Bidirectional monte carlo method for thermal radiation transfer in participating medium

Xiaofeng Zhang; Qing Ai; Kuilong Song; Heping Tan; Xiaofeng Zhang; Qing Ai; Kuilong Song; Heping Tan

doi:10.3934/energy.2021029

AIMS Energy

2021, Volume 9, Issue 3: 603-622. doi: 10.3934/energy.2021029

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Research article Special Issues

Bidirectional monte carlo method for thermal radiation transfer in participating medium

1.
Harbin Institute of Technology, 92 West DaZhi Street, Harbin, 150001, China
2.
Chinese Academy of Sciences, Shanghai, 201200, China

Received: 09 March 2021 Accepted: 05 May 2021 Published: 03 June 2021

A bidirectional Monte Carlo (BDMC) method based on reversibility of bundle trajectory and reciprocity of thermal radiative energy exchange was developed to solve radiative heat transfer in absorbing and scattering medium. Two types of sampling models were introduced into the Monte Carlo (MC) simulation, namely the equivalent sampling and the weight sampling, respectively. Mathematical formula for the sampling models and the statistical calculation of sampling bundles were derived. Furthermore, the reciprocity error correlation of radiative exchange factors between the BDMC method and the traditional Monte Carlo (TMC) method were demonstrated and analyzed. Radiative heat transfer in a two-dimensional rectangular domain with absorbing and scattering media was solved by using both the BDMC method and the TMC method. Radiative exchange factors and radiative equilibrium temperature profiles predicted by the BDMC method were compared with those predicted by the TMC method. The performance parameter P, defined to evaluate the performance of MC methods, was computed and compared between the BDMC and the TMC methods. The results showed the superiority of BDMC method compared with the TMC method for radiative heat transfer, in addition, the weight sampling was proved to be more flexible than the equivalent sampling in the BDMC method.

Keywords:

Citation: Xiaofeng Zhang, Qing Ai, Kuilong Song, Heping Tan. Bidirectional monte carlo method for thermal radiation transfer in participating medium[J]. AIMS Energy, 2021, 9(3): 603-622. doi: 10.3934/energy.2021029

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Abstract

1. Introduction

Nonlinear partial differential equations (PDEs) impart multi-scale characteristics to the system, thereby allowing for a more accurate prediction of the transmission process of soliton solutions. In practical uses, nonlinear PDEs and soliton solutions are vital for characterizing various phenomena in science and engineering such as biology, physics, ocean engineering, and many more ^[1,2,3]. Various types of soliton solutions have been reported for integrable systems. For instance, horse-shoe like soliton and lump chain solitons have been studied for the elliptic cylindrical Kadomtsev–Petviashvili equation ^[4]. Yang et al. analyzed degenerating lump chains into anomalously scattered lumps for the Mel'nikov equation ^[5]. In literature ^[6], a series of ripple waves with decay modes for the (3+1)‑dimensional Kadomtsev–Petviashvili equation have been reported. Rogue wave solutions to the (3+1)-dimensional Korteweg-de Vries Benjamin-Bona-Mahony equation were studied via the Hirota bilinear approach ^[7]. The propagation features and interactions of Rossby waves soliton of the geophysical equation were studied ^[8]. Breather, lump, and its interaction solutions for the higher dimensional evolution equation were studied^[9]. Multisoliton solutions for the variable coefficient Schrödinger equation has been explored in the literature ^[10]. Some other solitons solutions have been reported for the regularized long-wave equation ^[11], the Sharma-Tasso-Olver-Burgers equation ^[12], the modified Schrödinger's equation ^[13], the complex Ginzburg–Landau equation ^[14], the (2+1) dimensional Chaffee–Infante equation ^[15], and many more ^[16,17,18].

Stochastic differential equations (DEs) deal with phenomena having randomness or uncertainties. Stochastic DEs can be used in various field of science and engineering ^[19,20,21]. Solving stochastic nolinear PDEs is very challenging and hard due to randomness. Therefore, various methods have been introduced and implemented to derive solutions of stochatics PDEs such as the modified tanh method ^[22], the modified Kudrayshov technique ^[23], the Sardar subequation method ^[24], and many more ^[25,26].

Fractional operators (FOs) have been frequently used for modelling the physical phenomena in various fields due to its memory process ^[27,28,29]. In literature, several FOs have been constructed by researchers and scientists ^[30,31,32]. Most of them do not satisfy some properties such as the chain and quotient rules. A few years ago, Atangana ^[33] defined a local FO called beta derivative, which generalized the classical operator. The {beta derivative (BD)} is defined as follows:

$\mathscr{D}_{x}^{\beta}\Psi\left(x\right) = \frac{d^{\beta}\Psi}{dx^{\beta}} = \lim\limits_{h_{0}\rightarrow0}\frac{\Psi\left(x+h_{0}\left(x+\frac{1}{\Gamma(\beta)}\right)^{1-\beta}-\Psi\left(x\right)\right)}{h_{0}}, \thinspace\thinspace0 < \beta\leq1.$

Here, the $BD$ has the following characteristics: For every real numbers, $m$ and $n$ :

$\begin{align*} &(1)\; \mathscr{D}_{x}^{\beta}\Psi\left(x\right) = \left(x+\frac{1}{\Gamma(\beta)}\right)^{1-\beta}\frac{d\Psi}{dx}.\\ &(2)\; \mathscr{D}_{x}^{\beta}\left(m\Psi+n\Phi\right) = m\left(x+\frac{1}{\Gamma(\beta)}\right)^{1-\beta}\frac{d\Psi}{dx}+n\left(x+\frac{1}{\Gamma(\beta)}\right)^{1-\beta}\frac{d\Phi}{dx}.\\ &(3)\; \mathscr{D}_{x}^{\beta}\left(\Psi\circ \Phi\left(x\right)\right) = \left(x+\frac{1}{\Gamma(\beta)}\right)^{1-\beta}\frac{d\Psi}{dx}\Phi^{\prime}\left(x\right)\left(\Psi^{\prime}\left(x\right)\right).\\ &(4)\; \mathscr{D}_{x}^{\beta}\Psi\left(m\right) = 0. \end{align*}$

The $BD$ has been used for the analysis of soliton solutions with the fractional behavior of nonlinear PDEs ^[34,35,36]. This work modifies the Benjamin-Bona-Mahony equation (BBME) as follows:

$\begin{equation} \mathcal{M}_{t}+6\mathcal{M}\mathscr{D}_{x}^{\beta}\mathcal{M}+\mathscr{D}_{xxx}^{\beta}\mathcal{M}-\rho\mathscr{D}_{xx}^{\beta}\mathcal{M}_{t} = \tau\left(\mathcal{M}-\rho\mathscr{D}_{xx}^{\beta}\mathcal{M}\right)\frac{d\mathcal{P}}{dt}, \end{equation}$

(1.1)

where $\rho$ is real parameter, $\mathcal{M} = \mathcal{M}\left(x, t\right)$ is a real valued wave profile, $\tau$ is the intensity of sound, and $\mathcal{P} = \mathcal{P}\left(t\right)$ is a white noise having the following properties:

$(i) \mathcal{P}$ possesses constant trajectories.

$(ii) \mathcal{P}\left(0\right) = 0.$

$(iii) \mathcal{P}\left(t_{j+1}\right)-\mathcal{P}\left(t_{j}\right)$ has a normal standard distribution.

When we consider $\tau = 0$ and $\beta = 1,$ we get the BBME as follows:

$\begin{equation} \mathcal{M}_{t}+6\mathcal{M}\mathcal{M}_{x}+\mathcal{M}_{xxx}-\rho\mathcal{M}_{xxt} = 0. \end{equation}$

(1.2)

Benjamin, Bona, and Mahony examined equation (1.2) as an adjustment to the KdV equation. The BBME has been used to analyze the prorogation of long surface gravity pulses with small amplitudes. There are several studies on the BBME. For instance, BBME was studied by using the variational method ^[37], the deep learning method ^[38], the generalized exp-function method ^[39], and many more ^[40,41]. In ^[42], the authors have used the F-expansion method to study the solitary waves BBME under BD with white noise. In this paper, we use two advanced analytical methods to deduce more solitary waves solutions and to study the influence of the BD and the white noise.

2. The general procedure of the proposed approaches

This section provides the general procedure of the suggested approaches that one can use to find solitary and other waves solutions.

2.1. $\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}$ -expansion method

Here, we present the general procedure of the $\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}$ -expansion technique. Consider a PDE under space $BD$ as follows

$\begin{equation} \mathcal{A}_{1}\left(\mathcal{M}, \partial_{x}^{\beta}\mathcal{M}, \partial_{t}\mathcal{M}, \partial_{x}^{\beta}\partial_{x}^{\beta}\mathcal{M}, \partial_{x}^{\beta}\partial_{t}\mathcal{M}, \partial_{t}\partial_{t}\mathcal{M}, \cdots\right) = 0, \end{equation}$

(2.1)

where $\mathcal{A}_{1}$ is a polynomial in $\mathcal{M} = \mathcal{M}\left(x, t\right)$ and its partial derivatives. To use the proposed procedure, one should abide by the following:

Step 1. First using the wave transformation, one can obtain ODE as follows:

$\begin{equation} \mathcal{M}\left(x, t\right) = \mathcal{M}\left(\omega_{1}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}, \end{equation}$

(2.2)

where $\omega_{1} = \frac{\xi_{1}}{\beta}\left(x+\frac{1}{\Gamma(\beta)}\right)^{\beta}+\xi_{2}t$ . Additionally, $\xi_{1}$ and $\xi_{2}$ are referred to as the wave speed and the wave number, respectively. By inserting Eq (2.2) in Eq (2.1), the following will be obtained:

$\begin{equation} \mathcal{A}_{1}\left(\mathcal{M}, \mathcal{M}^{\prime}, \mathcal{M}^{\prime\prime}, \mathcal{M}^{\prime\prime\prime}\thinspace\thinspace\thinspace\thinspace\right) = 0, \end{equation}$

(2.3)

where the ordinary derivatives of different orders are indicated by primes.

Step 2. According to the proposed strategy, we examine the following form for the solution to Eq (2.3):

$\begin{equation} \mathcal{M}\left(\omega_{1}\right) = \sum\limits_{i = 0}^{\aleph}\mathcal{F}_{i}\left(\frac{\mathcal{G}^{\prime}\left(\omega_{1}\right)}{\mathcal{G}^{\prime}\left(\omega_{1}\right)+\mathcal{G}\left(\omega_{1}\right)+\mathcal{A}}\right)^{i}, \end{equation}$

(2.4)

where $\mathcal{F}_{i}$ is the function of the polynomial's coefficients $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)^{i}, \thinspace\thinspace i = 0, 1, 2, \ldots , \aleph.$ Assume that $\mathcal{G}\left(\omega_{1}\right)$ is a function that fulfills the subsequent relation:

$\begin{equation} \mathcal{G}^{\prime\prime}+\mathcal{A}\mathcal{G}^{\prime}+\mathcal{B}\mathcal{G}+\mathcal{B}\mathcal{A} = 0. \end{equation}$

(2.5)

The value of $\aleph$ can be determined using the homogeneous balance rule (HBR) between the highest nonlinear term and the highest order derivative in Eq (2.3).

Step 3. In this step, the result obtained from the substitution of Eq (2.4) into Eq (2.3) and the coefficients of various powers of $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)$ should be compared in terms of $\mathcal{A}, \mathcal{B}, \xi_{1}, \xi_{2}$ , and $i = 0, 1, 2, \ldots , \aleph$ . Using Mathematica or any other mathematical package, one can determine the solution's values $\mathcal{G}$ in the term $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)$ , and ultimately for the principles of $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)$ , $\mathcal{F}_{i}$ and $\omega_{1}$ . In doing so, the solution of Eq (2.2) can be obtained.

2.2. The modified $\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}$ -expansion approach

Here, we present the general procedure of applying the modified $\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}$ -expansion approach to obtain the wave solutions of a nonlinear PDE. This approach contains the following expansion:

$\begin{equation} \mathcal{M}\left(\omega_{1}\right) = \mathcal{F}_{0}+\sum\limits_{i = 1}^{\aleph}\left(\mathcal{F}_{i}\left(\frac{\mathcal{G}^{\prime}\left(\omega_{1}\right)}{\mathcal{G}\left(\omega_{1}\right)^{2}}\right)^{i}+\mathcal{S}_{i}\left(\frac{\mathcal{G}^{\prime}\left(\omega_{1}\right)}{\mathcal{G}\left(\omega_{1}\right)^{2}}\right)^{-i}\right), \end{equation}$

(2.6)

where $\mathcal{G}\left(\omega_{1}\right)$ satisfies the following the equation:

$\begin{equation} \mathcal{G}^{\prime\prime}\left(\omega_{1}\right) = \frac{\varPsi\mathcal{G}^{\prime}\left(\omega_{1}\right)^{2}}{\mathcal{G}\left(\omega_{1}\right)^{2}}+\psi\mathcal{G}^{\prime}\left(\omega_{1}\right)+\frac{2\mathcal{G}^{\prime}\left(\omega_{1}\right)^{2}}{\mathcal{G}\left(\omega_{1}\right)}+\varpi\mathcal{G}\left(\omega_{1}\right)^{2}, \end{equation}$

(2.7)

where $\varPsi, \psi$ , and $\varpi$ are the arbitrary constants. Next, one should find the value of $\aleph$ as previously mentioned. Then, substituting Eq (2.6) and using Eq (2.7) into Eq (2.3), one can obtain a differential equation in $\mathcal{G}(\omega_1)$ . Then, collecting those terms which contain $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}\right)^{i},$ $\left(i = 0, 1, 2, \ldots, n\right),$ and setting all the coefficients of $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}\right)^{i}$ equal to zero, one can acquire a system of algebraic equations. Solving the obtained system can possibly result in the following families.

Family 1. If $\varPsi\varpi > 0$ and $\psi = 0,$ the we have the following:

$\begin{equation} \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}} = \frac{\sqrt{\varPsi\varpi}\left(p_{1}cos\left(\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{2}sin\left(\omega_{1}\sqrt{\varPsi\varpi}\right)\right)}{\varpi\left(p_{2}cos\left(\omega_{1}\sqrt{\varPsi\varpi}\right)-p_{1}sin\left(\omega_{1}\sqrt{\varPsi\varpi}\right)\right)}, \end{equation}$

(2.8)

where $p_{1}, p_{2}, \varPsi$ , and $\varpi$ are arbitrary constants.

Family 2. If $\varPsi\varpi < 0$ and $\psi = 0,$ then we have the following:

$\begin{equation} \frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}} = -\frac{\sqrt{\varPsi\varpi}\left(p_{1}sinh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{1}cosh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{2}\right)}{\varpi\left(p_{1}sinh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{1}cosh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{2}\right)}. \end{equation}$

(2.9)

3. Traveling wave solutions of considered equation

Here, we explore the wave solutions for the proposed stochastic BBME under BD as given in Eq (1.1) with the following procedure:

$\begin{equation} \mathcal{M}\left(x, t\right) = \mathcal{M}\left(\omega_{1}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}. \end{equation}$

(3.1)

Furthermore, we have the following:

$\begin{equation} \mathcal{M}_{t} = \left(\xi_{2}\mathcal{M}^{\prime}+\tau\mathcal{M}\mathcal{P}_{t}+\frac{1}{2}\tau^{2}\mathcal{M}-\frac{1}{2}\tau^{2}\mathcal{M}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}, \end{equation}$

(3.2)

and

$\begin{align} &\mathscr{D}_{xx}^{\beta}\mathcal{M}_{t} = \left(\xi_{1}^{2}\xi_{2}\mathcal{M}^{\prime\prime\prime}+\tau\mathcal{P}_{t}\xi_{1}^{2}\mathcal{M}^{\prime\prime}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}, \\ &\mathscr{D}_{x}^{\beta}\mathcal{M} = \left(\xi_{1}\mathcal{M}^{\prime}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}, \thinspace\thinspace\thinspace\mathscr{D}_{xxx}^{\beta}\mathcal{M} = \left(\xi_{1}^{3}\mathcal{M}^{\prime\prime\prime}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}. \end{align}$

(3.3)

Inserting Eq (3.1) into Eq (1.1) and using Eqs (3.2) and (3.3), we obtain the following:

$\begin{equation} \xi_{2}\mathcal{M}^{\prime}+\left(\xi_{1}^{3}-\rho\xi_{1}^{2}\xi_{2}\right)\mathcal{M}^{\prime\prime\prime}+6\xi_{1}\mathcal{M}\mathcal{M}^{\prime}e^{-\frac{1}{2}\tau^{2}t}\mathcal{E}e^{\tau\mathcal{P}\left(t\right)} = 0. \end{equation}$

(3.4)

By considering $\mathcal{P}\left(t\right)$ , the Gaussian process, and $\mathcal{E}e^{\tau\mathcal{P}\left(t\right)} = e^{\frac{1}{2}\tau^{2}t},$ then, Eq (3.4) becomes:

$\begin{equation} \xi_{2}\mathcal{M}^{\prime}+\left(\xi_{1}^{3}-\rho\xi_{1}^{2}\xi_{2}\right)\mathcal{M}^{\prime\prime\prime}+6\xi_{1}\mathcal{M}\mathcal{M}^{\prime} = 0. \end{equation}$

(3.5)

Integrating Eq (3.5) one time while considering the integration constant to be zero, we obtain the following:

$\begin{equation} \zeta\mathcal{M}+\mathcal{M}^{\prime\prime}+\eta\mathcal{M}^{2} = 0, \end{equation}$

(3.6)

where

$\zeta = \frac{\xi_{2}}{\xi_{1}^{3}-\rho\xi_{1}^{2}\xi_{2}}, \thinspace\thinspace\thinspace\thinspace\eta = \frac{3}{\xi_{1}^{2}-\rho\xi_{1}\xi_{2}}.$

In Eq (3.6), by using the homogeneous balance principle, we obtain $\aleph = 2$ . Now, we have Eq (2.4) in the following form:

$\begin{equation} \mathcal{M}_{1}\left(\omega_{1}\right) = \mathcal{F}_{0}+\mathcal{F}_{1}\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)+\mathcal{F}_{2}\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)^{2}. \end{equation}$

(3.7)

Inserting the solution of Eq (3.7) with Eq (2.5) into Eq (3.6), the polynomial of the left side will be in $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)^{i}, \thinspace\thinspace i = 0, 1, 2\cdots\aleph.$ By further equating the coefficients of various powers of $\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{\prime}+\mathcal{G}+\mathcal{A}}\right)$ to zero, we obtain a system of algebraic equations. Using Mathematica to solve the system of equations, we obtain the following sets:

$\begin{equation} \begin{cases} \mathcal{F}_{0} = \frac{\xi_{1}\xi_{2}\left(\mathcal{A}^{2}-12\mathcal{A}\mathcal{B}+4\mathcal{B}(3\mathcal{B}+2)\right)-\xi_{2}\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}{6\xi_{1}\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}, \\ \mathcal{F}_{1} = \mp\frac{2\xi_{2}(\mathcal{A}-2\mathcal{B})(\mathcal{A}-\mathcal{B}-1)}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}, \mathcal{F}_{2} = \frac{2\xi_{2}(-\mathcal{A}+\mathcal{B}+1)^{2}}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}, \\ \rho = \frac{\frac{\xi_{1}^{2}}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}+\frac{\xi_{1}^{4}}{\xi_{2}}}{\xi_{1}^{3}}. \end{cases} \end{equation}$

(3.8)

Now, inserting the parameter values presented in Eq (3.8) into Eq (3.7), we get the exact solutions of Eq (3.6) in the following two cases:

Set 1. For $\mathcal{D} = \mathcal{A}^{2}-4\mathcal{B} > 0$ , we have the following:

$\begin{align} & \mathcal{M}\left(\omega_{1}\right) = \Biggr(\frac{\xi_{1}\xi_{2}\left(\mathcal{A}^{2}-12\mathcal{A}\mathcal{B}+\left(12\mathcal{B}^{2}+8\mathcal{B}\right)\right)-\xi_{2}\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}{6\xi_{1}\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}} \\ & -\frac{(2\xi_{2}(\mathcal{A}-2\mathcal{B})(\mathcal{A}-\mathcal{B}-1))\left(\nu_{2}e^{\sqrt{\mathcal{D}}\omega_{1}}\left(\mathcal{A}-\sqrt{\mathcal{D}}\right)+\nu_{1}\left(\sqrt{\mathcal{D}}+\mathcal{A}\right)\right)}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}\left(\nu_{2}e^{\sqrt{\mathcal{D}}\omega_{1}}\left(-\sqrt{\mathcal{D}}+\mathcal{A}-2\right)+\nu_{1}\left(\sqrt{\mathcal{D}}+\mathcal{A}-2\right)\right)} \\ & \frac{\left(2\xi_{2}(-\mathcal{A}+\mathcal{B}+1)^{2}\right)\left(\frac{\nu_{2}e^{\sqrt{\mathcal{D}}\omega_{1}}\left(\mathcal{A}-\sqrt{\mathcal{D}}\right)+\nu_{1}\left(\sqrt{\mathcal{D}}+\mathcal{A}\right)}{\nu_{2}e^{\sqrt{\mathcal{D}}\omega_{1}}\left(-\sqrt{\mathcal{D}}+\mathcal{A}-2\right)+\nu_{1}\left(\sqrt{\mathcal{D}}+\mathcal{A}-2\right)}\right)^{2}}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}\Biggr)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}, \end{align}$

(3.9)

where $\nu_{1}$ and $\nu_{2}$ remain constants.

Set 2. For $\mathcal{\mathcal{\mathcal{D}}} = \mathcal{A}^{2}-4\mathcal{B} < 0,$ we have the following:

$\begin{align} & \mathcal{\mathcal{M}}\left(\omega_{1}\right) = \Biggr(\frac{\xi_{1}\xi_{2}\left(\mathcal{A}^{2}-12\mathcal{A}\mathcal{B}+\left(12\mathcal{B}^{2}+8\mathcal{B}\right)\right)-\xi_{2}\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}}{6\xi_{1}\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}} \\ & -\frac{(2\xi_{2}(\mathcal{A}-2\mathcal{B})(\mathcal{A}-\mathcal{B}-1))}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}} \\ & \frac{\left(\mathcal{A}\nu_{2}+\nu_{1}\sqrt{-\mathcal{D}}\right)sin\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)+\left(\mathcal{A}\nu_{1}-\nu_{2}\sqrt{-\mathcal{D}}\right)cos\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)}{\left(\left(\mathcal{A}-2\right)\nu_{2}+\nu_{1}\sqrt{-\mathcal{D}}\right)sin\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)+\left(\left(\mathcal{A}-2\right)\nu_{1}-\nu_{2}\sqrt{-\mathcal{D}}\right)cos\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)} \\ & \frac{\left(2\xi_{2}(-\mathcal{A}+\mathcal{B}+1)^{2}\right)}{\sqrt{\xi_{1}^{2}\left(\mathcal{A}^{2}-4\mathcal{B}\right)^{2}}} \\ & \left(\frac{\left(\mathcal{A}\nu_{2}+\nu_{1}\sqrt{-\mathcal{D}}\right)sin\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)+\left(\mathcal{A}\nu_{1}-\nu_{2}\sqrt{-\mathcal{D}}\right)cos\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)}{\left(\left(\mathcal{A}-2\right)\nu_{2}+\nu_{1}\sqrt{-\mathcal{D}}\right)sin\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)+\left(\left(\mathcal{A}-2\right)\nu_{1}-\nu_{2}\sqrt{-\mathcal{D}}\right)cos\left(\frac{\sqrt{-\mathcal{D}}}{2}\right)}\right)^{2}\Biggr) \\ & e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}. \end{align}$

(3.10)

4. Application of modified $\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}$ -expansion approach

Since the highest-order nonlinear term and the highest-order derivative term are balanced according to the homogenous balance principle in Eq (3.6), we know that the balance number is $\aleph = 2$ . Therefore, we have the following:

$\begin{equation} \mathcal{M}\left(\omega_{1}\right) = \mathcal{F}_{0}+\mathcal{F}_{1}\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}+\mathcal{F}_{2}\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}\right)^{2}+\frac{\mathcal{S}_{1}}{\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}}+\frac{\mathcal{S}_{2}}{\left(\frac{\mathcal{G}^{\prime}}{\mathcal{G}^{2}}\right)^{2}}. \end{equation}$

(4.1)

Inserting Eq (4.1) with aid of Eq (2.7) into Eq (3.6), and following the same procedure as earlier, we obtain the following:

$\begin{align} \mathcal{F}_{1} = -\frac{2\varPsi\xi_{1}^{2}\psi}{4\rho\varPsi\xi_{1}^{2}\varpi+\rho\xi_{1}^{2}\left(-\psi^{2}\right)+1}, \mathcal{F}_{2} = -\frac{2\varPsi^{2}\xi_{1}^{2}}{4\rho\varPsi\xi_{1}^{2}\varpi+\rho\xi_{1}^{2}\left(-\psi^{2}\right)+1}, \\ \mathcal{S}_{1} = 0, \mathcal{S}_{2} = 0, \xi_{2} = \frac{\xi_{1}^{3}\left(4\varPsi\varpi-\psi^{2}\right)}{4\rho\varPsi\xi_{1}^{2}\varpi+\rho\xi_{1}^{2}\left(-\psi^{2}\right)+1}, \mathcal{F}_{0} = -\frac{2\varPsi\xi_{1}^{2}\varpi}{\rho\xi_{1}^{2}\left(4\varpi\varPsi-\psi^{2}\right)+1}. \end{align}$

(4.2)

Putting the values of the parameters presented in Eq (4.1) into Eq (3.6) and making use of Eqs (2.8) and (2.9), we obtain the following exact solutions.

Family 1. If $\varPsi\varpi > 0$ and $\psi = 0,$ then we have the following:

$\begin{equation} \mathcal{M}\left(\omega_{1}\right) = \left(-\frac{\left(2\varPsi^{2}\xi_{1}^{2}\right)\left(\frac{\sqrt{\varPsi\varpi}\left(p_{1}cos\left(\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{2}sin\left(\omega_{1}\sqrt{\varPsi\varpi}\right)\right)}{\varpi\left(p_{2}cos\left(\omega_{1}\sqrt{\varPsi\varpi}\right)-p_{1}sin\left(\omega_{1}\sqrt{\varPsi\varpi}\right)\right)^{2}}\right)^{2}}{4\varPsi\xi_{1}^{2}\varpi\rho+1}-\frac{2\varPsi\xi_{1}^{2}\varpi}{4\varPsi\xi_{1}^{2}\varpi\rho+1}\right)e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}. \end{equation}$

(4.3)

Family 2. If $\varPsi\varpi < 0$ and $\psi = 0,$ then we have the following:

$\begin{equation*} \mathcal{M}\left(\omega_{1}\right) = \left(-\frac{\left(2\varPsi^{2}\xi_{1}^{2}\right)\left(-\frac{\sqrt{\varPsi\varpi}\left(p_{1}sinh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{1}cosh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{2}\right)}{\left(\varpi\left(p_{1}sinh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{1}cosh\left(2\omega_{1}\sqrt{\varPsi\varpi}\right)+p_{2}\right)\right)^{2}}\right)^{2}}{4\varPsi\xi_{1}^{2}\varpi\rho+1}-\frac{2\varPsi\xi_{1}^{2}\varpi}{4\varPsi\xi_{1}^{2}\varpi\rho+1}\right) \label{eq:25}\\ e^{\tau\mathcal{P}\left(t\right)-\frac{1}{2}\tau^{2}t}.\nonumber \end{equation*}$

5. Graphical analysis

This portion of the present work graphically visualize the obtained solutions and presents some physical interpretations and discussions on the obtained results. In , solution (3.9) with particular values (i.e, $\nu_{1} = 5, \; \nu_{2} = -.5, \; \xi_{1} = -.2, \; \xi_{2} = -1, \; \mathcal{A} = 3, \; \mathcal{B} = 2.6, \; \tau = 0, \; \mathcal{P} = 0$ ) is visualized. In , the value of $\beta$ is varied while the noise intensity $\tau$ is considered as zero. The $\beta$ is used as 1, 0.9, and 0.8 for subfigures (1a, 1d), (1b, 1e), and (1c, 1f), respectively. Here, we observed the dark soliton wave, where we see that as the fractional order decreases when the wave separation is increased.

Figure 1. The visualization of exact solution (2.2) with

$\nu_{1} = .5, \nu_{2} = 1, \xi_{1} = -.7, \xi_{2} = .5, p_{1} = 2, p_{2} = 1, \mathcal{A} = -3, \mathcal{B} = 0, \tau = 0, \mathcal{P} = 0, \tau = 0$ and varying

$\beta$ .

DownLoad: Full-Size Img PowerPoint

Furthermore, shows the dynamics of the exact solution (2.2) by varying the noise intensity while keeping the $\beta = 0.95$ . Other parameters are used for the simulation of . The $\tau$ is used as 0.1, 0.4, and 0.9 for subfigures (2a, 2d), which is (2b, 2e), and (2c, 2f), respectively. In Figure 2, one can observe the affects of noise on the dynamics of the solution, which is simulated here. Furthermore, the dynamics of the exact solution (3.10) are visualized in and by varying $\beta$ and $\tau$ , respectively. In the simulation of these figures, the parameters are selected in the form $\nu_{1} = .5, \nu_{2} = 1, \xi_{1} = -.7, \xi_{2} = .5, p_{1} = 2, p_{2} = 1, \mathcal{A} = -4, \mathcal{B} = 0, \tau = 0, \mathcal{P} = 0$ ; alternatively in , the $\tau$ is considered as zero. and in . the $\beta$ is fixed as $0.95$ . The $\beta$ is used as 1, 0.9, and 0.8 for subfigures (3a, 3d), (3b, 3e), and (3c, 3f), respectively. Similarly, $\tau$ is used as 0.2, 0.5, and 0.8 for subfigures (4a, 4d), (4b, 4e), and (4c, 4f), respectively. Here, we observed the interaction of the bright wave with a kink wave, where the amplitude of the bright wave decreases as the $\beta$ decreases in the negative region of the spatial coordinate.

Figure 2. The visualization of exact solution (2.2) with

$\nu_{1} = .5, \nu_{2} = 1, \xi_{1} = -.7, \xi_{2} = .5, p_{1} = 2, p_{2} = 1, \mathcal{A} = -3, \mathcal{B} = 0, \mathcal{P} = 0.5, \beta = 0.95.$ and varying

$\tau$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. The visualization of solution

$\nu_{1} = .5, \nu_{2} = 1, \xi_{1} = -.7, \xi_{2} = .5, p_{1} = 2, p_{2} = 1, \mathcal{A} = -4, \mathcal{B} = 0, \mathcal{P} = 0, \tau = 0$ and different values of

$\beta$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. The visualization of solution

$\nu_{1} = .5, \nu_{2} = 1, \xi_{1} = -.7, \xi_{2} = .5, p_{1} = 2, p_{2} = 1, \mathcal{A} = -4, \mathcal{B} = 0, \mathcal{P} = 0, \beta = 0.95$ and different values of

$\tau$ .

DownLoad: Full-Size Img PowerPoint

In , the solution (3.9) with particular values (i.e, $\nu_{1} = 5, \; \nu_{2} = -.5, \; \xi_{1} = -.2, \; \xi_{2} = -1, \; \mathcal{A} = 3, \; \mathcal{B} = 2.6, \; \tau = 0,$ and $\mathcal{P} = 0$ ) is visualized. In , the various values for $\beta$ are considered, while the noise intensity $\tau$ is supposed to be zero. The $\beta$ is considered as 1, 0.95, and 0.9 for subfigures (5a, 5d), (5b, 5e), and (5c, 5f), respectively. Here, we observed the hybrid bright-dark soliton wave, where we see that as the fractional order decreases when then amplitude of the dark solitons increases and the bright soliton is decreases.

Figure 5. The visualization of solution with

$\rho = 1, \varpi = -.1, \xi_{1} = 1, \varPsi = 1, p_{1} = 1, p_{2} = 1, \mathcal{P} = 0, \tau = 0,$ and varying

$\beta$ .

DownLoad: Full-Size Img PowerPoint

Moreover, shows the dynamics of the exact solution (3.9) by varying the noise intensity while keeping the $\beta = 0.95$ . Other parameters are used for the simulation of . The $\tau$ is used as 0.5, 0.6, and 0.9 for subfigures (6a, 6d), (6b, 6e), and (6c, 6f), respectively. In , one can observe the affects of noise on the dynamics of the solution, which is simulated here; it can be seen that the highest and lowest amplitude areas become more random as $\tau$ increases.

Figure 6. The visualization of solution with

$\rho = 1, \varpi = -.1, \xi_{1} = 1, \varPsi = 1, p_{1} = 1, p_{2} = 1, \mathcal{P} = 0, \beta = 0.95$ and varying

$\tau$ .

DownLoad: Full-Size Img PowerPoint

Furthermore, the dynamics of the exact solution (3.10) are visualized in and by varying $\beta$ and $\tau$ , respectively. In the simulation of these figures, the parameters are selected in the form $\rho = 1, \varpi = -.1, \xi_{1} = 1, \varPsi = 1, p_{1} = 1, p_{2} = 1, \mathcal{P} = 0,$ and $\tau = 0$ ; alternatively, in , the $\tau$ is considered as zero, and in , the $\beta$ is fixed as $0.95$ . The $\beta$ is used as 1, 0.9, and 0.8 for subfigures (7a, 7d), (7b, 7e), and (7c, 7f), respectively. Similarly, $\tau$ is used as 0.05, 0.3, and 0.6 for subfigures (8a, 8d), (8b, 8e), and (8c, 4f), respectively. Here, we observed the periodic wave solution, where the amplitude of the periodic waves decreases as the $\beta$ decreases in the negative region of the spatial coordinate. Furthermore, we see that the wave profile behaves more randomly in areas where the amplitude is either low or high. Thus, from these analyses, it can be noticed that the obtained results are more generalized than the solutions reported in previous papers. Indeed, when the $BD$ operators equals one, the solution converges to the stochastic integer order solutions. If the intensity of the white noise is zero, then the solutions converge to a deterministic case. When $\beta = 1$ and $\tau = 0$ , the obtained solutions converge to the determinsitic case.

Figure 7. The visualization of solution with

$\rho = 1, \varpi = -.1, \xi_{1} = 1, \varPsi = 1, p_{1} = 1, p_{2} = 1, \mathcal{P} = 0, \tau = 0$ and varying

$\beta$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. The visualization of solution with

$\rho = 1, \varpi = -.1, \xi_{1} = 1, \varPsi = 1, p_{1} = 1, p_{2} = 1, \mathcal{P} = 0, \beta = 0.95$ and varying

$\tau$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions

This study has explored the stochastic BBME with the BD, thereby incorporating multiplicative noise in the Itô sense. We have derived various analytical soliton solutions for these equations by utilizing two distinct expansion methods, both within the framework of beta derivatives. A fractional multistep transformation was employed to convert the equations into nonlinear forms with respect to an independent variable. After performing algebraic manipulations, the solutions were found to be trigonometric and hyperbolic trigonometric functions. Our analysis demonstrated that the wave behavior was influenced by the fractional-order derivative in the proposed equations, thus providing deeper insights into the wave composition as the fractional order increases or decreases. Additionally, we examined the effect of white noise on the propagation of wave solutions. This study has underscored the computational robustness and adaptability of the proposed approach to investigate various phenomena in the physical sciences and engineering.

Author contributions

Conceptualization: M.S.D.S. Methodology: K.A.A. Software: S.S. Validation: A.K. Formal analysis: A.K. Investigation: M.H. Writing-original draft preparation: K.A.A. Writing-review and editing: H.S., A.M.

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1). The authors wish to extend their sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah.

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	Farmer JT, Howell JR (1998) Comparison of monte carlo strategies for radiative transfer in participating media. Adv Heat Transfer 31: 333-429. doi: 10.1016/S0065-2717(08)70243-0
[2]	Evans TM, Urbatsch TJ, Lichtenstein H, et al. (2003) A residual monte carlo method for discrete thermal radiative diffusion. J Comput Phys 189: 539-556. doi: 10.1016/S0021-9991(03)00233-X
[3]	Zhou HC, Chen DL, Cheng Q (2004) A new way to calculate radiative intensity and solve radiative transfer equation through using the monte carlo method. J Quant Spectrosc Radiat Transfer 83: 459-481. doi: 10.1016/S0022-4073(03)00031-1
[4]	Xia XL, Ren DP, Tan HP (2006) A curve monte carlo method for radiative heat transfer in absorbing and scattering gradient-Index medium. Numer Heat Transfer, Part B 50: 181-192. doi: 10.1080/10407790500459387
[5]	Wang A, Modest MF (2007) Spectral monte carlo models for nongray radiation analyses in inhomogeneous participating media. Int J Heat Mass Transfer 50: 3877-3889. doi: 10.1016/j.ijheatmasstransfer.2007.02.018
[6]	Mazumder S (2019) Application of a variance reduction technique to surface-to-surface monte carlo radiation exchange calculations. Int J Heat Mass Transfer 131: 424-431. doi: 10.1016/j.ijheatmasstransfer.2018.11.050
[7]	Hsu PF, Farmer JT (1997) Benchmark solutions of radiative heat transfer within nonhomogeneous participating media using the monte carlo and YIX method. J Heat Transfer 119: 185-188. doi: 10.1115/1.2824087
[8]	Ruan LM, Tan HP (2002) Solutions of radiative heat transfer in three-dimensional inhomogeneous, scattering media. J Heat Transfer 124: 985-988. doi: 10.1115/1.1495519
[9]	Hohn RH (1998) The monte carlo method in tadiative heat transfer. J Heat Transfer 120: 547. doi: 10.1115/1.2824310
[10]	Lu Z, Zhang D (2003) On importance sampling monte carlo approach to uncertainty analysis for flow and transport in porous media. Adv Water Resour 26: 1177-1188. doi: 10.1016/S0309-1708(03)00106-4
[11]	Wang X (2000) Improving the rejection sampling method in quasi-monte carlo methods. J Comput Appl Math 114: 231-246. doi: 10.1016/S0377-0427(99)00194-6
[12]	Peplow DE, Verghese K (2000) Differential sampling for the monte carlo practitioner. Prog Nucl Energy 36: 39-75. doi: 10.1016/S0149-1970(99)00024-4
[13]	Xia XL, Ren DP, Dong SK, et al. (2004) Radiative heat flux characteristics of coupled heat transfer in tubes and comparison of random sampling modes. J Eng Thermophys 25: 287-289.
[14]	Walters DV, Buckius RO (1992) Rigorous development for radiation heat transfer in nonhomogeneous absorbing, emitting and scattering media. Int J Heat Mass Transfer 35: 3323-3333. doi: 10.1016/0017-9310(92)90219-I
[15]	Cherkaoui M, Dufresne JL, Fournier R, et al. (1996) Monte carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation. J Heat Transfer 118: 401-407. doi: 10.1115/1.2825858
[16]	Cherkaoui M, Dufresne JL, Fournier R, et al. (1998) Radiative net exchange formulation within one-dimensional gas enclosures with reflective surfaces. Trans Am Soc Mech Eng 120: 275-278.
[17]	De Lataillade A, Dufresne JL, El Hafi M, et al. (2002) A net-exchange monte carlo approach to radiation in optically thick systems. J Quant Spectrosc Radiat Transfer 74: 563-584. doi: 10.1016/S0022-4073(01)00272-2
[18]	Eymet V, Fournier R, Blanco S, et al. (2005) A boundary-based net-exchange monte carlo method for absorbing and scattering thick media. J Quant Spectrosc Radiat Transfer 19: 27-46. doi: 10.1016/j.jqsrt.2004.05.049
[19]	Lionel Tessé, Francis Dupoirieux, Bernard Zamuner, et al. (2002) Radiative transfer in real gases using reciprocal and forward monte carlo methods and a correlated-k approach. Int J Heat Mass Transfer 45: 2797-2814. doi: 10.1016/S0017-9310(02)00009-1
[20]	Modest MF (2003) Backward monte carlo simulations in radiative heat transfer. J Heat Transfer 125: 57-62. doi: 10.1115/1.1518491
[21]	Lu XD, Hsu PF (2004) Reverse monte carlo method for transient radiative transfer in participating media. J Heat Transfer 126: 621-627. doi: 10.1115/1.1773587
[22]	Tan HP, Shuai Y, Dong SK (2005) Analysis of rocket plume base heating by using backward monte-carlo method. J Thermophysics Heat Transfer 19: 125-127. doi: 10.2514/1.10519
[23]	Shuai Y, Dong SK, Tan HP (2005) Simulation of the infrared radiation characteristics of high-temperature exhaust plume including particles using the backward monte carlo method. J Quant Spectrosc Radiat Transfer 195: 231-240. doi: 10.1016/j.jqsrt.2004.11.001
[24]	Lu X, Hsu PF (2005) Reverse monte carlo simulations of light pulse propagation in nonhomogeneous media. J Quant Spectrosc Radiat Transfer 93: 349-367. doi: 10.1016/j.jqsrt.2004.08.029
[25]	Kovtanyuk AE, Botkin ND, Hoffmann KH (2012) Numerical simulations of a coupled radiative-conductive heat transfer model using a modified monte carlo method. Int J Heat Mass Transfer 55: 649-654. doi: 10.1016/j.ijheatmasstransfer.2011.10.045
[26]	Soucasse L, Rivière P, Soufiani A (2013) Monte carlo methods for radiative transfer in quasi-isothermal participating media. Eurotherm Semin Comput Thermal Radiat Participating Media IV 128: 34-42.
[27]	Ruan LM, Qi H, Liu LH, et al. (2004) The radiative transfer in cylindrical medium and partition allocation method by overlap regions. J Quant Spectrosc Radiat Transfer 86: 343-352. doi: 10.1016/j.jqsrt.2003.08.011
[28]	Shuai Y, Zhang HC, Tan HP (2008) Radiation symmetry test and uncertainty analysis of monte carlo method based on radiative exchange factor. J Quant Spectrosc Radiat Transfer 109: 1281-1296. doi: 10.1016/j.jqsrt.2007.10.001
[29]	Yang WJ (1995) Radiative heat transfer by the monte carlo method. Adv Heat Transf 27: 45-91. doi: 10.1016/S0065-2717(08)70315-0
[30]	Dupoirieux F, Tessé L, Avila S, et al. (2006) An optimized reciprocity monte carlo method for the calculation of radiative transfer in media of various optical thicknesses. Int J Heat Mass Transfer 49: 1310-1319. doi: 10.1016/j.ijheatmasstransfer.2005.10.009
[31]	Ruan LM, Tan HP, Yan YY (2002) A monte carlo method applied to the medium with nongray absorbing-emitting-anisotropic scattering particles and gray approximation. Numer Heat Transfer, Part A 42: 253-268. doi: 10.1080/10407780290059530
[32]	Howell JR, Mengüç MP, Daun K, et al. (2020) Thermal radiation heat transfer. Boca Raton: CRC Press. doi: 10.1201/9780429327308

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