On exact special solutions for the stochastic regularized long wave-Burgers equation

In this paper, we will analyze the Regularized Long Wave-Burgers equation with conformable derivative (cd). Some white noise functional solutions for this equation are obtained by using white noise analysis, Hermite transforms, and the modified sub-equation method. These solutions include exact stochastic trigonometric functions, hyperbolic functions solutions and wave solutions. This study emphasizes that the modified fractional sub-equation method is sufficient to solve the stochastic nonlinear equations in mathematical physics.


Introduction
Recently, fractional calculus gained considerable interests and significant theoretical developments in many fields and many studies have been achieved in this field [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. Due to the fact that the stochastic models are more realistic than the deterministic models, we concentrate our study in this paper on the Wick-type stochastic time-fractional Regularized Long Wave-Burgers equation (RLWBE) with conformable derivative (cd). A lot of research on stochastic fractional differential equations has been done recently [15][16][17][18]. Ghany and Hyder [15] obtained analytical solutions to stochastic time-fractional KdV equations of the Wick-type, Ghany and Zakarya [16] obtained exact traveling wave solutions to a stochastic Schamel KdV equation of Wick-type, in [17] is analyzed a stochastic fractional KdV equation with cd, in [18] is used a white noise functional approach for the fractional coupled KdV equations and are obtained new soliton solutions. In this paper, we will analyze the time-fractional RLWBE.
The cd operator was exposed in [25]. This derivative operator can reform the failures of the other definitions. This important operator is the easiest, most natural and effectual definition of the fractional derivative for order η ∈ (0, 1). We should note that the definition can be generalized to involve any η. All the same, the order η ∈ (0, 1) is the most influential order.
We say that the conformable fractional differentiability of a function f : [0, ∞) → R is nothing else than the classical differentiability. Clearly, the conformable η-derivative of f at some point x > 0, where 0 < η < 1 is the pointwise product x 1-η f (x) [26].
The cd of order η ∈ (0, 1) is described by the following statement [25]: The definition represents a natural formation of normal derivatives. Furthermore, the expression of the definition represents that it is the most natural, and the most effectual definition. The definition for 0 ≤ η < 1 gives the classical expressions on polynomials.
The stochastic model of Eq. (1.1) in the Wick sense with conformable derivatives can be given in the following process: where "♦" is the Wick product on the Kondratiev distribution space (S) -1 , ε(τ ) and λ(τ ) are (S) -1 -valued functions [18]. In order to obtain the exact solutions of the random RLWBE with conformable derivative, we only consider it in a white noise environment, that is, we will discuss the Wick-type stochastic RLWBE (1.2).
Our aim in this work is to obtain a new stochastic soliton and periodic wave solutions of the Wick-type stochastic RLWBE with the aid of cd. We use the modified sub-equation method [32,33], white noise theory, and Hermite transform to produce a new set of exact soliton and periodic wave solutions for the RLWBE with cd. Moreover, we apply the inverse Hermite transform to obtain stochastic soliton and periodic wave solutions of the Wicktype stochastic RLWBE with the aid of cd. Finally, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions.

Exact solutions of Eq. (1.1)
In this part, we will investigate exact solutions of RLWBE. Using the Hermite transform of Eq. (1.1), we use the deterministic equation where k, are arbitrary constants and θ is a nonzero function to be determined. Hence, Eq. (2.1) can be converted to the following NODE: • Considering the solution of Eq. (2.2), we can write it as a series expansion solution as follows: where α i (i = 0, 1, . . . , n), β i (i = 1, 2, . . . , n) are functions to be determined later and G(ξ ) satisfies the Riccati equation as follows: where σ is an arbitrary constants. • N is obtained with the aid of a balance between the highest order derivatives and the nonlinear terms in Eq. (2.2). A few special solutions of Eq. (2.4) are given by [34]: (2) When σ > 0, (3) When σ = 0, ρ = const., Remark. The generalized trigonometric and hyperbolic functions are defined as [2] tan is the Mittag-Leffler function. By balancing p dp dξ with d 3 p dξ 3 in Eq. (2.2), it is found that N = 2. Then we can choose the solution of Eq. (2.2) to be given by where G(ξ ) satisfies Eq. (2.4). Now, replacing (2.7) and (2.4) into (2.2), by equating all coefficients of G(ξ ), we can solve the equations. Then we obtain the following groups of solutions.

White noise functional solutions of Eq. (1.2)
In this section, we apply the inverse Hermite transform and Theorem 4. (I) Exact stochastic hyperbolic solutions: (II) Exact stochastic trigonometric solutions: (III) Exact stochastic wave solutions: (3.5)

Example
In this section, we investigate a special application example to represent the availability of our results and to confirm the real assistance of these results. We explain that the solutions of Eq. (1.2) are strongly dependent on the form of the given functions ε(τ ) and λ(τ ). So, for dissimilar forms of ε(τ ) and λ(τ ), we can find dissimilar solutions of Eq. (1.2) which come from Eqs. (3.1)-(3.5). We illustrate this by giving the following example. When η = 1, Suppose λ(τ ) = ∂ ε(τ ) and ε(τ ) = f (τ ) + ρW τ , where ∂ and ρ are arbitrary constants, f (τ ) is a limited mensurable function on R + and W τ is the Gaussian white noise which is the time derivative (in the strong sense in (S) -1 ) of the Brownian motion B τ . The Hermite transform of W τ is given by [36]. Using the definition of W τ (z), Eqs. (3.1)-(3.5) yield the white noise functional solution of Eq. (1.1) as follows: