# White Noise Functional Solutions for Wick-Type Stochastic Fractional Mixed KdV-mKdV Equation Using ExtendedG′/G-Expansion Method

Authors: Zhao Li; Peng Li; Tianyong Han
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, white noise functional solutions of Wick-type stochastic fractional mixed KdV-mKdV equations have been obtained by using the extendedG′/G-expansion method and the Hermite transform. Firstly, the Hermite transform is used to transform Wick-type stochastic fractional mixed KdV-mKdV equations into deterministic fractional mixed KdV-mKdV equations. Secondly, the exact traveling wave solutions of deterministic fractional mixed KdV-mKdV equations are constructed by applying the extended G′/G-expansion method. Finally, a series of white noise functional solutions are obtained by the inverse Hermite transform.


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## Body

## 1. Introduction

The study of stochastic differential equation (SDE) can be traced back to Einstein’s classical paper in 1905, which proposed the microscopic random motion of particles and macrodiffusion equations. In Einstein’s research, the exact dynamics of the system is quite uncertain and which can be modeled by SDE. In 1949, Itô [1], a Japanese mathematician, first defined a random integral named Itô stochastic integral, which laid the theoretical foundation for the research of SDEs. In fact, the physical quantities of the objective world generally change with time and space, which are usually simulated by partial differential equation. Thus, stochastic partial differential equation (SPDE) [2] is usually used to simulate mathematical problems in the fields of science and engineering.Fractional calculus was proposed before the birth of SPDE. In 1695, Leibniz and L’Hospital have discussed the definition and significance of derivative when the order of derivative is1/2. In recent years, with the combination of fractional calculus theory and SPDE, it is gradually found that fractional SPDE can describe some nonlinear phenomena in the fields of natural science and engineering applications [3, 4]. In recent years, white noise functional solution is a very important topic in the research of fractional SPDEs. Many researchers have proposed many methods to construct the white noise functional solutions of fractional SPDEs, such as the Exp-function method [5], the Kudryashov method [6], improved computational method [7], and computerized symbolic [8]. The biggest obstacle in finding the white noise functional solution of SPDE is that the nonlinear ordinary differential equation obtained by the Hermite transform and random traveling wave transform is a nonlinear ordinary differential equation with variable coefficients. Therefore, many methods of constructing the white noise functional solutions of fractional partial differential equations are not applicable. So, it is particularly important to find a new method to construct the solution of nonlinear differential equation with variable coefficients. As early as 2008, Professor Wang [9] proposed a method named G′/G-expansion method to construct the exact traveling wave solution of partial differential equation. Later, many experts and scholars [10, 11] further expanded the method to enrich the solutions of partial differential equations. In this paper, we intend to find the solutions of fractional SPDES by the extended G′/G-expansion method.In recent years, with the development of fractional derivative, the Wick-type stochastic fractional mixed KdV-mKdV equation (see [5, 6, 8, 12, 13]), a very important class of fractional SPDE, has been widely concerned by many researchers. This model can be described as follows:(1)DtαU+Θ1t⋄U⋄DxαU+Θ2t⋄U⋄2⋄DxαU+Dx3αU=0,where U=Ut,x, t,x∈ℝ+×ℝ, and 0<α≤1. DtαU, DxαU, and Dx3αU are the conformable fractional derivative. Θ1t and Θ2t are integrable white noise functionals from ℝ+ to the Kondrative distribution space D−1. The operator ⋄ represents the Wick product on D−1.Based on Eq. (1), we consider the fractional mixed KdV-mKdV equation which is a very important fractional partial differential equation and usually used to simulate shallow water surface waves phenomena [14–19].(2)Dtαu+θ1tuDxαu+θ2tu2Dxαu+Dx3αu=0,0<α<1,where ut,x stands for the wave profile. θ1t and θ2t are integrable function on ℝ+. The fractional derivatives are considered in the sense of conformable fractional derivatives [20–23].Our work is as follows. In Section2, we review the white noise theory and briefly introduce the extend G′/G-expansion method of Wick-type fractional SPDE. In Section 3, white noise functional solutions of Eq. (1) are constructed by apply Hermite transform, fractional traveling wave transformation, and the extend G′/G-expansion method. In Section 4, we give a summary.


## 2. Preliminaries

### 2.1. White Noise Theory

Assume that the riggingDℝN⊂L2ℝN⊂D∗ℝN, where DℝN represents the Schwartz test functions space. D∗ℝN stands for the tempered distributions space. In Ref. [24], Holden et al. have proved that there is a unique measure of white noise, that is measure μ on D∗ℝN,BD∗ℝN, where BD∗ℝN is the family of all Borel sets in D∗ℝN. The Hermite function enx is defined by enx=e−1/2x2hn2x/πn−1!1/2, where n≥1, hnx, is the Hermite polynomial. Define(3)Hmω=∏i=1∞hmiω,ei,ω∈D∗ℝN,where hmii≥1 represents the Hermite polynomials. eii≥1 denotes the orthonormal basis in L2ℝN. Let D1n be the Kondrative space of stochastic test functions space, and then D−1n stands for the Kondrative space of stochastic distributions. Let F=∑mamHm,G=∑m¯am¯Hm¯∈D−1n, where am,bm¯∈ℝn. The wick product is defined by(4)F⋄G=∑m,m¯am,bm¯Hm+m¯.Then, the Hermite transform ofF is given by(5)HFz=F~z=∑mamzm∈ℂn,where z=z1,z2,⋯,zn∈ℂn, F=∑mamHm∈D−1n. Next, we define the Hermite transform(6)F⋄G~z=F~zG~z,where F~z and G~z exist.Forq<∞, ρ>0, then we define the neighborhoods Nqρ(7)Nqρ=z1,z2,⋯∈ℂN:∑m≠0zα22ℕqα<ρ2.Theorem 1 (see [24]).
Supposeu:D×Nqρ⟶ℝ be a strong solution of the following equation(8)Pt,x,Dtα,Dx1α,Dx2α,⋯,Dxnα,u,z=0,where t,x in open bounded set in D⊂ℝ+×ℝn, z∈Nqr. Then, there exists Ut,x∈D−1n such that ut,x,z=U~t,xz and solves the stochastic equation(9)P⋄t,x,Dtα,Dx1α,Dx2α,⋯,Dxnα,U,ω=0,inD−1n.


### 2.2. ExtendedG′/G-Expansion Method

Now, we consider a wick-type stochastic fractional partial differential equation(10)P⋄t,x,U,Dt⋄αU,Dx⋄αU,Dx⋄2αU,⋯,ω=0,where Dt⋄αU,Dx⋄αU,Dx⋄2αU,⋯ are the conformable fractional derivatives of u in the wick-type sense. Applying the Hermite transform, we can get a fractional partial differential equation as follows(11)P~t,x,U~,DtαU~,DxαU~,Dx2αU~,⋯,z=0,where U~=HU is the Hermite transform of U. z=z1,z2,⋯,zn∈ℂn. Then, we introduced the transformation:(12)U~t,x,z=uξ,ξt,x,z=kxαα+c∫atθη,zη1−αdη,where a>0. k and c are constants. θ is a nonzero function. Next, substituting Eq. (12) into Eq. (11), we can obtain the following of the ordinary differential equation(13)Qu,dudξ,d2udξ2,d3udξ3,⋯=0.Next, we briefly introduce the extendedG′/G-expansion method. Firstly, we assume that the traveling wave solution of Eq. (13) can be described as follows:(14)uξ=∑i=0NaiG′Gi+∑i=1NbiG′G−i,where G=Gξ satisfies(15)GG′′=AGG′+BG′2+CG2,where A, B, and C are real number. Secondly, substituting Eq. (14) into Eq. (13), and balancing the highest order derivative term and the highest order nonlinear term, then, we can determine the positive integer N. Next, substituting Eq. (14) together with (15) into Eq. (13) with the value of N determined in the previous step, we can get the polynomials about G′/G−i (i=0,1,2,⋯) and G′/Gi (i=0,1,2,⋯), then we set all coefficients of the polynomial to zero, and we have a set of algebraic equation. Finally, we solve the algebraic equations, and we can get the values of ai and bi. The readers can refer to [25] for details of this method.


## 2.1. White Noise Theory

Assume that the riggingDℝN⊂L2ℝN⊂D∗ℝN, where DℝN represents the Schwartz test functions space. D∗ℝN stands for the tempered distributions space. In Ref. [24], Holden et al. have proved that there is a unique measure of white noise, that is measure μ on D∗ℝN,BD∗ℝN, where BD∗ℝN is the family of all Borel sets in D∗ℝN. The Hermite function enx is defined by enx=e−1/2x2hn2x/πn−1!1/2, where n≥1, hnx, is the Hermite polynomial. Define(3)Hmω=∏i=1∞hmiω,ei,ω∈D∗ℝN,where hmii≥1 represents the Hermite polynomials. eii≥1 denotes the orthonormal basis in L2ℝN. Let D1n be the Kondrative space of stochastic test functions space, and then D−1n stands for the Kondrative space of stochastic distributions. Let F=∑mamHm,G=∑m¯am¯Hm¯∈D−1n, where am,bm¯∈ℝn. The wick product is defined by(4)F⋄G=∑m,m¯am,bm¯Hm+m¯.Then, the Hermite transform ofF is given by(5)HFz=F~z=∑mamzm∈ℂn,where z=z1,z2,⋯,zn∈ℂn, F=∑mamHm∈D−1n. Next, we define the Hermite transform(6)F⋄G~z=F~zG~z,where F~z and G~z exist.Forq<∞, ρ>0, then we define the neighborhoods Nqρ(7)Nqρ=z1,z2,⋯∈ℂN:∑m≠0zα22ℕqα<ρ2.Theorem 1 (see [24]).
Supposeu:D×Nqρ⟶ℝ be a strong solution of the following equation(8)Pt,x,Dtα,Dx1α,Dx2α,⋯,Dxnα,u,z=0,where t,x in open bounded set in D⊂ℝ+×ℝn, z∈Nqr. Then, there exists Ut,x∈D−1n such that ut,x,z=U~t,xz and solves the stochastic equation(9)P⋄t,x,Dtα,Dx1α,Dx2α,⋯,Dxnα,U,ω=0,inD−1n.


## 2.2. ExtendedG′/G-Expansion Method

Now, we consider a wick-type stochastic fractional partial differential equation(10)P⋄t,x,U,Dt⋄αU,Dx⋄αU,Dx⋄2αU,⋯,ω=0,where Dt⋄αU,Dx⋄αU,Dx⋄2αU,⋯ are the conformable fractional derivatives of u in the wick-type sense. Applying the Hermite transform, we can get a fractional partial differential equation as follows(11)P~t,x,U~,DtαU~,DxαU~,Dx2αU~,⋯,z=0,where U~=HU is the Hermite transform of U. z=z1,z2,⋯,zn∈ℂn. Then, we introduced the transformation:(12)U~t,x,z=uξ,ξt,x,z=kxαα+c∫atθη,zη1−αdη,where a>0. k and c are constants. θ is a nonzero function. Next, substituting Eq. (12) into Eq. (11), we can obtain the following of the ordinary differential equation(13)Qu,dudξ,d2udξ2,d3udξ3,⋯=0.Next, we briefly introduce the extendedG′/G-expansion method. Firstly, we assume that the traveling wave solution of Eq. (13) can be described as follows:(14)uξ=∑i=0NaiG′Gi+∑i=1NbiG′G−i,where G=Gξ satisfies(15)GG′′=AGG′+BG′2+CG2,where A, B, and C are real number. Secondly, substituting Eq. (14) into Eq. (13), and balancing the highest order derivative term and the highest order nonlinear term, then, we can determine the positive integer N. Next, substituting Eq. (14) together with (15) into Eq. (13) with the value of N determined in the previous step, we can get the polynomials about G′/G−i (i=0,1,2,⋯) and G′/Gi (i=0,1,2,⋯), then we set all coefficients of the polynomial to zero, and we have a set of algebraic equation. Finally, we solve the algebraic equations, and we can get the values of ai and bi. The readers can refer to [25] for details of this method.


## 3. Explicit Solutions of System (1)

Applying Hermite transform, Eq. (1) can be transformed into fractional partial differential equation in the sense of conformable fractional derivatives(16)DtαU~t,x,z+Θ~1t,zU~t,x,zDxαU~t,x,z+Θ~2t,zU~2t,x,zDxαU~t,x,z+Dx3αU~t,x,z=0,where z=z1,z2,⋯∈ℂN. Next, we introduce the transformation Θ~1t,z=θ1t,z, Θ~2t,z=θ2t,z, and U~t,x,z=uξt,x,z with(17)ξt,x,z=kxαα+c∫atθη,zη1−αdη,where a>0. k and c are nonzero constants. θ is a nonzero function.Substituting (16) into (3.1), we obtain(18)cθdudξ+kθ1ududξ+kθ2u2dudξ+k3d3udξ3=0.Integrating Eq. (18) with respect to ξ and assuming the integral constant to zero, we obtain(19)cθu+kθ12u2+kθ23u3+k3d2udξ2=0.By making the homogeneous balance betweenu3 and d2u/dξ2 in Eq. (19), we have N=1. Thus, the solution of Eq. (19) can be written as follows:(20)uξ=a0+a1G′G+b1G′G−1,where a0, a1, and b1 are undetermined parameters.Substituting Eq. (20) together with Eq. (15) into Eq. (19), Eq. (19) is transformed into polynomials in G′/G−i(i=1,2,3) and G′/Gi(i=0,1,2,3). Collecting each coefficient of the polynomials yields a set of algebraic equations. By solving the algebraic equations, we can get following.Set 1:a1=±−6k2A−12/θ2, a0=±BA−1−3k2/2A−12θ2−θ1/2θ2, b1=0, and c=−kθ1/4θa0−3k3BCa1/2θa0+k3/2θa0B2+2CA−1.Set 2:a1=0, a0=±BC−3k2/2C2θ2−θ1/2θ2, b1=±−6k2C2/θ2, and c=−kθ1/4θa0−3k3BA−1b1/2θa0+k3/2θa0B2+2CA−1.Substituting the solutions into (20), we have(21)uξ=±BA−1−3k22A−12θ2−θ12θ2±−6k2A−12θ2G′G,uξ=±BC−3k22C2θ2−θ12θ2±−6k2C2θ2G′G−1.


### 3.1. The Solutions of Eq. (19)

Next, we can obtain the solutions of Eq. (19) as follows.Family 1. WhenB2−4A−1C>0 and A≠1, we obtain(22)u1ξ=−θ12θ2±11−A−3k2A−12B2+4C−4AC2θ2H1,(23)u2ξ=±BC−3k22C2θ2−θ12θ2±−6k2C2θ2B2+4C−4AC21−AH1+B21−A−1,where ξt,x,z=kxα/α+c∫atθη,z/η1−αdη, H1=C1sinhB2+4C−4AC/2ξ+C2coshB2+4C−4AC/2ξ/C1coshB2+4C−4AC/2ξ+C2sinhB2+4C−4AC/2ξ. C1 and C2 are constants.Fixed parameters are as follows:A=2, B=4, C=3, C1≠0, C2=0, θ1=−12, θ2=−6, θ=2, k=1, and α=1/2; then, three-dimensional portrait can be drawn in Figure 1.Figure 1
The solutionu1t,x of Equation (22) for differential parameter.Family 2. When4A−1C−B2>0 and A≠1, we have(24)u3ξ=−θ12θ2±11−A−3k2A−124A−1C−B22θ2H2,(25)u4ξ=±BC−3k22C2θ2−θ12θ2±−6k2C2θ24A−1C−B221−AH2+B21−A−1,where ξt,x,z=kxα/α+c∫atθη,z/η1−αdη, H2=−C1sin4AC−4C−B2/2ξ+C2cos4AC−4C−B2/2ξ/C1cos4AC−4C−B2/2ξ+C2sin4AC−4C−B2/2ξ. C1 and C2 are constants.Fixed parameters are as follows:A=2, B=4, C=5, C1≠0, C2=0, θ1=−12, θ2=−6, θ=2, k=1, and α=1/2; then, three-dimensional portrait can be drawn in Figure 2.Figure 2
The solutionu3t,x of Equation (24) for differential parameter.Family 3. WhenB2−4A−1C=0 and A≠1, we obtain(26)u5ξ=−θ12θ2±11−A−6k2A−12θ2·C1C1ξ+C2,(27)u6ξ=±BC−3k22C2θ2−θ12θ2±1−A−6k2C2θ2C1C1ξ+C2+B2−1,where ξt,x,z=kxα/α+c∫atθη,z/η1−αdη. C1 and C2 are constants.Fixed parameters are as follows:A=2, B=4, C=4, C1=1, C2=2, θ1=−12, θ2=−6, θ=2, k=1, and α=1/2; then, three-dimensional portrait can be drawn in Figure 3.Figure 3
The solutionu5t,x of Equation (26) for differential parameter.


### 3.2. White Noise Functional Solutions of Eq. (1)

In order to construct the white noise functional solutions of Eq. (1), we apply the inverse Hermite transform and Theorem 1 to the solutions u1ξ,u2ξ,⋯,u6ξ. Then, we obtain six white noise functional solution.Family 1. WhenB2−4A−1C>0 and A≠1, we obtain(28)U1t,x=−Θ1t2Θ2t±11−A−3k2A−12B2+4C−4AC2Θ2t⋄12⋄H1,U2t,x=±BC−3k22C2Θ2t⋄12−Θ1t2Θ2t±−6k2C2Θ2t⋄12⋄B2+4C−4AC21−AH1+B21−A−1,where H1=C1sinh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη+C2cosh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη/C1cosh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη+C2sinh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη. C1 and C2 are constants.Fixed parameters are as follows:A=2,B=1,C=0,k=1,c=1,θ1t=0,θ2t=−t/sin20.5t; then, three-dimensional portrait and two-dimensional portrait can be drawn in Figure 4.Figure 4
Wave profiles of exact solutionU1t,x with A=2,B=1,C=0,k=1,c=1,θ1t=0,θ2t=−t/sin20.5t.
(a)
Perspective view of the wave(b)
The wave alongtFamily 2. When4A−1C−B2>0 and A≠1, we have(29)U3t,x=−Θ1t2Θ2t±11−A−3k2A−124A−1C−B22Θ2t⋄12⋄H2,U4t,x=±BC−3k22C2Θ2t⋄12−Θ1t2Θ2t±−6k2C2Θ2t⋄12⋄4A−1C−B221−AH2+B21−A⋄−1,where H2=−C1sin⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη+C2cos⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη/C1cos⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη+C2sin⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη. C1 and C2 are constants.Family 3. WhenB2−4A−1C=0 and A≠1, we obtain(30)U5t,x=−Θ1t2Θ2t±11−A−6k2A−12Θ2t⋄12⋄C1C1kxα/α+c∫atθη/η1−αdη+C2,U6t,x=±BC−3k22C2Θ2t⋄12−Θ1t2Θ2t±1−A−6k2C2Θ2t⋄12⋄C1C1kxα/α+c∫atθη/η1−αdη+C2+B2⋄−1,where C1 and C2 are constants.


## 3.1. The Solutions of Eq. (19)

Next, we can obtain the solutions of Eq. (19) as follows.Family 1. WhenB2−4A−1C>0 and A≠1, we obtain(22)u1ξ=−θ12θ2±11−A−3k2A−12B2+4C−4AC2θ2H1,(23)u2ξ=±BC−3k22C2θ2−θ12θ2±−6k2C2θ2B2+4C−4AC21−AH1+B21−A−1,where ξt,x,z=kxα/α+c∫atθη,z/η1−αdη, H1=C1sinhB2+4C−4AC/2ξ+C2coshB2+4C−4AC/2ξ/C1coshB2+4C−4AC/2ξ+C2sinhB2+4C−4AC/2ξ. C1 and C2 are constants.Fixed parameters are as follows:A=2, B=4, C=3, C1≠0, C2=0, θ1=−12, θ2=−6, θ=2, k=1, and α=1/2; then, three-dimensional portrait can be drawn in Figure 1.Figure 1
The solutionu1t,x of Equation (22) for differential parameter.Family 2. When4A−1C−B2>0 and A≠1, we have(24)u3ξ=−θ12θ2±11−A−3k2A−124A−1C−B22θ2H2,(25)u4ξ=±BC−3k22C2θ2−θ12θ2±−6k2C2θ24A−1C−B221−AH2+B21−A−1,where ξt,x,z=kxα/α+c∫atθη,z/η1−αdη, H2=−C1sin4AC−4C−B2/2ξ+C2cos4AC−4C−B2/2ξ/C1cos4AC−4C−B2/2ξ+C2sin4AC−4C−B2/2ξ. C1 and C2 are constants.Fixed parameters are as follows:A=2, B=4, C=5, C1≠0, C2=0, θ1=−12, θ2=−6, θ=2, k=1, and α=1/2; then, three-dimensional portrait can be drawn in Figure 2.Figure 2
The solutionu3t,x of Equation (24) for differential parameter.Family 3. WhenB2−4A−1C=0 and A≠1, we obtain(26)u5ξ=−θ12θ2±11−A−6k2A−12θ2·C1C1ξ+C2,(27)u6ξ=±BC−3k22C2θ2−θ12θ2±1−A−6k2C2θ2C1C1ξ+C2+B2−1,where ξt,x,z=kxα/α+c∫atθη,z/η1−αdη. C1 and C2 are constants.Fixed parameters are as follows:A=2, B=4, C=4, C1=1, C2=2, θ1=−12, θ2=−6, θ=2, k=1, and α=1/2; then, three-dimensional portrait can be drawn in Figure 3.Figure 3
The solutionu5t,x of Equation (26) for differential parameter.


## 3.2. White Noise Functional Solutions of Eq. (1)

In order to construct the white noise functional solutions of Eq. (1), we apply the inverse Hermite transform and Theorem 1 to the solutions u1ξ,u2ξ,⋯,u6ξ. Then, we obtain six white noise functional solution.Family 1. WhenB2−4A−1C>0 and A≠1, we obtain(28)U1t,x=−Θ1t2Θ2t±11−A−3k2A−12B2+4C−4AC2Θ2t⋄12⋄H1,U2t,x=±BC−3k22C2Θ2t⋄12−Θ1t2Θ2t±−6k2C2Θ2t⋄12⋄B2+4C−4AC21−AH1+B21−A−1,where H1=C1sinh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη+C2cosh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη/C1cosh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη+C2sinh⋄B2+4C−4AC/2kxα/α+c∫atθη/η1−αdη. C1 and C2 are constants.Fixed parameters are as follows:A=2,B=1,C=0,k=1,c=1,θ1t=0,θ2t=−t/sin20.5t; then, three-dimensional portrait and two-dimensional portrait can be drawn in Figure 4.Figure 4
Wave profiles of exact solutionU1t,x with A=2,B=1,C=0,k=1,c=1,θ1t=0,θ2t=−t/sin20.5t.
(a)
Perspective view of the wave(b)
The wave alongtFamily 2. When4A−1C−B2>0 and A≠1, we have(29)U3t,x=−Θ1t2Θ2t±11−A−3k2A−124A−1C−B22Θ2t⋄12⋄H2,U4t,x=±BC−3k22C2Θ2t⋄12−Θ1t2Θ2t±−6k2C2Θ2t⋄12⋄4A−1C−B221−AH2+B21−A⋄−1,where H2=−C1sin⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη+C2cos⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη/C1cos⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη+C2sin⋄4AC−4C−B2/2kxα/α+c∫atθη/η1−αdη. C1 and C2 are constants.Family 3. WhenB2−4A−1C=0 and A≠1, we obtain(30)U5t,x=−Θ1t2Θ2t±11−A−6k2A−12Θ2t⋄12⋄C1C1kxα/α+c∫atθη/η1−αdη+C2,U6t,x=±BC−3k22C2Θ2t⋄12−Θ1t2Θ2t±1−A−6k2C2Θ2t⋄12⋄C1C1kxα/α+c∫atθη/η1−αdη+C2+B2⋄−1,where C1 and C2 are constants.


## 4. Conclusion

In this paper, we constructed the white noise functional solutions of Wick-type stochastic fractional mixed KdV-mKdV equation by using the extendedG′/G-expansion method and the Hermite transform. Compared with the existing literature [5, 6, 8, 12, 13], the negative power solutions U2t,x, U4t,x, and U6t,x obtained in the paper are not reported. The method discussed in the paper is not only applicable to Eq. (1), but also can help mathematicians and physicists find the white noise functional solutions of Wick-type fractional SPDEs. In the future, our work will mainly focus on the white noise functional solutions of SPDEs.



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