The new stochastic solutions for the fractional stochastic (3+1)-model of fluid with gas bubbles

Abstract

This study investigates the fractional (3 + 1)-dimensional equation for fluids with gas bubbles through conformable fractional operator, incorporating a noise term through the modified Sardar sub-equation method. This equation describes nonlinear wave propagation in compressible fluids containing distributed gas microbubbles, which is ubiquitous in biological tissues, petroleum engineering, acoustics, and fluid-structure interactions. The fractional derivative incorporates memory effects and anomalous dispersion, enhancing the model’s accuracy in reflecting real-world phenomena. The mathematical system is transformed into a nonlinear ordinary differential equation by applying a complex traveling wave transformation. We provide some innovative stochastic solutions through free physical parameters for the suggested model via Brownian motion process. The novel stochastic solutions investigated how noise affects frequencies in nonlinear systems such as biomedical ultrasonography, nuclear science, fluid-structure interaction, and more. To demonstrate the behavior of the offered stochastic solutions, a variety of profile graphs were created utilizing the Matlab release’s capabilities. The proposed methodology can ultimately be modified for use with a range of other models in applied science.

Introduction

In recent years, mathematicians and physicists have focused increasingly on the study of complex wave propagation, which is represented by a certain type of nonlinear fractional partial differential equations (NFPDEs), such as a fractal modification of the Boussinesq equation, ¹ fractional oscillation equation arising from a micro/nanobeam-based micro-electromechanical system under the magnetic force, ² fractal oscillation model for a doubly clamped micro-electromechanical system, ³ fractional Phi-4 and space-time fractional simplified MCH equations, ⁴ fractional symmetric regularized long wave equation, ⁵ etc. These equations are regarded as extensions of nonlinear partial differential equations (NPDEs) as they use fractional derivatives in space and time rather than integer-order derivatives. These equations go beyond classical models by integrating nonlocal memory and anomalous transport behavior, making them suited for current complex systems in science and engineering. Fractional derivatives frequently serve as a modeling instrument to achieve enhanced accuracy in applications represented by differential equations. The NFPDEs have emerged as the suitable mathematical framework for modeling and analyzing natural phenomena across various scientific disciplines, owing to the significant emphasis placed on their interpretation. There are, however, no standardized approaches capable of yielding all exact solutions to NFPDEs, as these equations exhibit various forms and characteristics that pose unique challenges in the pursuit of their exact analytical solutions. Several effective and powerful techniques have been documented that they can identify analytical solutions by utilizing various characteristics, such as variational iteration method, ⁶ homotopy perturbation method, ⁷ improved modified extended tanh-function method, ⁸ planar dynamical systems approach, ⁹ among others, as stated in the references to those research.

Numerous significant definitions exist for various types of fractional derivatives, including Grunwald–Letnikov fractional derivative, ¹⁰ He’s fractional derivative, ¹¹ local fractional derivative, ¹² Riemann–Liouville fractional derivative, ¹³ etc. The conformable fractional derivative has attracted interest because it keeps closer ties to the classical derivative while extending it to fractional orders. ¹⁴ Unlike other definitions that may lose standard calculus properties, Khalil’s derivative preserves linearity, product rule, chain rule, and other essential features. Its simplicity, locality, and physical interpretability make it especially appealing for differential equations, dynamical systems, and applied modeling. ¹⁴

Definition 1

[14] Let $\mathfrak{Q}:(0,\infty )\to \mathbb{R}$ be a function, then the α order of the conformable derivative of it is

$D_t^{\alpha }(\mathfrak{Q}(t))=\underset{\kappa \to 0}{\lim }\frac{\mathfrak{Q}(t+\kappa t^{1-\alpha })-\mathfrak{Q}(t)}{\kappa },t>0,0<\alpha \le 1.$

This definition satisfies:

(i) $D_t^{\alpha }(K_1{\mathfrak{Q}}_1+K_2{\mathfrak{Q}}_2)=K_1{D}_t^{\alpha }({\mathfrak{Q}}_1)+K_2{D}_t^{\alpha }({\mathfrak{Q}}_2),K_1;K_2\in \mathbb{R}$ ,

(ii) $D_t^{\alpha }(t^{\iota })=\iota t^{\iota -\alpha },\iota \in \mathbb{R}$ ,

(iii) $D_t^{\alpha }({\mathfrak{Q}}_1{\mathfrak{Q}}_2)={\mathfrak{Q}}_2{D}_t^{\alpha }({\mathfrak{Q}}_1)+{\mathfrak{Q}}_1{D}_t^{\delta }({\mathfrak{Q}}_2)$ ,

(vi) $D_t^{\alpha }({\mathfrak{Q}}_1/{\mathfrak{Q}}_2)={\mathfrak{Q}}_2{D}_t^{\alpha }({\mathfrak{Q}}_1)-{\mathfrak{Q}}_1{D}_t^{\alpha }({\mathfrak{Q}}_2)/{\mathfrak{Q}}_2^2$ .

For further information regarding the attributes of a conformable fractional definition, see Ref. 14.

The NPDEs and NFPDEs are recommended for researching and assessing real-world models since their behavior is impacted by their preceding states.^15,16 They offer more comprehensive and precise models applicable to various physical, fluids, and engineering systems, particularly those that encompass nonlocal interactions, time delays, or multiscale phenomena. For instance, the studies in Refs. 17 and 18 present refined differential equation-based approaches that are successfully used to address real-world problems involving fluids and granular materials. Nonlinear wave propagation in compressible fluids embedded with scattered gas microbubbles, which are frequently seen in biological tissues, acoustics, and fluid-structure interactions, is modeled using the fractional (3 + 1)-dimensional equation for fluids with gas bubbles. The fractional derivative incorporates memory effects or anomalous dispersion, increasing the model’s accuracy in capturing real-world occurrences. The space fractional stochastic (3 + 1)-dimensional wave equation for fluids with gas bubbles that we examine in this paper has the following form^19,20:

$D_x^{\alpha }\left(Q_t+{\lambda }_1{D}_x^{\alpha }Q+{\lambda }_2{D}_{\mathit{xxx}}^{\alpha }Q+{\lambda }_{3}Q{D}_x^{\alpha }Q\right)+{\lambda }_4{D}_{yy}^{\alpha }Q+{\lambda }_5{D}_{zz}^{\alpha }Q=\sigma B_t{D}_x^{\alpha }Q,0<\alpha \le 1,$

(1)

where Q denotes mixture velocity function of the scaled spatial coordinates x, y, z, characterizing the transverse y & z perturbations of waves propagating in the x direction; t is the temporal coordinate; $and D_x^{\alpha }$ denotes the conformable fractional-order derivative of the fractional order α. The parameter σ denotes the noise intensity; while B(t) is the Brownian motion. The parameters λ₁ is the bubble liquid viscosity and λ₂ is the bubble liquid dispersion, λ₃ is the bubble liquid nonlinearity, λ₄ is the z transverse perturbation, and λ₅ is the transverse perturbation. The authors considered the bifurcation of the analytical solutions for model (1). ¹⁹ Elbrolosy et al. introduced some new stochastic solutions for model (1) based on the extension of the auxiliary function method. ²⁰

One common example of a stochastic process that has the characteristics of both a Markov and a martingale process is the Brownian motion (Wiener) process. ²¹ This process is necessary for explaining stochastic processes since it is the basis of stochastic calculus. It is important in stochastic dynamics and stochastic processes and provides a vital basis for the validation of molecular dynamics models. Moreover, the stochastic disruptions that take place in the physical environment have a number of unidentified sources. It is commonly known that the latest advancements in stochastic calculus made possible by SPDEs will provide a foundation for comprehensively modeling complex equations.^22,23 More than anybody else, mathematicians continue to feel comfortable using stochastic processes and SPDEs in natural models. The following definitions present the Brownian motion process and explore some of its main features.

Definition 2

[14] Brownian motion refers to a one-dimensional continuous time stochastic process {B(t)}_t≥0 that meets the following conditions:

1. B(t) is continuous for t ≥ 0;

2. For 0 ≤ s < t < u < k, B(t) − B(s) and B(k) − B(u) are independent;

3. For all 0 ≤ t₁ < t₂ B (t₂) − B (t₁) has a normal distribution N (0, t₂ − t₁);

4. B (0) = 0.

Our objective is to apply the modified Sardar sub-equation (MSSE) method ²⁴ to the space fractional stochastic (3 + 1)-dimensional wave equation concerning fluids containing gas bubbles, incorporating multiplicative noise in the Itô sense. Specifically, we introduce novel stochastic solutions based on physical characteristics. The presented solutions have significant potential applications in fields such as biological tissues, petroleum engineering, acoustics, and fluid-structure interactions. The proposed method is straightforward, efficient, and accurate. Additionally, we explore the nonlinear dynamic behavior of selected stochastic solutions. A particularly noteworthy aspect is the demonstration of how the influence of noise affects these proposed solutions. To the best of our knowledge, this is the first study to utilize the suggested approach to tackle the fractional stochastic (3 + 1)-dimensional wave equation for fluids with gas bubbles under the influence of multiplicative noise.

The remaining sections of this article are planned as follows. Section 2 offers a brief overview of the modified Sardar sub-equation (MSSE) approach. Sec. 3 introduces some new solitary waves for the space fractional stochastic (3 + 1)-dimensional wave equation for fluids with gas bubbles. Section 4 provides a physical interpretation of the obtained results. Lastly, a conclusion on the obtained findings is provided in Section 5.

An explanation of the method

The modified Sardar sub-equation (MSSE) technique is described here. ²⁴ Consider the NFPDE given as follows:

$F_1(Q,Q_t,D_x^{\alpha }Q,D_y^{\alpha }Q,D_z^{\alpha }Q,D_{xt}^{2\alpha }Q,D_{xx}^{2\alpha }Q,D_{yy}^{2\alpha }Q,\dots )=0,$

(2)

where F₁ is a polynomial in Q (x, y, z, t). Then, applying the transformation

$Q(x,t)=q(\zeta )e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\gamma t,$

(3)

to convert equation (2) to the following ODE

$F_2(q,q^{\prime },q^{″},q‴,\dots )=0,$

(4)

q′ = dq/dζ. The MSSE approach gives its solution in the following form:

$q(\zeta )=\sum _{n=0}^K{r}_n{H}^n(\zeta ),$

(5)

where r _n ∀n are calculated later. The value of K is obtained by balancing between the nonlinear term with the highest derivative. The function H is evaluated by

$H^{\prime }(\zeta )=\sqrt{s_0+s_1{H}^2+s_2{H}^4}.$

(6)

The constants s _j ∀ j are given under some restrictions. These conditions and the function H are given in the following cases.

1. If s₀ = 0, s₁ > 0, s₂ ≠ 0. Then,

$H(\zeta )=\left\{\begin{array}{c}\pm \sqrt{-\frac{s_1}{s_2}}sech(\sqrt{s_1}(\zeta +k)),\hfill \\ \pm \sqrt{\frac{s_1}{s_2}}csch(\sqrt{s_1}(\zeta +k)).\hfill \end{array}\right.$

• 2. If s₀ = 0, s₁ > 0, s₂ = ±4l₁l₂. Then,

$H(\zeta )=\pm \frac{4l_1\sqrt{s_1}}{(4l_1^2-s_2)cosh(\sqrt{s_1}(\zeta +k))\pm (4l_1^2+s_2)sinh(\sqrt{s_1}(\zeta +k))}.$

• 3. If $s_0=\frac{s_1^2}{4s_2},s_1<0,s_2>0$ , with l₁ and l₂ are arbitrary parameters. Then,

$H(\zeta )=\left\{\begin{array}{c}\pm \sqrt{-\frac{s_1}{2s_2}}tanh\left(\sqrt{-\frac{s_1}{2}}(\zeta +k)\right),\hfill \\ \pm \sqrt{-\frac{s_1}{2s_2}}coth\left(\sqrt{-\frac{s_1}{2}}(\zeta +k)\right),\hfill \\ \pm \sqrt{-\frac{s_1}{8s_2}}\left(tanh\left(\sqrt{-\frac{s_1}{8}}(\zeta +k)\right)+coth\left(\sqrt{-\frac{s_1}{8}}(\zeta +k)\right)\right),\hfill \\ \pm \sqrt{-\frac{s_1}{8s_2}}\left(\frac{\pm \sqrt{l_1^2+l_2^2}-l_{1}cosh(\sqrt{-2s_1}(\zeta +k))}{l_{1}sinh(\sqrt{-2s_1}(\zeta +k))+l_2}\right).\hfill \end{array}\right.$

• 4. If s₀ = 0, s₁ < 0, s₂ ≠ 0. Then,

$H(\zeta )=\left\{\begin{array}{c}\pm \sqrt{-\frac{s_1}{s_2}}sec(-\sqrt{s_1}(\zeta +k)),\hfill \\ \pm \sqrt{-\frac{s_1}{s_2}}csc(\sqrt{-s_1}(\zeta +k)).\hfill \end{array}\right.$

• 5. If $s_0=\frac{s_1^2}{4s_2},s_1>0,s_2>0$ , with $l_1^2-l_2^2$ where l₁ and l₂ are arbitrary parameters. Then,

$H(\zeta )=\left\{\begin{array}{c}\pm \sqrt{\frac{s_1}{2s_2}}tan\left(\sqrt{\frac{s_1}{2}}(\zeta +k)\right),\hfill \\ \pm \sqrt{\frac{s_1}{2s_2}}cot\left(\sqrt{\frac{s_1}{2}}(\zeta +k)\right),\hfill \\ \pm \sqrt{\frac{s_1}{2s_2}}\left(tan\left(\sqrt{2s_1}(\zeta +k)\right)\pm sec\left(\sqrt{2s_1}(\zeta +k)\right)\right),\hfill \\ \pm \sqrt{\frac{s_1}{8s_2}}\left(tan\left(\sqrt{\frac{s_1}{8}}(\zeta +k)\right)-cot\left(\sqrt{\frac{s_1}{8}}(\zeta +k)\right)\right),\hfill \\ \pm \sqrt{\frac{s_1}{2s_2}}\left(\frac{\pm \sqrt{l_1^2-l_2^2}-l_{1}cos(\sqrt{2s_1}(\zeta +k))}{l_{1}sin(\sqrt{2s_1}(\zeta +k))+l_2}\right),\hfill \\ \pm \sqrt{\frac{s_1}{2s_2}}\left(\frac{cos(\sqrt{2s_1}(\zeta +k))}{sin(\sqrt{2s_1}(\zeta +k))\pm 1}\right).\hfill \end{array}\right.$

Next, put equations (5) and (6) into equation (4) to obtain an equation. Using the coefficients of the function H ⁱ ∀i yields a system of equations whose solutions are the values of r _n ∀n.

Mathematical analysis

Using the wave transformation^19,20:

$Q(x,y,z,t)=q(\zeta )e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\gamma t,$

(7)

where c₁, c₂, c₃, γ are constants. Substituting equation (7) into equation (1) and using the following:

$Q_t=\gamma q^{\prime }+\sigma q{B}_t{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(8)

$\begin{array}{c}{D}_x^{\alpha }{Q}_t=(c_1\gamma q^{″}+\sigma c_1{q}^{\prime })e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},D_x^{\alpha }Q=c_1{q}^{\prime }{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \\ D_{xx}^{\alpha }Q=c_1^2{q}^{″}{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},D_{\mathit{xxxx}}^{\alpha }Q=c_1^4{q}^{⁗}{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \\ D_{yy}^{\alpha }Q=c_2^2{q}^{″}{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},D_{zz}^{\alpha }Q=c_3^2{q}^{″}{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(9)

Putting equation (9) into equation (1), gives

${\lambda }_2{c}_1^4{q}^{⁗}+(c_1\gamma +{\lambda }_1{c}_1^2+{\lambda }_4{c}_2^2+{\lambda }_5{c}_3^2)q^{″}+{\lambda }_3{c}_1^2{(q{q}^{\prime })}^{\prime }{e}^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t}=0.$

(10)

Taking expectation Ξ(.), yields

${\lambda }_2{c}_1^4{q}^{⁗}+(c_1\gamma +{\lambda }_1{c}_1^2+{\lambda }_4{c}_2^2+{\lambda }_5{c}_3^2)q^{″}+{\lambda }_3{c}_1^2{(q{q}^{\prime })}^{\prime }{e}^{-\frac{1}{2}{\sigma }^{2}t}\mathrm{\Xi }(e^{\sigma B(t)})=0.$

(11)

Since $\mathrm{\Xi }(e^{\sigma B(t)})=e^{1/2{\sigma }^{2}t}$ , equation (11) becomes

${\lambda }_2{c}_1^4{q}^{⁗}+(c_1\gamma +{\lambda }_1{c}_1^2+{\lambda }_4{c}_2^2+{\lambda }_5{c}_3^2)q^{″}+{\lambda }_3{c}_1^2{(q{q}^{\prime })}^{\prime }=0.$

(12)

Integrating equation (12) twice and removing the integral constant results in

${\lambda }_2{c}_1^4{q}^{″}+(c_1\gamma +{\lambda }_1{c}_1^2+{\lambda }_4{c}_2^2+{\lambda }_5{c}_3^2)q+\frac{{\lambda }_3{c}_1^2}{2}{q}^2=0.$

(13)

${\mathrm{\Gamma }}_1{q}^{″}+{\mathrm{\Gamma }}_2{q}^2+{\mathrm{\Gamma }}_{3}q=0,$

(14)

where

${\mathrm{\Gamma }}_1={\lambda }_2{c}_1^4,{\mathrm{\Gamma }}_2=\frac{{\lambda }_3{c}_1^2}{2},{\mathrm{\Gamma }}_3=c_1\gamma +{\lambda }_1{c}_1^2+{\lambda }_4{c}_2^2+{\lambda }_5{c}_3^2.$

(15)

Taking the balance between q″ and q² gives K = 2. Hence, equation (5) becomes

$q(\zeta )=r_0+r_{1}H(\zeta )+r_2{H}^2(\zeta ).$

(16)

Next, substitute equation (16) along with equation (6) into equation (13) to have an equation. Then, taking the coefficients of H ⁱ ∀i gives a system of algebraic equations. The solutions of this system are given in the following cases.

If s₀ = 0, s₁ > 0, s₂ ≠ 0, then there are two families of the solutions of the algebraic system. These solutions are given in the following families as follows.

The first family is given by

$r_0=r_1=0,r_2=-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3},\gamma =\frac{-c_1^2{\lambda }_1-4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}.$

(17)

Consequently, the traveling wave solutions of equation (1) are expressed as follows.

$q_1(\zeta )=\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}sec{h}^2(\sqrt{s_1}(\zeta +k)),$

$q_2(\zeta )=-\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}csc{h}^2(\sqrt{s_1}(\zeta +k)).$

Thus,

$Q_1(x,y,z,t)=\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}sec{h}^2(\sqrt{s_1}(\zeta +k))e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(18)

$Q_2(x,y,z,t)=-\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}csc{h}^2(\sqrt{s_1}(\zeta +k))e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(19)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1-4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

The second family is given by

$r_0=-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3},r_1=0,r_2=-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3},\gamma =\frac{-c_1^2{\lambda }_1+4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}.$

(20)

Hence, the traveling wave solutions of equation (1) are expressed as follows.

$q_3(\zeta )=-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}+\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}sec{h}^2(\sqrt{s_1}(\zeta +k)),$

$q_4(\zeta )=-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}-\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}csc{h}^2(\sqrt{s_1}(\zeta +k)).$

Thus,

$Q_3(x,y,z,t)=\left(-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}+\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}sec{h}^2(\sqrt{s_1}(\zeta +k))\right)e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(21)

$Q_4(x,y,z,t)=\left(-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}-\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}csc{h}^2(\sqrt{s_1}(\zeta +k))\right)e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(22)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1+4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

If s₀ = 0, s₁ > 0, s₂ = ±4l₁l₂, then there are two families of the solutions of the algebraic system. The first family is given by the same solutions presented in equation (17). Therefore, the traveling wave solutions of equation (1) are given by

$q_{\mathrm{5,6}}(\zeta )=-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3}{\left(\frac{4l_1\sqrt{s_1}}{(4l_1^2-s_2)cosh(\sqrt{s_1}(\zeta +k))\pm (4l_1^2+s_2)sinh(\sqrt{s_1}(\zeta +k))}\right)}^2.$

Thus,

$\begin{array}{ll}\hfill Q_{\mathrm{5,6}}(x,y,z,t)& =-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3}{\left(\frac{4l_1\sqrt{s_1}}{(4l_1^2-s_2)cosh(\sqrt{s_1}(\zeta +k))\pm (4l_1^2+s_2)sinh(\sqrt{s_1}(\zeta +k))}\right)}^2\hfill \\ \hfill & \times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(23)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1-4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

The second family is given by the same solutions presented in equation (20). Therefore, the traveling wave solutions of equation (1) are given by

$q_{\mathrm{7,8}}(\zeta )=-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3}{\left(\frac{4l_1\sqrt{s_1}}{(4l_1^2-s_2)cosh(\sqrt{s_1}(\zeta +k))\pm (4l_1^2+s_2)sinh(\sqrt{s_1}(\zeta +k))}\right)}^2.$

Thus,

$\begin{array}{ll}\hfill Q_{\mathrm{7,8}}(x,y,z,t)& =\left(-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3}{\left(\frac{4l_1\sqrt{s_1}}{(4l_1^2-s_2)cosh(\sqrt{s_1}(\zeta +k))\pm (4l_1^2+s_2)sinh(\sqrt{s_1}(\zeta +k))}\right)}^2\right)\hfill \\ \hfill & \times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(24)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1+4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

If $s_0=\frac{s_1^2}{4s_2},s_1<0,s_2>0$ , then the solutions of the algebraic system are given as follows.

$\begin{array}{ll}\hfill r_0& =\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2},r_1=0,r_2=-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3},\hfill \\ \hfill \gamma & =\frac{-c_1^2{\lambda }_1\pm \frac{4c_1^2\sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2\left(s_1^2-3s_0{s}_2\right)}}{{\lambda }_3}-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}.\hfill \end{array}$

As a result, the traveling wave solutions of equation (1) can be expressed by

$q_{\mathrm{9,10}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}tan{h}^2\left(\sqrt{-\frac{s_1}{2}}(\zeta +k)\right),$

$q_{\mathrm{11,12}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}cot{h}^2\left(\sqrt{-\frac{s_1}{2}}(\zeta +k)\right),$

$q_{\mathrm{13,14}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\frac{12c_1^2{\lambda }_2{s}_1}{8{\lambda }_3}{\left(tanh\left(\sqrt{-\frac{s_1}{8}}(\zeta +k)\right)+coth\left(\sqrt{-\frac{s_1}{8}}(\zeta +k)\right)\right)}^2,$

$q_{\mathrm{15,16}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\frac{12c_1^2{\lambda }_2{s}_1}{8{\lambda }_3}{\left(\frac{\pm \sqrt{c_1^2+c_2^2}-c_{1}cosh(\sqrt{-2s_1}(\zeta +k))}{c_{1}sinh(\sqrt{-2s_1}(\zeta +k))+c_2}\right)}^2.$

Thus,

$\begin{array}{ll}\hfill Q_{\mathrm{9,10}}(x,y,z,t)& =\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}tan{h}^2\left(\sqrt{-\frac{s_1}{2}}(\zeta +k)\right)\right)\hfill \\ \hfill & \times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(25)

$\begin{array}{ll}\hfill Q_{\mathrm{11,12}}(x,y,z,t)& =\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}cot{h}^2\left(\sqrt{-\frac{s_1}{2}}(\zeta +k)\right)\right)\hfill \\ \hfill & \times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(26)

$\begin{array}{c}{Q}_{\mathrm{13,14}}(x,y,z,t)=\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\right.\hfill \\ \left.\frac{12c_1^2{\lambda }_2{s}_1}{8{\lambda }_3}{\left(tanh\left(\sqrt{-\frac{s_1}{8}}(\zeta +k)\right)+coth\left(\sqrt{-\frac{s_1}{8}}(\zeta +k)\right)\right)}^2\right)\times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(27)

$\begin{array}{c}{Q}_{\mathrm{15,16}}(x,y,z,t)=\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}+\right.\hfill \\ \left.\frac{12c_1^2{\lambda }_2{s}_1}{8{\lambda }_3}{\left(\frac{\pm \sqrt{c_1^2+c_2^2}-c_{1}cosh(\sqrt{-2s_1}(\zeta +k))}{c_{1}sinh(\sqrt{-2s_1}(\zeta +k))+c_2}\right)}^2\right)\times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(28)

where l₁ and l₂ are arbitrary parameters and

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1\pm \frac{4c_1^2\sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2\left(s_1^2-3s_0{s}_2\right)}}{{\lambda }_3}-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

Next, if s₀ = 0, s₁ < 0, s₂ ≠ 0, then the solutions of the algebraic system are given as follows.

The first family is given by the same solutions presented in equation (17). Consequently, the traveling wave solutions of equation (1) are given by

$q_{17}(\zeta )=\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}se{c}^2(\sqrt{-s_1}(\zeta +k)),$

$q_{18}(\zeta )=\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}cs{c}^2(\sqrt{-s_1}(\zeta +k)).$

Thus,

$Q_{17}(x,y,z,t)=\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}se{c}^2(\sqrt{-s_1}(\zeta +k))e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(29)

$Q_{18}(x,y,z,t)=\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}cs{c}^2(\sqrt{-s_1}(\zeta +k))e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(30)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1-4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

The second family is given by the same solutions presented in equation (20). Therefore, the traveling wave solutions of equation (1) are given by

$q_{19}(\zeta )=-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}+\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}se{c}^2(\sqrt{-s_1}(\zeta +k)),$

$q_{20}(\zeta )=-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}+\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}cs{c}^2(\sqrt{-s_1}(\zeta +k)).$

Thus,

$Q_{19}(x,y,z,t)=\left(-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}+\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}se{c}^2(\sqrt{-s_1}(\zeta +k))\right)e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(31)

$Q_{20}(x,y,z,t)=\left(-\frac{8c_1^2{\lambda }_2{s}_1}{{\lambda }_3}+\frac{12c_1^2{\lambda }_2{s}_1}{{\lambda }_3}cs{c}^2(\sqrt{-s_1}(\zeta +k))\right)e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},$

(32)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1+4c_1^4{\lambda }_2{s}_1-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

If $s_0=s_1^2/4s_2,s_1>0,s_2>0$ , then the solutions of the algebraic system are given as follows.

$\begin{array}{ll}\hfill r_0& =\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2},r_1=0,r_2=-\frac{12c_1^2{\lambda }_2{s}_2}{{\lambda }_3},\hfill \\ \hfill \gamma & =\frac{-c_1^2{\lambda }_1\pm \frac{4c_1^2\sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2\left(s_1^2-3s_0{s}_2\right)}}{{\lambda }_3}-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}.\hfill \end{array}$

Hence, the traveling wave solutions of equation (1) are given by

$q_{\mathrm{21,22}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}ta{n}^2\left(\sqrt{\frac{s_1}{2}}(\zeta +k)\right),$

$q_{\mathrm{23,24}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}co{t}^2\left(\sqrt{\frac{s_1}{2}}(\zeta +k)\right),$

$q_{\mathrm{25,26}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}{\left(tan\left(\sqrt{2s_1}(\zeta +k)\right)\pm sec\left(\sqrt{2s_1}(\zeta +k)\right)\right)}^2,$

$q_{\mathrm{27,28}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{8{\lambda }_3}{\left(tan\left(\sqrt{\frac{s_1}{8}}(\zeta +k)\right)-cot\left(\sqrt{\frac{s_1}{8}}(\zeta +k)\right)\right)}^2,$

$q_{\mathrm{29,30}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}{\left(\frac{\pm \sqrt{l_1^2-l_2^2}-l_{1}cos(\sqrt{2s_1}(\zeta +k))}{l_{1}sin(\sqrt{2s_1}(\zeta +k))+l_2}\right)}^2,$

$q_{\mathrm{31,32}}(\zeta )=\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}{\left(\frac{cos(\sqrt{2s_1}(\zeta +k))}{sin(\sqrt{2s_1}(\zeta +k))\pm 1}\right)}^2.$

Thus,

$\begin{array}{ll}\hfill Q_{\mathrm{21,22}}(x,y,z,t)& =\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}ta{n}^2\left(\sqrt{\frac{s_1}{2}}(\zeta +k)\right)\right)\hfill \\ \hfill & \times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(33)

$\begin{array}{ll}\hfill Q_{\mathrm{23,24}}(x,y,z,t)& =\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}co{t}^2\left(\sqrt{\frac{s_1}{2}}(\zeta +k)\right)\right)\hfill \\ \hfill & \times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(34)

$\begin{array}{c}{Q}_{\mathrm{25,26}}(x,y,z,t)=\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\right.\hfill \\ \left.\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}{\left(tan\left(\sqrt{2s_1}(\zeta +k)\right)\pm sec\left(\sqrt{2s_1}(\zeta +k)\right)\right)}^2\right)\times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(35)

$\begin{array}{c}{Q}_{\mathrm{27,28}}(x,y,z,t)=\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\right.\hfill \\ \left.\frac{12c_1^2{\lambda }_2{s}_1}{8{\lambda }_3}{\left(tan\left(\sqrt{\frac{s_1}{8}}(\zeta +k)\right)-cot\left(\sqrt{\frac{s_1}{8}}(\zeta +k)\right)\right)}^2\right)\times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(36)

$\begin{array}{c}{Q}_{\mathrm{29,30}}(x,y,z,t)=\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\right.\hfill \\ \left.\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}{\left(\frac{\pm \sqrt{l_1^2-l_2^2}-l_{1}cos(\sqrt{2s_1}(\zeta +k))}{l_{1}sin(\sqrt{2s_1}(\zeta +k))+l_2}\right)}^2\right)\times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(37)

$\begin{array}{c}{Q}_{\mathrm{31,32}}(x,y,z,t)=\left(\frac{4\left(\pm \sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2{s}_1^2-3c_1^4{\lambda }_2^2{\lambda }_3^2{s}_0{s}_2}-c_1^2{\lambda }_2{\lambda }_3{s}_1\right)}{{\lambda }_3^2}-\right.\hfill \\ \left.\frac{12c_1^2{\lambda }_2{s}_1}{2{\lambda }_3}{\left(\frac{cos(\sqrt{2s_1}(\zeta +k))}{sin(\sqrt{2s_1}(\zeta +k))\pm 1}\right)}^2\right)\times e^{\sigma B(t)-\frac{1}{2}{\sigma }^{2}t},\hfill \end{array}$

(38)

where

$\zeta =\frac{1}{\alpha }(c_1{x}^{\alpha }+c_2{y}^{\alpha }+c_3{z}^{\alpha })+\left(\frac{-c_1^2{\lambda }_1\pm \frac{4c_1^2\sqrt{c_1^4{\lambda }_2^2{\lambda }_3^2\left(s_1^2-3s_0{s}_2\right)}}{{\lambda }_3}-c_2^2{\lambda }_4-c_3^2{\lambda }_5}{c_1}\right)t.$

Results and discussion

The fractional (3 + 1)-dimensional fluid-bubble equation offers a more precise and adaptable framework for examining wave dynamics in bubbly environments. It effectively incorporates critical phenomena, including nonlinearity, dispersion, and long-term memory. This model has significant applications in fields such as engineering, biomedical imaging, and acoustics. This work’s nonlinear wave equation, derived as a fractional derivative, can be a useful model for liquids with gaseous bubbles, which exhibit complex wave interactions with their surroundings. This makes fractional derivatives important for describing memory and nonlocality in fluids, as well as for describing bubbles that contain liquids. The model produces solitary wave solutions with localized waveform structure that can travel without losing form or amplitude, similar to pressure waves in media.

The investigation into stochastic solutions for the fractional stochastic (3 + 1)-model of fluid with gas bubbles is significantly driven by the inherently random and memory-dependent characteristics of bubble dynamics within multiphase media. In real-world applications, such as cavitation, biomedical ultrasound, underwater acoustics, and geophysical flows, the existence of gas bubbles leads to nonlinear resonance, energy transfer, and dissipation processes that cannot be sufficiently represented by purely deterministic models. Fractional derivatives effectively capture the long-memory and hereditary influences of the medium, whereas stochastic perturbations represent the inevitable fluctuations caused by thermal agitation and turbulent background noise. Developing stochastic solutions enables researchers to systematically quantify how random disturbances affect wave amplitude decay, phase changes, stability, and blow-up occurrences in bubbly media. Furthermore, the interaction of fractional damping and stochastic forcing sheds light on the essential thresholds of noise intensity that distinguish stable propagation from instability or resonance-driven amplification. Such analyses are critical for both theoretical comprehension of nonlinear dispersive waves in complex fluids and practical design of acoustic and engineering systems in noisy, heterogeneous settings.

We now analyze the individual impact of noise strength and the fractional derivative order α, as well as their combined effect. Figure 1 illustrates the impact of noise on the solution (18) for integer order (α = 1) and t = 2. The graphic illustrates how the wave amplitude decreases monotonically as the noise level rises. This attenuation happens without a phase shift or reversal in the propagation direction, suggesting that the stochastic perturbation simply affects the amplitude scaling while maintaining the traveling wave’s structural and directional characteristics. Figure 2 illustrates the impact of fractional derivative order α on the solution (18) for noise intensity σ = 0. The graphic shows that while causing a phase shift, raising the order of the fractional derivative maintains the stability of the wave amplitude. Interestingly, the direction of propagation does not change, suggesting that the fractional effects are more likely to appear as spatial or temporal phase modulation than as amplitude attenuation or motion reversal. Figure 3 illustrates the behavior of the stochastic solutions with σ = 1 for large values of time t, namely, t = 11, 12, 13. 14. One observes that the amplitude of the localized traveling wave solution exhibits a monotonic decay with respect to time t. This decay is a direct consequence of the stochastic modulation factor appearing in the solution, which typically takes the form of an exponential damping term $e^{-1/2{\sigma }^{2}t}$ . The stochastic contribution causes an exponential attenuation of the wave amplitude since the exponent is negative and increases linearly in t. Figure 4 shows 3D soliton wave (18) with σ = 0 and α = 1. Figure 5 shows 3D soliton wave (18) with σ = 0 and α = 0.4. Figure 6 shows 3D soliton wave (18) with σ = 1. Figure 7 shows 2D soliton wave (25) with different values of t with phase shift. Figure 8 illustrates the impact of noise on the solution (24) for integer order (α = 1) and t = 2. The figure indicates that the introduction of noise leads to a significant flattening of the wave surface, accompanied by a reduction in its overall strength. This demonstrates that stochastic perturbations act as a smoothing mechanism, diminishing the sharpness and intensity of the wave profile. Figure 9 shows 3D soliton wave (18) with σ = 0.5 and α = 0.8. These characteristics are lost, specifically that it becomes rough, as the noise levels increase and the fractional-order derivative falls from one.

Figure 1.

2D soliton wave (18) with different values of σ at t = 2.

Figure 2.

2D soliton wave (18) with different values of the fractional order α.

Figure 3.

2D soliton wave (18) with different values of t.

Figure 4.

3D soliton wave (18) with σ = 0 and α = 1.

Figure 5.

3D soliton wave (18) with σ = 0 and α = 0.4.

Figure 6.

2D soliton wave (18) σ = 1.

Figure 7.

2D soliton wave (25) with different values of t.

Figure 8.

2D soliton wave (25) with different values of σ at t = 2.

Figure 9.

3D soliton wave (25) with σ = 0.5 and α = 0.8.

Future research may focus on further refining the analytical method discussed, incorporating more complex models with higher dimensions and various forms of nonlinearity. Additionally, validating these theoretical models through experimental evidence would represent a significant milestone, particularly in relation to practical applications in realistic fluid dynamics, advanced ultrasound technologies, Biological tissues, and medical diagnostics. Exploring the relationships between fractional-order parameters and physical properties could enhance our understanding of wave phenomena in different media, leading to the development of more sophisticated models in the future. Such progress could invigorate fields of engineering and technology that necessitate the manipulation and control of wave behavior.

Conclusions

This study successfully employed the MSSE method to derive innovative stochastic solitary wave solutions for the fractional (3 + 1)-dimensional equation governing fluids containing gas bubbles, utilizing a conformable fractional operator and incorporating a noise term. The nonlinear propagation of waves in compressible fluids with dispersed gas microbubbles is described by this equation. This phenomenon is frequently seen in acoustics, fluid-structure interactions, petroleum engineering, and biological tissues. By taking memory effects and anomalous dispersion into account, a fractional derivative enhances the model’s accuracy in simulating real-world situations. The mathematical model is transformed into a nonlinear ordinary differential equation by using a complex traveling wave transformation. Using a Brownian motion process and free physical parameters, some novel stochastic solutions for the suggested model were presented. These new stochastic solutions investigate how noise affects frequencies in nonlinear systems, such as superfluid, nuclear research, and biomedical ultrasonography. Numerous profile graphs were produced using the Matlab release’s capabilities to illustrate how the given stochastic solutions behaved.

There are certain limitations to the MSSE method, despite the fact that it offers a flexible foundation for creating precise traveling wave solutions. First, because it is ansatz-driven, the method is inevitably dependent on particular types of beginning and boundary conditions; it is unable to capture all potential solution families, especially those that involve discontinuities or complex physical constraints. Second, its applicability to fractional-order models is typically limited to specific ranges of the fractional parameter α, as closed-form solutions may only be available for rational or simplified values, necessitating numerical or approximate methods for more general cases. Third, many analytic solutions derived by MSSE are still difficult to test experimentally because to non-ideal phenomena such as dissipation, perturbations, or measurement noise, which can conceal the precise solution structure. Given these limits, MSSE is best considered as a complement to numerical simulations and physical modeling, rather than a stand-alone approach.

Future research could benefit from fractal modifications to the (3 + 1)-dimensional equation that governs fluids containing gas bubbles, since such techniques may better represent the multiscale, irregular, and chaotic nature of bubbly flows. The incorporation of fractal and fractal-fractional operators into the governing equations could provide deeper insights into anomalous diffusion, bubble oscillations, and turbulence phenomena that classical models struggle to predict. Recent studies have already highlighted the effectiveness of fractal modeling in nonlinear systems and fluid dynamics, see Refs. 25–28.