−1cn2−Λ22m2−1ζe−σBt−1/2σ2t,for Λ2/2m2−1<0, Λ1>0, and
(47)ux,y,t=±−2m2Λ22m2−1Λ1dn−Λ22m2−1ζeiℏ−σBt−1/2σ2t,(48)vx,y,t=2γ4ζ2m2Λ2γ3ζ1Λ12−m2dn2−Λ22−m2ζe−σBt−1/2σ2t,for Λ2<0 and Λ1>0, respectively. If m⟶1, then the solutions (45) and (46) and (47) and (48) turn to:
(49)ux,y,t=±−2Λ2Λ1sech−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=k2m2Λ2sech2−Λ2ζe−σBt−1/2σ2t,for Λ2<0 and Λ1>0.


## 4.1. Sine-Cosine Method

Assume the solutionφ of Equation (16) has the form
(18)φζ=AYn,where
(19)Y=cosBζorY=sinBζ.Setting Equation (18) into Equation (16) we get
(20)−AB2−n2Yn+nn−1Yn−2−Λ1A3Y3n−Λ2AYn=0,rewriting the above equation
(21)Λ2A−AB2n2Yn+nn−1AB2Yn−2+Λ1A3Y3n=0.Equalizing the term ofY in Equation (21), we attain
(22)n−2=3n⇒n=−1.Substituting Equation (22) into Equation (21)
(23)Λ2A−AB2Y−1+Λ1A3+2AB2Y−3=0.Equating each coefficient ofY−3 and Y−1 to zero, we have
(24)Λ2A−AB2=0,Λ1A3+2AB2=0.By solving these equations, we get(25)B=Λ2,A=−2Λ2Λ1.Hence, the solution of Equation (16) is
(26)φζ=AsecBζorφζ=AcscBζ.There are many cases depending on the sign ofΛ1 and Λ2.Case 1.
IfΛ2>0 and Λ1<0, then the exact solutions of the NCFSSS ((1) and (2)) are
(27)ux,y,t=−2Λ2Λ1secΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sec2Λ2ζe−σBt−1/2σ2t,or
(28)ux,y,t=−2Λ2Λ1cscΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1csc2Λ2ζe−σBt−1/2σ2t.Case 2.
IfΛ2<0 and Λ1<0, then the exact solutions of the NCFSSS ((1) and (2)) are
(29)ux,y,t=i2Λ2Λ1sech−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sech2−Λ2ζe−σBt−1/2σ2t,or
(30)ux,y,t=2Λ2Λ1csch−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=−2Λ2γ4ζ2Λ1γ3ζ1csch2−Λ2ζe−σBt−1/2σ2t.Case 3.
IfΛ2<0 and Λ1>0, then the exact solutions of the NCFSSS ((1) and (2)) are
(31)ux,y,t=−2Λ2Λ1sech−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sech2−Λ2ζe−σBt−1/2σ2t,or
(32)ux,y,t=−i−2Λ2Λ1csch−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=−2Λ2γ4ζ2Λ1γ3ζ1csch2−Λ2ζe−σBt−1/2σ2t.Case 4.
IfΛ2>0 and Λ1>0, then the exact solution of the NCFSSS ((1) and (2)) are
(33)ux,y,t=i2Λ2Λ1secΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1sec2Λ2ζe−σBt−1/2σ2t,or
(34)ux,y,t=i2Λ2Λ1cscΛ2ζeiℏ−σBt−1/2σ2t,vx,y,t=2Λ2γ4ζ2Λ1γ3ζ1csc2Λ2ζe−σBt−1/2σ2t,where Λ1 and Λ2 are defined in (17) and ζ=ζ1/αxα+ζ2/αyα−ζ3t and h=h1/αxα+h2/αyα+h3t.


## 4.2. The Jacobi Elliptic Function Method

We suppose that the solution to Equation (16) has the type
(35)φζ=a+bsnρζ,where a,b, and ρ are undefined constants and snρζ=snρζ,m is the Jacobi elliptic sine function (Latin: sinus amplitudinis) for 0<m<1. Differentiate Equation (35) two times, we have
(36)φ′′ζ=−m2+1bρ2snρζ+2m2bρ2sn3ρζ.Plugging Equations (35) and (36) into Equation (16), we attain
(37)2m2bρ2−Λ1b3sn3ρζ−3Λ1ab2sn2ρζ−m2+1bρ2+3Λ1a2b+Λ2bsnρζ−Λ1a3+aΛ2=0.Putting each coefficient ofsnρζn to be zero for n=0,1,2,3, we have
(38)Λ1a3+aΛ2=0,m2+1bρ2+3Λ1a2b+Λ2b=0,3Λ1ab2sn2=0,2m2bρ2−Λ1b3=0.When we solve the previous equations, we get(39)a=0,b=±−2m2Λ2m2+1Λ1,ρ2=−Λ2m2+1.As a result, using (35), the solution of Equation (16) is
(40)φζ=±−2m2Λ2m2+1Λ1sn−Λ2m2+1ζ.Therefore, the exact solution of the NCFSSS ((1) and (2)) is
(41)ux,y,t=±−2m2Λ2m2+1Λ1sn−Λ2m2+1ζeiℏ−σBt−1/2σ2t,(42)vx,y,t=γ4ζ2m2Λ2m2+1γ3ζ1Λ1sn2−Λ2m2+1ζe−σBt−1/2σ2t,for Λ2<0 and Λ1>0. If m⟶1, then the solutions (41) and (42) turn to:
(43)ux,y,t=±−Λ2Λ1tanh−Λ22ζeiℏ−σBt−1/2σ2t,vx,y,t=γ4ζ2Λ22γ3ζ1Λ1tanh2−Λ22ζe−σBt−1/2σ2t.In the same way, we can substitutesn in (35) with cnξ=cnξ,m (wherecnis the elliptic cosine (Latin: cosinus amplitudinis)) and dnξ,m=dnξ,m (where dn is the delta amplitude (Latin: delta amplitudinis)) to derive the solutions of Equation (16) as follows:
(44)φζ=±−2m2Λ22m2−1Λ1cn−Λ22m2−1ζ,φζ=±2m2Λ22−m2Λ1dn−Λ22−m2ζ.Therefore, the solutions of the NCFSSS ((1) and (2)) are as follows:
(45)ux,y,t=±−2m2Λ22m2−1Λ1cn−Λ22m2−1ζeiℏ−σBt−1/2σ2t,(46)vx,y,t=2γ4ζ2m2Λ2γ3ζ1Λ12m2−1cn2−Λ22m2−1ζe−σBt−1/2σ2t,for Λ2/2m2−1<0, Λ1>0, and
(47)ux,y,t=±−2m2Λ22m2−1Λ1dn−Λ22m2−1ζeiℏ−σBt−1/2σ2t,(48)vx,y,t=2γ4ζ2m2Λ2γ3ζ1Λ12−m2dn2−Λ22−m2ζe−σBt−1/2σ2t,for Λ2<0 and Λ1>0, respectively. If m⟶1, then the solutions (45) and (46) and (47) and (48) turn to:
(49)ux,y,t=±−2Λ2Λ1sech−Λ2ζeiℏ−σBt−1/2σ2t,vx,y,t=k2m2Λ2sech2−Λ2ζe−σBt−1/2σ2t,for Λ2<0 and Λ1>0.


## 5. The Effect of BM on NCFSSS Solutions

The effect of BM on the exact solutions of the NCFSSS ((1) and (2)) is discussed here. Fix the parameters γ1=γ2=−1,γ3=1,γ4=−2,ℏ1=ℏ2=ζ1=ζ2=1,ℏ3=−1, and m=0.5. Hence, ζ3=−2,Λ1=2, and Λ2=−2. Now, for various values of α (the fractional derivative order) and σ (noise intensity), we provide a number of graphs for t∈0,5 and x∈0,6. To draw these graphs, we use the MATLAB tools. In the following Figure 1, if σ=0, we can observe how the surface fluctuates as the value of α changes:Figure 1
3D diagrams of Equation (41).While in Figures2 and 3, we can observe that after small transit patterns, the surface smooths significantly when noise is incorporated, and its intensity increases σ=1,2 for different value of α.Figure 2
3D diagrams of Equation (41) with α=1.Figure 3
3D diagrams of Equation (41) with α=0.5.Figure4 shows 2D graphs of Equation (41) with σ=0,0.5,1,2 and with α=1, which highlight the above results.Figure 4
2D diagrams of Equation (41).We may infer from Figures1–4 the following:
(1)
As fractional-orderα decreases, the surface shrinks(2)
The solutions of NCFSSS are stabilized by BM around zero


## 6. Conclusions

In this paper, we considered the (2+1)-dimensional nonlinear conformable fractional stochastic Schrödinger system ((1) and (2)) which has never been examined before with stochastic term and fractional space at the same time. We employed two different methods such as the sine-cosine and the Jacobi elliptic function methods to get elliptic, trigonometric, rational, and hyperbolic fractional stochastic solutions. These obtained solutions are useful in describing some of interesting physical phenomena due to the importance of the NCFSSS in plasma physics and fluid dynamics. Finally, the effect of BM on the exact solution of the NCFSSS ((1) and (2)) is demonstrated by introducing 3D and 2D graphs for some analytical fractional stochastic solutions. In future study, we can address the NCFSSS ((1) and (2)) with multidimensional multiplicative noise.



---

*Source: 6306220-2022-06-20.xml*