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Article

Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients

by
Mohammed Zakarya
1,2,*,
Manal Al-Qarni
1 and
Tahani Al-Qahtani
1
1
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia
2
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
*
Author to whom correspondence should be addressed.
Symmetry 2024, 16(12), 1572; https://doi.org/10.3390/sym16121572
Submission received: 17 October 2024 / Revised: 13 November 2024 / Accepted: 20 November 2024 / Published: 24 November 2024

Abstract

:
In this work, we obtain non-Gaussian (NG) stochastic solutions to χ -Wick-type stochastic ( χ -Wk-TS) Burgers’ equations with variable coefficients. An Exp-function method, the connection between white noise theory and hypercomplex systems (HCSs), the χ -Wick product ( χ -Wk-product) and an χ -Hermite transform ( χ -Hr-transform) are proposed. We provide a new set of non-Gaussian solitary wave solutions (NG-SWSs) to Burgers’ equations with variable coefficients. NG white noise functional solutions (NG-WNFSs) to χ -Wk-TS Burgers’ equations with variable coefficients are shown. The symmetry coefficients of partial differential equations and the symmetrical properties of SPDEs are critical in determining the best solution.

1. Introduction

In this work, we examine NG Burgers’ equations with variable coefficients.
Let λ be an NG probability measure in a locally compact space Q * . We consider a quasinuclear chain H q χ L 2 Q * , d λ ( x ) H q χ , where the zero space L 2 Q * , d λ ( x ) is the space of square integrable functions defined in a commutative normal HCS L 1 Q * , d m r ( x ) with the basis Q * and the multiplicative measure m r [1,2,3,4]. H q χ is the space of generalised functions, and H q χ is the space of test functions that are constructed using Delsarte’s characters χ n : χ n C Q * [5]. This work takes as its fundamental concern an χ -Wk-TS Burgers’ equation with a variable coefficient and NG parameters of the following form [6]:
V t F 1 ( t ) χ V x x + F 2 ( t ) χ V χ V x = 0 ,
where “ χ ” is an operation called the χ -Wk-product on the generalised distribution space H q χ , and F 1 ( t ) , F 2 ( t ) are non-Gaussian on H q χ 1-valued functions from R + to the generalised distribution space H q χ [7].
Furthermore, we obtain stochastic forms of Burgers’ equations with variable coefficients when the χ -Wk-product “ χ ” is commuted by the ordinary product in Equation (1) as follows:
v t f 1 ( t ) v x x + f 2 ( t ) v v x = 0 ,
where ( x , t ) R × R + , f 1 ( t ) and f 2 ( t ) are integrable functions on R + .
In this sense, studying nonlinear equations in random environments is very important. Furthermore, variable-coefficient nonlinear equations, as well as constant-coefficient equations, cannot term factual physical phenomena precisely. In many branches of nonlinear science, Burgers’ equations with variable coefficients have various applications. The study of Burgers’ equations with variable coefficients is thriving, as this type of equation can model veritable features in a vast array of fields of science, technology, electrodynamics, engineering, fluid mechanics and wave propagation [7,8,9,10]. If Equation (2) is considered in a random environment, we have a random Burgers’ equation with a variable coefficient. In order to obtain the travelling SWS to a random Burgers’ equation with a variable coefficient, we only consider it in a white noise environment; that is, we will discuss χ -Wk-TS Burgers’ equations with variable coefficients (1).
Recently, Ghany [11,12,13] and Ghany and Hyder [14,15,16,17] studied WNFSs to nonlinear stochastic partial differential equations (SPDEs) more intensively. Furthermore, Okb El Bab, Ghany, Hyder and Zakarya [1,3] investigated significant subjects connected to the construction of NG-WNA using the theory of HCSs and certain applications. For more studies, see [18,19,20,21].
In our work, we employ the Exp-function method to look for new travelling SWSs to variable-coefficient Burgers’ equations. Then, with the aid of WNA and the χ -Hr-transform, we use these solutions to obtain solitary solutions to the χ -Wk-TS Burgers’ equations with variable coefficients.
This work is structured as follows: The second section is dedicated to the fundamental idea of constructing an NG-WNA. In the third section, we use the outcomes procured in the second section and the Exp-function method to obtain the travelling SWS to Equation (2). In the fourth section, we obtain solitary NG-WNFSs to χ -Wk-TS Burgers’ Equation (1). The last section involves a non-Wk version of the solution to Equation (1). Moreover, we give the conclusion.

2. The Basic Idea of Constructing an NG-WNA

According to NG space—for more details, see [1,22]—we can give definitions for the χ -Wk-product and the χ -Hr-transform in the space H q χ , with respect to the NG measure λ , as follows:
Definition 1
(The χ -Wk-product). Suppose that ζ 1 = m = 0 ξ m q m χ , and ζ 2 = n = 0 η n q n χ H q χ with ξ m , η n C . Then, the χ-Wk-product of ζ 1 , ζ 2 indicated by ζ 1 χ ζ 2 is defined by the following:
ζ 1 χ ζ 2 = m , n = 0 ξ m η n q m + n χ .
It is worth noting that the spaces H q χ and H q χ are closed under the χ -Wk-product.
The χ -Wk-product “ χ ” satisfies the basic algebraic properties “closed, commutative, associative and distributive law”, which are given directly as the following proposition:
Proposition 1.
For ζ 1 , ζ 2 , ζ 3 H q χ , we obtain
(i)
ζ 1 χ ζ 2 H q χ
(ii)
ζ 1 χ ζ 2 = ζ 2 χ ζ 1 ,
(iii)
ζ 1 χ ( ζ 2 χ ζ 3 ) = ( ζ 1 χ ζ 2 ) χ ζ 3 ,
(iv)
ζ 1 χ ( ζ 2 + ζ 3 ) = ζ 1 χ ζ 2 + ζ 1 χ ζ 3 .
Lemma 1.
If ζ 1 , ζ 2 H q χ and η 1 , η 2 H q χ , we have
(i)
ζ 1 χ ζ 2 H q χ ,
(ii)
η 1 χ η 2 H q χ .
Definition 2
(The χ -Hr-transform). Suppose that ζ = n = 0 ξ n q n χ H q χ with ξ n C . Then, the χ-Hr-transform of ζ indicated by H r χ ζ is defined as the following:
H r χ ζ ( Θ ) = ζ ( Θ ) = n = 0 ξ n Θ n C ( when convergent ) ,
where Θ = ( Θ 1 , Θ 2 , ) C n .
Now, we define for 0 < M , q < the neighbourhoods of zero in C , which are indicated by N q , M ( 0 ) :
N q , M ( 0 ) = Θ C : n = 0 | Θ n | 2 K q n < M 2 .
Proposition 2
([1]). If ζ H q χ , then H r χ ζ converges for each Θ N q , M ( 0 ) for each q , M < .
One of the significant advantages of the χ -Hr-transform is the following result:
Proposition 3
([1]). For ζ 1 , ζ 2 H q χ , we have
H r χ ζ 1 χ ζ 2 ( Θ ) = H r χ ζ 1 ( Θ ) . H r χ ζ 2 ( Θ ) ,
for all Θ such that H r χ ζ 1 and H r χ ζ 2 exist.
Note that the above introduces a helpful advantage of the χ -Hr-transform, which is that it turns the χ -Wk-product into an ordinary product (complex); see [1].
For χ -Brownian motion, we have W χ ( t ) = d d t B r χ ( t ) = B ˙ r χ ( t ) in H q χ .

SPDEs Driven by NG-WNA

Let us assume that our modelling considerations lead to us considering an SPDE expressed as follows:
F x , t , D t , D x , V , q = 0 ,
where F is a function, V = V ( x , t , q ) is an unknown (generalized) stochastic process and D t , D x are operators.
Firstly, we interpret all the products as χ -Wk-products and all functions as χ -Wk versions, indicated as follows:
F χ x , t , D t , D x , V , q = 0 .
Secondly, through the effect of the χ -Hr-transform in Equation (8), the χ -Wk-products are turned into ordinary products (from complex numbers), and the equation takes the following form:
F ( x , t , D t , D x , V , Θ ) = 0 ,
where V = H r χ ( V ) is the χ -Hr-transform of V, and Θ = ( Θ 1 , Θ 2 , ) C n .
Thirdly, suppose that we can obtain a solution v = v ( x , t , Θ ) to Equation (9) for each Θ = Θ 1 , Θ 2 , N q , M ( 0 ) for some q < , M > 0 where
N q , M ( 0 ) = Θ = Θ 1 , Θ 2 , C n and n = 0 Θ n 2 K q n < M 2 .
Fourthly, by taking the inverse χ -Hr-transform V = ( H r χ ) 1 v H q χ , we obtain a solution V to the original χ -Wk version of Equation (8).

3. A Travelling SWS to Equation (2)

In this section, we can employ white noise theory, the χ -Hr-transform and the Exp-function method to obtain a travelling SWS to Equation (2). By using the χ -Hr-transform for Equation (1), we obtain the deterministic equation as follows:
V t ( x , t , Θ ) F 1 ( t , Θ ) V x x ( x , t , Θ ) + F 2 ( t , Θ ) V ( x , t , Θ ) V x ( x , t , Θ ) = 0 ,
where Θ = ( Θ 1 , Θ 2 , ) C n is a vector parameter. To look for the travelling SWS to Equation (10), we perform the transformations
F 1 ( t , Θ ) : = f 1 ( t , Θ ) , F 2 ( t , Θ ) : = f 2 ( t , Θ ) ,
and
V ( x , t , Θ ) : = v ( x , t , Θ ) = v ( γ ( x , t , Θ ) ) ,
with the form
γ ( x , t , Θ ) = l x + 0 t λ ( τ , Θ ) d τ + c ,
where l 0 and c are arbitrary constants, and λ ( τ , Θ ) , a nonzero function of the denoted variables, will be calculated later. Then, Equation (10) can be converted into the following ordinary differential equation:
λ v f 1 l 2 v + f 2 l v v = 0 .
In view of the Exp-function method, the solution to Equation (11) can be expressed as follows:
v ( x , t , Θ ) = n = c d a n exp ( n γ ) m = p q b m exp ( m γ ) ,
where a n , b m are unknown constants that will be calculated later, while c , d , p and q are positive integers that can be freely selected. Hence, Equation (12) can be rewritten into the form
v ( x , t , Θ ) = a c exp ( c γ ) + + a d exp ( d γ ) b p exp ( p γ ) + + b q exp ( q γ ) .
To calculate the values of c and p, we counterweight the highest-order linear and nonlinear terms from Equation (11). Through simple calculation, we have
v = c 1 exp [ ( c + 3 p ) γ ] + c 2 exp [ 4 p γ ] + ,
v v = c 3 exp [ ( 2 c + 2 p ) γ ] + c 4 exp [ 4 p γ ] + ,
where c i are the calculated coefficients only for simplicity. Balancing the highest-order exponential functions in Equations (14) and (15), we have p = c . Likewise, to calculate the values of d and q, we counterweight the lowest-order linear and nonlinear terms in Equation (11):
v = + d 1 exp [ ( d + 3 q ) γ ] + d 2 exp [ 4 q γ ] ,
v v = + d 3 exp [ ( 2 d + 2 q ) γ ] + d 4 exp [ 4 q γ ] ,
where d i are the calculated coefficients only for simplicity. Hence, we can obtain d = q . Now, we solve Equation (10) for certain particular cases for the constants p , c , d and q.
  • Case A.
If we establish c = p = 1 and q = d = 1 , then Equation (13) becomes
v ( x , t , Θ ) = a 1 exp ( γ ( x , t , Θ ) ) + a 0 + a 1 exp ( γ ( x , t , Θ ) ) exp ( γ ( x , t , Θ ) ) + b 0 + b 1 exp ( γ ( x , t , Θ ) ) .
Substituting Equation (18) into Equation (11) and equating the coefficients of all of thepowers of e x p ( n γ ) to zero in Mathematica results in a system of algebraic equations in the unknowns a 0 , b 0 , a 1 , a 1 , b 1 and λ as shown below:
1 A [ C 2 exp 2 γ ( x , t , Θ ) + C 1 exp γ ( x , t , Θ ) + C 0 + C 1 exp γ ( x , t , Θ ) + C 2 exp 2 γ ( x , t , Θ ) ] = 0 ,
where
A = ( exp ( γ ( x , t , Θ ) ) + b 0 + b 1 exp ( γ ( x , t , Θ ) ) ) 3 , C 2 = ( a 0 a 1 b 0 ) ( a 1 f 2 l + f 1 l 2 + λ ) , C 1 = l a 0 2 + a 0 a 1 b 0 + 2 a 1 ( a 1 + a 1 b 1 ) f 2 + 4 a 1 + b 0 ( a 0 a 1 b 0 ) + 4 a 1 b 1 f 1 l + 2 a 1 a 0 b 0 + a 1 ( b 0 2 + 2 b 1 ) λ , C 0 = 3 b 0 ( a 1 + a 1 b 1 ) f 1 l 2 + a 0 l ( a 1 f 2 b 1 ( a 1 f 2 + 2 f 1 l ) ) + b 0 ( a 1 a 1 b 1 ) λ , C 1 = 2 a 1 2 f 2 l a 1 l a 0 b 0 f 2 2 a 1 b 1 f 2 + ( b 0 2 4 b 1 ) f 1 l a 1 ( b 0 2 + 2 b 1 ) λ + b 1 a 0 2 f 2 l + 2 a 1 b 1 ( 2 f 1 l 2 + λ ) + a 0 b 0 ( f 1 l 2 + λ ) , C 2 = ( a 1 b 0 a 0 b 1 ) a 1 f 2 l + b 1 ( f 1 l 2 + λ ) .
By solving the above system with the use of Mathematica, we obtain the following values:
a 1 = a 1 3 b 0 2 f 2 2 ( t , Θ ) + 4 a 1 4 b 0 4 l f 1 2 ( t , Θ ) f 2 2 ( t , Θ ) + 4 a 1 b 0 2 l f 1 2 ( t , Θ ) 4 l 2 f 1 2 ( t , Θ ) , b 1 = ( a 1 2 b 0 2 f 2 2 ( t , Θ ) + 2 a 1 b 0 2 l f 1 ( t , Θ ) f 2 ( t , Θ ) ) 4 l 2 f 1 2 ( t , Θ ) , λ = ( a 1 l f 2 ( t , Θ ) + l 2 f 1 ( t , Θ ) ) , a 0 = 0 , a 1 = a 1 , b 0 = b 0 .
Substituting these coefficients into Equation (18), we obtain the SWS to Equation (10) as follows:
v 1 ( x , t , Θ ) = a 1 + 2 l a 1 b 0 f 1 ( t , Θ ) 2 l f 1 ( t , Θ ) exp γ 1 ( x , t , Θ ) a 1 b 0 f 2 ( t , Θ ) ,
where
γ 1 ( x , t , Θ ) = l x 0 t a 1 f 2 ( τ , Θ ) + l f 1 ( τ , Θ ) d τ .
  • Case B.
If we establish c = p = 1 and q = d = 2 , then Equation (13) become s
v ( x , t , Θ ) = a 1 exp ( γ ( x , t , Θ ) ) + a 0 + a 1 exp ( γ ( x , t , Θ ) ) + a 2 exp ( 2 γ ( x , t , Θ ) ) b 1 exp ( γ ( x , t , Θ ) ) + b 0 + b 1 exp ( γ ( x , t , Θ ) ) + exp ( 2 γ ( x , t , Θ ) ) .
Substituting Equation (23) into Equation (11) and equating the coefficients of all of the powers of exp ( n γ ) to zero in Mathematica results in a system of algebraic equations in the unknowns a 0 , b 0 , a 1 , b 1 , b 1 , a 1 , a 2 , λ as shown below:
1 A [ C 5 exp 5 γ ( x , t , Θ ) + C 4 exp 4 γ ( x , t , Θ ) + C 3 exp 3 γ ( x , t , Θ ) + C 2 exp 2 γ ( x , t , Θ ) + C 1 exp γ ( x , t , Θ ) + C 0 + C 1 exp γ ( x , t , Θ ) + C 2 exp 2 γ ( x , t , Θ ) + C 3 exp γ ( x , t , Θ ) + C 4 exp γ ( x , t , Θ ) + C 5 exp γ ( x , t , Θ ) ] = 0 ,
where
A = ( exp ( 2 γ ( x , t , Θ ) ) + b 1 exp ( γ ( x , t , Θ ) ) + b 0 + b 1 exp ( γ ( x , t , Θ ) ) ) 3 , C 5 = 0 , C 4 = 0 , C 3 = 0 , C 2 = ( a 1 b 0 a 0 b 1 ) a 1 f 2 l + b 1 ( f 1 l 2 + λ ) , C 1 = a 0 2 b 1 f 2 l + 2 a 1 2 b 1 f 2 l + a 1 l 2 a 1 b 1 f 2 b 0 2 f 1 l + 4 b 1 b 1 f 1 l + a 1 ( b 0 2 + 2 b 1 b 1 ) λ 2 a 1 b 1 2 ( 2 f 1 l 2 + λ ) + a 0 b 0 a 1 f 2 l + b 1 f 1 l 2 b 1 λ , C 0 = 3 ( a 1 2 f 2 l + a 1 l a 2 b 1 f 2 + a 0 b 1 f 2 + 3 b 1 f 1 l b 0 b 1 f 1 l b 1 l ( a 0 a 1 f 2 + ( a 1 b 0 + 3 a 2 b 1 2 a 0 b 1 ) f 1 l ) b 1 ( a 1 b 0 + a 2 b 1 ) λ + a 1 ( b 1 + b 0 b 1 ) λ ) , C 1 = l ( 2 a 0 2 a 1 ( 4 a 1 + a 1 b 0 + 5 a 2 b 1 ) + a 1 a 2 b 1 + a 0 ( 2 a 2 b 0 + a 1 b 1 ) f 2 + 4 a 0 b 0 4 a 2 b 0 2 + 13 a 1 b 1 11 a 1 b 1 + a 1 b 0 b 1 2 a 2 b 1 b 1 a 0 b 1 2 f 1 l ) + 5 a 1 b 1 a 1 ( b 1 + b 0 b 1 ) 2 a 2 ( b 0 2 + 2 b 1 b 1 ) + a 0 ( 2 b 0 + b 1 2 ) λ , C 2 = l ( 2 a 0 2 a 1 ( 4 a 1 + a 1 b 0 + 5 a 2 b 1 ) + a 1 a 2 b 1 + a 0 ( 2 a 2 b 0 + a 1 b 1 ) f 2 + ( 4 a 0 b 0 4 a 2 b 0 2 + 13 a 1 b 1 11 a 1 b 1 + a 1 b 0 b 1 2 a 2 b 1 b 1 a 0 b 1 2 ) f 1 l ) + 5 a 1 b 1 a 1 ( b 1 + b 0 b 1 ) 2 a 2 ( b 0 2 + 2 b 1 b 1 ) + a 0 ( 2 b 0 + b 1 2 ) λ , C 3 = 3 l ( a 0 a 1 + a 2 ( a 1 a 1 b 0 a 2 b 1 ) f 2 ( 3 a 1 2 a 1 b 0 3 a 2 b 1 + a 0 b 1 + a 2 b 0 b 1 ) f 1 l ) + 3 a 1 + a 0 b 1 a 2 ( b 1 + b 0 b 1 ) λ , C 4 = a 1 2 f 2 l + 2 a 0 ( a 2 f 2 l 2 f 1 l 2 + λ ) + a 1 b 1 ( a 2 f 2 l + f 1 l 2 + λ ) a 2 l 2 a 2 b 0 f 2 + ( 4 b 0 + b 1 2 ) f 1 l + ( 2 b 0 + b 1 2 ) λ , C 5 = ( a 1 a 2 b 1 ) a 2 f 2 l f 1 l 2 + λ .
By solving the above system, we obtain the following values:
a 1 = a 0 2 f 2 ( t , Θ ) 2 l b 1 f 1 ( t , Θ ) , a 2 = 2 l f 1 ( t , Θ ) f 2 ( t , Θ ) , b 1 = a 0 3 b 1 f 2 3 ( t , Θ ) 8 b 1 3 l 3 f 1 3 ( t , Θ ) 4 l 2 b 1 2 a 0 f 1 2 ( t , Θ ) f 2 ( t , Θ ) , λ = l 2 f 1 ( t , Θ ) , a 1 = b 0 = 0 , a 0 = a 0 , b 1 = b 1 .
Substituting these coefficients into Equation (23), we obtain the SWS to Equation (10) as follows:
v 2 ( x , t , Θ ) = 2 l a 0 f 1 ( t , Θ ) 2 b 1 l f 1 ( t , Θ ) exp γ 2 ( x , t , Θ ) + a 0 f 2 ( t , Θ ) ,
where
γ 2 ( x , t , Θ ) = l x l 0 t f 1 ( τ , Θ ) d τ .
  • Case C.
If we establish c = p = 2 and q = d = 1 , then Equation (13) takes the form
v ( x , t , Θ ) = a 2 exp ( 2 γ ( x , t , Θ ) ) + a 1 exp ( γ ( x , t , Θ ) ) + a 0 + a 1 exp ( γ ( x , t , Θ ) ) exp ( 2 γ ( x , t , Θ ) ) + b 1 exp ( γ ( x , t , Θ ) ) + b 0 + b 1 exp ( γ ( x , t , Θ ) ) .
Substituting Equation (28) into Equation (11) and equating the coefficients of all powers of exp ( n γ ) to zero in Mathematica results in a system of algebraic equations in the unknowns a 0 , a 1 , a 2 , a 1 , b 0 , b 1 , b 1 , and λ as shown below:
1 A [ C 5 exp 5 γ ( x , t , Θ ) + C 4 exp 4 γ ( x , t , Θ ) + C 3 exp 3 γ ( x , t , Θ ) + C 2 exp 2 γ ( x , t , Θ ) + C 1 exp γ ( x , t , Θ ) + C 0 + C 1 exp γ ( x , t , Θ ) + C 2 exp 2 γ ( x , t , Θ ) + C 3 exp γ ( x , t , Θ ) + C 4 exp γ ( x , t , Θ ) + C 5 exp γ ( x , t , Θ ) ] = 0 ,
where
A = ( exp 2 γ ( x , t , Θ ) + b 1 exp γ ( x , t , Θ ) + b 0 + b 1 exp γ ( x , t , Θ ) ) 3 , C 5 = ( a 1 a 2 b 1 ) a 2 f 2 l + f 1 l 2 + λ ,
C 4 = a 1 2 f 2 l + a 2 l 2 a 2 b 0 f 2 + 4 b 0 f 1 l b 1 2 f 1 l + a 1 b 1 a 2 f 2 l + f 1 l 2 λ + a 2 ( 2 b 0 + b 1 2 ) λ 2 a 0 a 2 f 2 l + 2 f 1 l 2 + λ , C 3 = 3 l ( a 0 a 1 + a 2 ( a 1 + a 1 b 0 + a 2 b 1 ) f 2 ( 3 a 1 2 a 1 b 0 + a 0 b 1 + a 2 b 0 b 1 3 a 2 b 1 ) f 1 l ) 3 a 1 + a 0 b 1 a 2 ( b 0 b 1 + b 1 ) λ ,
C 2 = l ( 2 a 0 2 + 4 a 1 a 1 a 1 2 b 0 2 a 0 a 2 b 0 + a 0 a 1 b 1 + a 1 a 2 b 1 5 a 1 a 2 b 1 f 2 + ( 4 a 0 b 0 4 a 2 b 0 2 11 a 1 b 1 + a 1 b 0 b 1 a 0 b 1 2 + 13 a 1 b 1 2 a 2 b 1 b 1 ) f 1 l ) + ( 5 a 1 b 1 + a 1 b 0 b 1 a 0 ( 2 b 0 + b 1 2 ) + a 1 b 1 + 2 a 2 ( b 0 2 + 2 b 1 b 1 ) ) λ , C 1 = l ( a 1 a 2 b 0 a 0 2 b 1 2 a 1 a 1 b 1 + 2 a 1 2 b 1 + a 0 ( 5 a 1 + a 1 b 0 + 4 a 2 b 1 ) f 2 + a 1 b 0 2 + a 0 b 0 b 1 2 a 1 ( b 0 + 2 b 1 2 ) + 13 a 0 b 1 11 a 2 b 0 b 1 + 4 a 1 b 1 b 1 f 1 l ) a 0 b 0 b 1 + 2 a 1 ( 2 b 0 + b 1 2 ) + a 0 b 1 5 a 2 b 0 b 1 a 1 b 0 2 + 2 b 1 b 1 λ , C 0 = 3 l ( a 1 ( a 1 + a 0 b 1 ) + ( a 0 a 1 + a 1 a 2 ) b 1 f 2 ( a 1 ( b 0 b 1 3 b 1 ) + b 1 ( a 1 b 0 2 a 0 b 1 + 3 a 2 b 1 ) ) f 1 l ) + 3 a 1 ( b 0 b 1 + b 1 ) + b 1 ( a 1 b 0 + a 2 b 1 ) λ ,
C 1 = 2 a 1 2 b 1 f 2 l + a 0 2 b 1 f 2 l + a 1 l 2 a 1 b 1 f 2 b 0 2 f 1 l + 4 b 1 b 1 f 1 l a 1 ( b 0 2 + 2 b 1 b 1 ) λ + 2 a 1 b 1 2 ( 2 f 1 l 2 + λ ) + a 0 b 0 a 1 f 2 l + b 1 ( f 1 l 2 + λ ) , C 2 = ( a 1 b 0 a 0 b 1 ) a 1 f 2 l + b 1 ( f 1 l 2 + λ ) , C 3 = 0 , C 4 = 0 , C 5 = 0 .
Solving the above system yields the following values:
a 1 = a 0 b 1 , b 1 = a 0 b 1 f 2 ( t , Θ ) a 2 f 2 ( t , Θ ) + 4 l f 1 ( t , Θ ) , a 1 = a 2 b 1 , b 0 = a 0 f 2 ( t , Θ ) a 2 f 2 ( t , Θ ) + 4 l f 1 ( t , Θ ) , λ = l ( a 2 f 2 ( t , Θ ) + 2 l f 2 ( t , Θ ) ) , a 0 = a 0 , a 2 = a 2 , b 1 = b 1 .
Substituting these coefficients into Equation (28), we obtain the SWS to Equation (10) as follows:
v 3 ( x , t , Θ ) = a 2 + 4 a 0 l f 1 ( t , Θ ) a 0 f 2 ( t , Θ ) + e x p ( 2 γ 3 ( x , t , Θ ) ) ( a 2 f 2 ( t , Θ ) + 4 l f 1 ( t , Θ ) ) ,
where
γ 3 ( x , t , Θ ) = l x 0 t ( a 2 f 2 ( τ , Θ ) + 2 l f 2 ( τ , Θ ) ) d τ .
Remark 1.
There are different solutions to Equation (10). These cases came from setting various values for the following coefficients: p , c , d and q . The cases above demonstrate the broad applicability of our technique.

4. NG Stochastic Travelling SWSs to Equation (1)

In this section, we applied the results of the above sections. By applying the χ -Hr-transform, we obtained solitary NG-WNFSs to χ -Wk-TS Burgers’ equations like that in Equation (1).
According to the theorem in Holden [11], a bounded open set F R × R + , q < , M > 0 exists such that the solutions v ( x , t , Θ ) to Equation (10) and all of its partial derivatives which are involved in Equation (10) are (uniformly) bounded for ( x , t , Θ ) F × N q , M ( 0 ) ; continuous with regard to ( x , t ) F for each Θ N q , M ( 0 ) ; and analytic with regard to Θ N q , M ( 0 ) for all ( x , t ) F . Therefore, V ( x , t ) H q χ exists such that V ( x , t ) = ( H r χ ) 1 v ( x , t , Θ ) . Also, V ( x , t ) is solved in Equation (1) in H q χ .
Then, by using the inverse χ -Hr-transform on the outcomes of Section 3, we obtain solitary NG-WNFSs to Equation (1) as follows:
V 1 ( x , t ) = a 1 + 2 l a 1 b 0 F 1 ( t ) 2 l F 1 ( t ) χ e x p χ ( Γ 1 ( x , t ) ) a 1 b 0 F 2 ( t ) ,
where
Γ 1 ( x , t ) = l [ x 0 t ( a 1 F 2 ( τ ) + l F 1 ( τ ) ) d τ ] ,
V 2 ( x , t ) = 2 l a 0 F 1 ( t ) 2 b 1 l F 1 ( t ) χ e x p χ ( Γ 2 ( x , t ) ) + a 0 F 2 ( t ) ,
where
Γ 2 ( x , t ) = l [ x 0 t l F 1 ( τ ) d τ ] ,
V 3 ( x , t ) = a 2 + 4 a 0 l F 1 ( t ) a 0 F 2 ( t ) + e x p χ ( 2 Γ 3 ( x , t ) ) χ ( a 2 F 2 ( t ) + 4 l F 1 ( t ) ) ,
where
Γ 3 ( x , t ) = l [ x 0 t ( a 2 F 2 ( τ ) + 2 l F 2 ( τ ) ) d τ ] .
Remark 2.
We notice that for various forms of F 1 ( t ) and F 2 ( t ) , we can obtain various travelling solitary NG-WNFSs to Equation (1) from Equations (33)–(38).

5. Non-Wk Versions of Solutions to Equation (1)

According to Holden ([11], Lemma 2.6.16), it is well known that the Wk version of a function is usually difficult to evaluate. So, in this section, we provide a non-Wk solution to Equation (1). Suppose that W χ ( t ) = B ˙ r χ ( t ) is the NG-WNA, where B r χ ( t ) is χ -Brownian motion. We have the χ -Hr-transform
W χ ( t , Θ ) = i = 1 Θ i 0 t η i ( τ ) d τ .
According to Holden ([11], Lemma 2.6.16), we have
exp χ B r χ ( t ) = exp B r χ ( t ) t 2 2 ,
Assume that F 1 ( t ) = π 1 F 2 ( t ) and F 2 ( t ) = α ( t ) + π 2 W χ ( t ) where π 1 and π 2 are optional constants and α ( t ) is an integrable function on R + . Hence, for F 1 ( t ) F 2 ( t ) 0 , the solution to Equation (1), a non-Wk version, becomes the following:
V 4 ( t , x ) = a 1 + 2 l a 1 b 0 π 1 2 l π 1 exp Γ 4 ( t , x ) a 1 b 0 ,
where
Γ 4 ( t , x ) = l x l π 1 + a 1 0 t [ α ( τ ) + π 2 W χ ( τ ) d τ ] ,
V 5 ( t , x ) = 2 l a 0 π 1 2 b 1 l π 1 exp Γ 5 ( x , t ) + a 0 ,
where
Γ 5 ( t , x ) = l x l π 1 0 t [ α ( τ ) + π 2 W χ d ( τ ) ] ,
V 6 ( t , x ) = a 2 + 4 a 0 l π 1 a 0 + exp 2 Γ 6 ( x , t ) ( a 2 + 4 l π 1 ) ,
where
Γ 6 ( t , x ) = l x a 2 + 2 l 0 t [ α ( τ ) + π 2 W χ d ( τ ) ] .
Note that we can obtain the above solutions using χ -Brownian motion as follows:
V 4 ( t , x ) = a 1 + 2 l a 1 b 0 π 1 2 l π 1 exp Γ 4 ( t , x ) a 1 b 0 ,
where
Γ 4 ( t , x ) = l x l π 1 + a 1 0 t α ( τ ) d τ + π 2 B r χ ( t ) t 2 2 ,
V 5 ( t , x ) = 2 l a 0 π 1 2 b 1 l π 1 exp Γ 5 ( x , t ) + a 0 ,
where
Γ 5 ( t , x ) = l x l π 1 0 t α ( τ ) d τ + π 2 B r χ ( t ) t 2 2 ,
V 6 ( t , x ) = a 2 + 4 a 0 l π 1 a 0 + exp 2 Γ 6 ( x , t ) ( a 2 + 4 l π 1 ) ,
where
Γ 6 ( t , x ) = l x a 2 + 2 l 0 t α ( τ ) d τ + π 2 B r χ ( t ) t 2 2 .
Remark 3.
We notice that for various forms of α ( t ) , π 1 and π 2 , we can obtain various travelling non-Wk versions of solutions to Equation (1) from Equations (45)–(50).

6. Conclusions

This study is devoted to the implementation of novel techniques that provide NG-WNFSs to χ -Wk-TS Burgers’ equations with variable coefficients. If the problem is considered in an NG stochastic environment, we can obtain NG stochastic Burgers’ equations with variable coefficients. In order to obtain travelling SWSs to stochastic Burgers’ equations with variable coefficients, we only consider this problem in an NG white noise environment; that is, we investigate χ -Wk-TS Burgers’ equations with variable coefficients (1). For this aim, we develop NG Wk calculus based on the theory of HCSs L 1 Q * , d m r ( x ) . Precisely, we use the direct connection between HCSs, WNA [1,2,3,4] and Delsarte’s characters χ n to introduce an χ -Wk-product and an χ -Hr-transform in the space of generalized functions H q χ , with the zero space L 2 Q * , d λ ( x ) , and discuss their properties. By means of the usual properties of complex analytic functions, we study SPDEs with NG parameters (for more details, see [1]). The primary techniques used in this work are the Exp-function method, WNA and the χ -Hr-transform, which are all used to obtain NG-WNFSs to Equation (1). The χ -Wk-TS Burgers’ equations with variable coefficients were solved using the Mathematica program. With the help of the inverse χ -Hr-transform, we obtained solitary NG-WNFSs to Equation (1). We also devoted a section to the non-Wk version of the solution tp Equation (1). Furthermore, the plan we present in this paper can be used for different nonlinear partial differential equations in mathematical physcis, such as Kersten–Krasil’shchik coupled KdV-mKdV equations, the modified Boussinesq equation, the Schamel–KdV equation and the Zakharov–Kuznetsov equation.

Author Contributions

Methodology, M.A.-Q., T.A.-Q. and M.Z.; Software, M.A.-Q., T.A.-Q. and M.Z.; Formal analysis, M.A.-Q., T.A.-Q. and M.Z.; Investigation, writing—original draft preparation, M.A.-Q., T.A.-Q. and M.Z.; writing—review and editing, M.A.-Q., T.A.-Q. and M.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This work is funded by King Khalid University for funding this work through large Research Project under grant number RGP 2/190/45.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through large Research Project under grant number RGP 2/190/45.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. El Bab, A.O.; Ghany, H.A.; Zakarya, M. A construction of non-Gaussian white noise analysis using the theory of hypercomplex systems. Glob. J. Sci. Front. Res. Math. Decis. Sci. 2016, 16, 11–25. [Google Scholar]
  2. Hyder, A.; Zakarya, M. Non-Gaussian Wick calculus based on hypercomplex systems. Int. J. Pure Appl. Math. 2016, 109, 539–556. [Google Scholar] [CrossRef]
  3. Zakarya, M. Hypercomplex Systems with Some Applications of White Noise Analysis; LAP LAMBERT Academic Publishing: Saarbrucken, Germany, 2017. [Google Scholar]
  4. Hyder, A. White noise analysis combined with hypercomplex systems for solving stochastic modified KdV equations with non-Gaussian parameters. Pioneer J. Adv. Appl. Math 2018, 24, 39–61. [Google Scholar]
  5. Berezansky, Y.M. A generalization of white noise analysis by means of theory of hypergroups. Rep. Math. Phys. 1996, 38, 289–300. [Google Scholar] [CrossRef]
  6. Chen, B.K.; Li, Y.; Chen, H.L.; Wang, B.H. Exp-function method for solving the Burgers-Fisher equation with variable coefficients. arXiv 2010, arXiv:1004.1815. [Google Scholar]
  7. Benjamin, T.B.; Bona, J.L.; Mahony, J.J. Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 1972, 272, 47–78. [Google Scholar]
  8. Dehghan, M.; Shokri, A. A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simul. 2008, 79, 700–715. [Google Scholar] [CrossRef]
  9. Hereman, W.; Martino, R.; Miller, J.; Hong, L.; Formenac, S.; Menz, A. Exact Solutions of Nonlinear Partial Differential Equations the Tanh/Sech Method; Mathematica Visiting Scholar Grant Program Wolfram Research Inc.: Champaign, IL, USA, 2000; Volume 25, pp. 1–26. [Google Scholar]
  10. Khalique, C.M. Exact solutions and conservation laws of a coupled integrable dispersionless system. Filomat 2012, 26, 957–964. [Google Scholar] [CrossRef]
  11. Holden, H.; Øksendal, B.; Ubøe, J.; Zhang, T. Stochastic Partial Differential Equations; Springer Science and Business, LLC: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
  12. Ghany, H.A. Exact solutions for stochastic generalized Hirota-Satsuma coupled KdV equations. Chin. J. Phys. 2011, 49, 926–940. [Google Scholar]
  13. Ghany, H.A. Exact solutions for stochastic fractional Zakharov-Kuznetsov equations. Chin. J. Phys. 2013, 51, 875–881. [Google Scholar]
  14. Ghany, H.A.; Hyder, A.A. White noise functional solutions for the Wick-type two-dimensional stochastic Zakharov-Kuznetsov equations. Int. Rev. Phys. 2012, 6, 153–157. [Google Scholar]
  15. Ghany, H.A.; El Bab, A.O.; Zabel, A.; Hyder, A.A. The fractional coupled KdV equations: Exact solutions and white noise functional approach. Chin. Phys. 2013, 22, 080501. [Google Scholar] [CrossRef]
  16. Ghany, H.A.; Hyder, A.A. Exact solutions for the wick-type stochastic time-fractional KdV equations. Kuwait J. Sci. 2014, 41, 75–84. [Google Scholar]
  17. Ghany, H.A.; Hyder, A.A. Abundant solutions of Wick-type stochastic fractional 2D KdV equations. Chin. Phys. B 2014, 23, 060503. [Google Scholar] [CrossRef]
  18. Fang, L.; Ma, L.; Ding, S.; Zhao, D. Finite-time stabilization for a class of high-order stochastic nonlinear systems with an output constraint. Appl. Math. Comput. 2019, 358, 63–79. [Google Scholar] [CrossRef]
  19. Yang, C.; Wang, J.; Miao, S.; Zhao, B.; Jian, M.; Yang, C. Boundary Coupling for Consensus of Nonlinear Leaderless Stochastic Multi-Agent Systems Based on PDE-ODEs. Mathematics 2022, 10, 4111. [Google Scholar] [CrossRef]
  20. Alam, M.N.; Tunc, C. New solitary wave structures to the (2 + 1)-dimensional KD and KP equations with spatio-temporal dispersion. J. King Saud-Univ.-Sci. 2020, 32, 3400–3409. [Google Scholar] [CrossRef]
  21. Almatrafi, M.B.; Alharbi, A.R.; Tunç, C. Constructions of the soliton solutions to the good Boussinesq equation. Adv. Differ. Equ. 2020, 2020, 629. [Google Scholar] [CrossRef]
  22. Ghany, H.A.; Hyder, A.; Zakarya, M. Non-Gaussian white noise functional solutions of χ-Wick-type stochastic KdV equations. Appl. Math. Inf. Sci. 2017, 11, 915–924. [Google Scholar] [CrossRef]
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Zakarya, M.; Al-Qarni, M.; Al-Qahtani, T. Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients. Symmetry 2024, 16, 1572. https://doi.org/10.3390/sym16121572

AMA Style

Zakarya M, Al-Qarni M, Al-Qahtani T. Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients. Symmetry. 2024; 16(12):1572. https://doi.org/10.3390/sym16121572

Chicago/Turabian Style

Zakarya, Mohammed, Manal Al-Qarni, and Tahani Al-Qahtani. 2024. "Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients" Symmetry 16, no. 12: 1572. https://doi.org/10.3390/sym16121572

APA Style

Zakarya, M., Al-Qarni, M., & Al-Qahtani, T. (2024). Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients. Symmetry, 16(12), 1572. https://doi.org/10.3390/sym16121572

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