White noise theory and general improved Kudryashov method for stochastic nonlinear evolution equations with conformable derivatives

The aim of this work is to investigate the Wick-type stochastic nonlinear evolution equations with conformable derivatives. The general Kudryashov method is improved by a new auxiliary equation. So, a new technique, which we call “the general improved Kudryashov method (GIKM)”, is introduced to produce exact solutions for the nonlinear evolution equations with conformable derivatives. By means of GIKM, white noise theory, Hermite transform, and computerized symbolic computation, a novel technique is presented to solve the Wick-type stochastic nonlinear evolution equations with conformable derivatives. This technique is applied to construct exact traveling wave solutions for Wick-type stochastic combined KdV–mKdV equation with conformable derivatives. Moreover, numerical simulations with 3D profiles are shown for the obtained results.


Introduction
Nonlinear evolution equations have a significant job in applied sciences, especially in physics. Obtaining traveling wave solutions of these equations has been of major benefit primarily within the context of mathematical physics. Such examinations have prompted many intriguing sorts of solutions in the past, for example, the soliton solutions, the periodic solutions, the cnoidal solutions, the peakon solutions. In any case, searching these solutions has not been simple at all as is showed in the literature. So, many powerful manners have been introduced, such as the homogeneous balance manner [1], the first integral manner [2], the tanh-coth manner [3], the modified tanh-coth manner [4], the inverse scattering manner [5], Hirota's bilinear manner [6], the RB sub-Ode manner [7,8], the sine-Gordon manner [9], the (G /G)-expansion manner [10], the (G /G, 1/G)-expansion manner [11], the Exp-function manner [12], F-expansion manner [13,14], and so on.
There are abundant and full treatises related to the fractional and conformable derivatives. Conformable fractional formulation of the fractional calculus was introduced in [15]. The conformable calculus of time-scale was evidenced in [16]. In [17], the fractional traditional mechanics was discussed by some conformable-type derivatives. Lately, the conformable-type differential equations have become a significant object in physics and mathematics. So, abundant experts focus their attention on the analytical and the approximate integrals to these equations [18,19]. Existence and uniqueness results for some conformable-type partial differential equations (PDEs) have been proved by Gokdogan et al. [20] and by Sania et al. [21]. In [22], a conformable sub-equation manner was suggested to create exact solutions to the space and time fractional nonlinear resonant Schrödinger equation. Also, fractional modulation to the Nipah virus was given by Markovian process and some local time differential maps [23]. Overall, many studies have been done about the solutions and properties of fractional and conformable-type PDEs [24][25][26][27][28][29].
Many researchers have been interested in the subject of random traveling wave, it is a very important topic in the field of stochastic partial differential equations (SPDEs). The stochastic KdV equation was proposed by Wadati [30] in 1983. He studied the diffusion of soliton of the equation due to KdV under the Gaussian noise effect. The stochastic traveling wave solutions for the local fractal KdV equation have been obtained by the Expfunction technique in [31] and [32], respectively. Moreover, on account of [14,[33][34][35][36][37][38][39], many kinds of Wick-type stochastic and fractional evolution equations have been studied by utilizing diverse expansion techniques and white noise analysis.
Consider a nonlinear PDE (NPDE) where (χ, α) ∈ R × R + is the freelance variable and u(χ, α) is its follower variable. Applying the one-variable transformation we change (1) to an ordinary and nonlinear differential equation (NODE) where := d dκ . In [40], Kudryashov proposed his manner to find analytical solutions to Eq. (1). He researched for the exact solutions taking into account the expression 1+e κ , which is the integral to the equation dX dκ = X 2 -X. A modified Kudryashov manner was presented by exchanging the ordinary exponential function e κ by means of the general sort of the exponential function a κ in [41][42][43][44]. In these contributions, experts got the exact solutions to the NPDE (1) by using the expansion u(κ) = x i=0 μ i X i , where X = 1 1±a κ , which is the integral to the equation dX dκ = ln a(X 2 -X). Thereafter, some authors [45][46][47][48] applied a general sort of the Kudryashov manner to rummage exact solutions of the NPDE (1). They have selected a rational expansion 1+Ce κ , which is the integral to the equation dX dκ = X 2 -X. Lately, Abdus Salam and Habiba [49] improved the general Kudryashov manner given in [45] by electing the auxiliary equation dX dκ = σ X 3 -X, 0 = σ ∈ R. This helpful equation has the comprehensive solution X = ±1 √ σ +Ce 2κ .
In this work, the general Kudryashov method [45] is improved by the novel auxiliary equation which has numerous general solutions depending on the natural number n (see Eq. (25)). Thus, a novel technique to build exact solutions for nonlinear evolution equations is obtained. This technique is called the GIKM. The major feature of the GIKM over the others lies in the way that it utilizes an especially clear and powerful algorithm to obtain exact solutions for large families of nonlinear evolution equations. Also, a large set of exact solutions can be determined effectively on picking the parameters that showed up. Besides, the proposed GIKM generalizes some previous techniques. It depends on improving the general Kudryashov technique by the general auxiliary equation (4), which has various general solutions. Moreover, we apply the GIKM and white noise topics to construct exact solutions for the Wick-type stochastic combined KdV-mKdV equation with conformable derivatives. Also, numerical simulations with 3D profiles are provided to the obtained exact solutions. The remnant of this work is structured as follows: Sect. 2 contains the needed topics about the conformable calculus and the Gaussian analysis of white noise. In Sect. 3, the GIKM is demonstrated. In Sect. 4, we apply the GIKM, jointly with the Gaussian analysis of white noise, to investigate the Wick-type stochastic combined KdV-mKdV equation with conformable derivatives. Section 5 gives discussions and numerical simulations for the obtained results. Section 6 presents a conclusion.

Basilar topics of white noise discipline
The Gaussian white noise discipline begins with the rigging D( Depending on the Bochner-Minlos theorem [52], we have a lonesome white noise measure μ w on (D * (R N ), β(D * (R N ))). Presume that ζ n (x) = π -1/4 ((n -1)!) -1/2 e -x 2 /2 h n-1 ( √ 2x), n ∈≥ 1 are the Hermite functions, where h n (x) denotes the Hermite polynomials. It is well known that the gathering (ζ n ) n∈N fabricates an orthonormal basis for L 2 (R). Let m = (m 1 , . . . , m N ) be N -dimensional multi-indices with m 1 , . . . , m N ∈ N, then the tensor multiplications ζ m := ζ (m 1 ,...,m N ) = χ m 1 ⊗ · · · ⊗ χ m N , m ∈ N N fabricate an orthonormal basis to L 2 (R N ). Introduce an ordering in N N by Using the above ordering, we Let n ∈ N, the Kondrative space of test stochastic functions (D) n 1 is defined by and the Kondrative space of distribution stochastic functions (D) n -1 is defined by The Wick product of two with a m , bm ∈ R n is known as where for all z such that F(z) and G(z) exist. The relation "•" indicates the bilinear multiplication in C n , which is known as (z 1 1 , . . . , Thus, the Hermite transform changes the Wick multiplication into the classical multiplication and convergence in (D) n -1 into bounded and pointwise convergence in a certain neighborhood of the origin in C n . For more specifics about Kondrative spaces, Hermite transformation, and Wick multiplication, we refer to [52].
In the following, the distribution stochastic process (or (D) n -1 -process) is a measurable map u from R N into (D) n -1 . Furthermore, if the (D) n -1 -valued function u is continuous, differentiable, C k , etc., then the (D) n -1 -process u has the same features, respectively. Now, for [52]. To study the stochastic conformable PDEs, we require the following facts.
. Then X(α, ϑ) has a -order conformable derivative for each α ∈ (s, t) and and X(α, z) is a continuous function for α ∈ [s, t] ∀z ∈ O π (ρ). Then the -order conformable integral operator of X(α) exists and is a solution (in the usual strong and pointwise sense) of the equation Moreover, suppose that u(χ, α, z) and all its conformable derivatives, which are implicated in Eq.
The integer numbers x and y can be appointed by balancing the highest order linear and nonlinear terms in Eq. (22). By inserting Eqs. (23) and (24) into Eq. (22), we get an algebraic-form equation in X and its powers. Placing the coefficients of all terms that include the similar power for X to be zero, gives a system of algebraic-form equations in μ i , ν j , and θ . By employing the symbolic system Mathematica, we can determine μ i , ν j , and θ . Lastly, by utilizing these values and Eq. (25), we can construct some exact and traveling wave solutions to Eq. (20).

Application to Wick-type stochastic combined KdV-mKdV equation with conformable derivatives
In this section, GIKM for n = 5, white noise theory, Hermite transform, and computerized symbolic computation are applied to find exact traveling wave solutions of Wicktype stochastic combined KdV-mKdV with conformable derivatives. The KdV and mKdV equations are solitary equations, which have been widely researched. For these equations, the nonlinear terms usually arise in abundant physical issues, like flow mechanics, quantum fields, and plasma physics. This section is devoted to constructing exact traveling wave solutions of Wick-type stochastic combined KdV-mKdV equation with conformable derivatives where (χ, α) ∈ R × R + and 0 < ≤ 1, while and Λ are real and integrable nonzero functions with values in (D) -1 . Equation (26) is the perturbation of the variable coefficients combined KdV-mKdV equation with conformable derivatives where δ, λ are nonzero integrable functions on R + . Moreover, if Eq. (27) is considered in some random ambience, we have a random combined KdV-mKdV equation. To construct exact solutions of the random combined KdV-mKdV equation, we only examine it in a white noise ambience, thus, we will investigate the Wick-type stochastic combined KdV-mKdV equation (26). By using Hermite transform and Eq. (26), we get a conformable deterministic equation where z = (z 1 , z 2 , . . .) ∈ (C N ) c . To construct traveling wave solutions to Eq. (28), we employ the transformations (α, z) = δ(α, z), where ω is a free constant and θ is a nonzero function to be specified. Hence, Eq. (28) can be transformed to the following NODE Integrating the NODE (30) and placing the integration constants to be zero give Considering the homogeneous balance for d 2 u dκ 2 and u 3 , we get xy -4 = 0. Let y = 1, then x = 5. So, we can set the wave solution of Eq. (31) as the form Substituting Eqs. (32) and (24) for n = 5 into Eq. (31) gives an algebraic-form equation in X and its powers. Equating the coefficients of the terms that contain the same power for X to zero gives a system of algebraic-form equations in μ i , ν j (i = 0, . . . , 5, j = 0, 1) and θ (see the Appendix). By treating this system via Mathematica, we obtain the following sets of values.
Case I.
Case II.
Case III.

Conclusion
In fact, the stochastic physical models are more sensible than the deterministic models. Thus, right now, we focus the investigation on the SPDEs with conformable derivatives. Foremost, the general Kudryashov method [45] is improved by the novel auxiliary equation (4), which has numerous general solutions depending on the natural number n (see  Eq. (25)). Thus, a novel technique to build exact solutions for nonlinear evolution equations is obtained. This technique is called the GIKM. The major feature of the GIKM over the others lies in the way that it utilizes an especially clear and powerful algorithm to obtain exact solutions for a large family of nonlinear evolution equations. Also, a large set of exact solutions can be determined effectively on picking the parameters that showed up. Besides, the proposed GIKM generalizes some previous techniques. It depends on im-proving the general Kudryashov technique by the general auxiliary equation (4) which has various general solutions. Moreover, we apply the GIKM and white noise topics to construct exact solutions for the Wick-type stochastic combined KdV-mKdV equation with conformable derivatives. Also, numerical simulations with 3D profiles are provided to the obtained exact solutions. Eventually, the overall approach proposed in this paper can be utilized for solving diverse nonlinear evolution equations in physics and engineering, both deterministic and stochastic types.