New probabilistic solutions of the generalized shallow water wave equation with dual random dispersion coefficients

Abstract

In this paper, some exact solutions of the stochastic generalized nonlinear shallow water wave equation are investigated. This equation is important in fluid mechanics’ fields since it can model the propagation of disturbances in water and other incompressible fluids. Opposite to what is usually considered in the literature, the two dispersion coefficients of the nonlinear terms are considered dependent random quantities as a more realistic case. The modified extended-tanh function (METF) method is combined with the random variable transformation (RVT) technique to get full probabilistic solutions of the problem via computing the probability density functions (PDFs) of the solution processes. Based on the probability density function, any statistical moment of the solution can be evaluated. Through two different applications for the input random variables (dispersion coefficients), my findings are applied efficiently. Finally, numerical results are presented graphically along the spatial dimension at a certain wave speed and time. The obtained results ratify that the proposed technique is efficient and powerful for obtaining analytical probabilistic solutions for the problem.

Keywords:

• Stochastic differential equations
• random coefficients
• random variable transformation
• modified extended tanh-function
• probability density function

Reviewing Editor:

• Marco Montemurro, Arts et Metiers ParisTech – Centre de Bordeaux-Talence, FRANCE

Subjects:

• Environment & Agriculture
• Marine & Aquatic Science
• Mathematics & Statistics
• Applied Mathematics
• Statistics & Probability
• Physical Sciences
• Physics

1. Introduction

The subject of stochastic differential equations (SDEs) (Ghanem and Spanos, 2003; Øksendal, 2010) has a rapidly developing life of its own as a fascinating research field with many interesting open questions. This is because the SDEs have a wide range of applications outside mathematics and there are many fruitful connections to other mathematical disciplines. The origin of this area of research is the observed uncertainties that are involved in many physical properties. For examples: infection rates, such as epidemic prevalence and social addiction in epidemiology, the diffusion and advection coefficients of mass transport processes in physics, the coefficient of viscosity in fluid mechanics, etc., are variables subject to substantial uncertainty. Thus, their deterministic modelling is clearly unrealistic. This promotes the search for mathematical models that consider randomness in their formulation. The consideration of uncertain quantities in deterministic differential equations (DDE) has demonstrated to be useful in mathematical representations for modelling numerous real problems. Uncertainty can be directly introduced into differential equations by considering coefficients, source function term and/or initial/boundary conditions are random variables (RVs) and/or stochastic processes (SPs). This treatment leads to an area commonly referred to as SDEs. Its substantial goal is to extend classical deterministic results to random forms.

Seeking analytical as well as numerical solutions for SDEs has continued to attract attention through the last few decades. In this area, Intensive studies have been conducted. Currently, they have a considerable impact on the analysis and results of several models in engineering and sciences (He, 2009; Hussein et al., 2008; Hussein and Selim, 2013). Most of these studies are established on mean square calculus and its generalizations (Cortés et al., 2012; Golmankhaneh et al., 2013). The main results of these contributions are the solution SP and its first and second statistical moments. On the other hand, the more desirable and complicated goal is the computing of the probability density function (PDF) associated to the solution SP. The PDF represents the complete stochastic solution of the problem since from it, all higher order statistical moments of the solution SP can be conducted. In this context, several recent articles have dealt with the computation of PDF in (Bevia et al., 2023; Casabán et al., 2015; El-Tawil et al., 2007; Hussein and Selim, 2019; Hussein and Selim, 2015; Hussein and Selim, 2021; Hussein and Selim, 2012; Hussein and Selim, 2009). In (Bevia et al., 2023), the stochastic model of the generalized logistic differential equation was investigated through conducting the sample-path and mean-square solution and computing its PDF by using the random variable transformation (RVT) technique and Liouville’s equation. In (Hussein and Selim, 2019), the PDF of the solution SP for the stochastic Milne problem was obtained by combining the Karhunen–Love expansion with the RVT technique. In (Hussein and Selim, 2015), a deterministic analytical technique (modified extended tanh-function) and the RVT technique were consolidated to calculate the PDF of the exact wave solutions for the shallow water wave equation with a single random nonlinear coefficient. In (El-Tawil et al., 2007), the authors combined a deterministic numerical technique (finite element method) with the RVT technique to evaluate a closed form PDF of the solution SP for a randomly excited ordinary differential equation with a random operator. In addition, through (Casabán et al., 2015; Hussein and Selim, 2009; Hussein and Selim, 2012; Hussein and Selim, 2021) comprehensive probabilistic behaviors have been presented, via computing the solution PDF using the RVT technique, for various stochastic nonlinear models like dust plasma model in (Hussein and Selim, 2021), random SI-type epidemiological model in (Casabán et al., 2015), stochastic radiative transfer equation with Rayleigh scattering in (Hussein and Selim, 2012) and the stochastic transport equation of neutral particles with anisotropic scattering in (Hussein and Selim, 2009).

This paper presents a study of an important application of SDEs in nonlinear dynamics which is the generalized shallow water wave (GSWW) equation with random coefficients. This equation is given by: (1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) where $\varphi (x,t)$ is the vertical displacement of the surface and $\alpha \mathrm{ }$ and $\beta$ are the dual random dispersion coefficients of the nonlinear terms. This equation is a stochastic nonlinear partial differential equation that is important in the fluid mechanics’ field. The shallow water wave (SWW) equations model the propagation of disturbances in water and other incompressible fluids. The underlying assumption is that the depth of the fluid is much smaller compared to the wavelength of the disturbance. The SWW equations were applied to atmospheric flows, tidal waves, tsunami prediction and waves in bathtubs (Hussein and Selim, 2015). The study of these equations is crucial for gaining insights into the behavior of waves and disturbances in fluid systems, and it has practical applications in various fields such as oceanography, meteorology, and civil engineering. Therefore, the established solutions of Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) can be used to represent a wide variety of significant physical phenomena.

The deterministic versions of SWW equations have been discussed extensively by many authors (El-Wakil et al., 2005; Krishnan et al., 2011; Li and Liu, 2018; Liu and He, 2017). On the other hand, the work, in literature, for the stochastic versions is still inadequate. Recently, a stochastic modified shallow water wave equation in the sense of the M-truncated derivative (Hamza et al., 2023) and with beta-derivative (Mohammed et al., 2023) was investigated. The stochasticity was considered in the added force term. Opposite to what is usually assumed in the literature, the authors in (Hussein and Selim, 2015) have studied the stochastic version of the problem where the nonlinear coefficients were considered random quantities. But for simplicity, they have assumed that the dual nonlinear coefficients are equal. This simplifying assumption makes the problem to be with a single random variable (RV). As a continuation of that work, the contribution in this article is to solve the more realistic version of the problem with two non-equal dependent random coefficients. The methodology used to solve the problem is based on combining a deterministic technique, to find closed form analytical wave solutions for the considered nonlinear partial differential equation (NLPDE), with the RVT technique (Casabán et al., 2016; Casabán et al., 2014; Walpole et al., 2011) as a stochastic technique. In this context, diverse analytical deterministic approaches for solving NLPDEs are available in the literature. For instance, the METF method (El-Wakil et al., 2005; El-Wakil and Abdo, 2007; El-Wakil et al., 2002), Lie symmetry analysis and the extended Jacobian elliptic function expansion method (Kumar et al., 2022), generalized exponential rational function method (Kumar et al., 2023), Hirota’s bilinear method (Akinyemi, 2023; Kumar and Mohan, 2021), the generalized Rickety equation mapping method (Kumar and Mann, 2023), the extended sine-Gordon expansion approach (Fahim et al., 2022), the extended direct algebraic method (Tasnim et al., 2023), modified Khater’s (mK) method (Tripathy et al., 2023), the improved F-expansion approach (Mirzazadeh et al., 2022) and the sub-equation method (Akinyemi et al., 2021).

Specifically in the present work, the METF method and the bivariate version of the RVT technique are incorporated to get some general probabilistic solutions to the GSWW equation (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) .

This article is organized as follows: in section 2, the bivariate RVT theorem and a general proposition of it, that will be applied in the subsequent sections, are presented. In section 3, the exact probabilistic solution of the GSWW equation (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) is constructed. The solution PDF and its statistical moments are acquired in general form utilizing the METF method incorporated with the RVT technique. The obtained general probabilistic solutions are investigated through two different types of variabilities for the random dispersion coefficients in Subsections (3.1) and (3.2). In section 4, graphs of the PDF and mean of the obtained rational, solitary, and periodic wave solutions are plotted for the two applications at certain wave speed, time and specific distribution’s parameters. The conclusions of this study are introduced in section 5. Finally, Appendix A is added for presenting the deterministic solutions of the GSWW equation using the METF method in detail.

2. The bivariate RVT technique

Theorem 1

(Hussein and Selim, 2019): Let $\mathit{X}={{(X}_1,X_2)}^T$ be a two-dimensional random vector with joint probability density function $f_{X_1,X_2}(x_1,x_2).$ Let $\mathit{Y}={(Y_1,Y_2)}^T={\mathit{r}\left(\mathit{X}\right)=\left(r_1{(X}_1,X_2\right),{r_2(X}_1,X_2))}^T$ be a two dimensional continuous mapping of $\mathit{X}$ with inverse is $\mathit{X}=\mathit{s}\left(\mathit{Y}\right)={\left(s_1{(Y}_1,Y_2\right),{s_2(Y}_1,Y_2))}^T.$ Assuming the Jacobian $J=\frac{\partial s_1}{\partial y_1}\frac{\partial s_2}{\partial y_2}-\frac{\partial s_2}{\partial y_1}\frac{\partial s_1}{\partial y_2}\ne 0$ and all derivatives are continuous. Then, the joint PDF of the random vector $\mathit{Y}={(Y_1,Y_2)}^T$ is given by: (2) $$f_{Y_1,Y_2}(y_1,y_2)=f_{X_1,X_2}\left(s_1{(y}_1,y_2),s_2{(y}_1,y_2)\right)\left|J\right|.$$(2)

Remark 1:

To apply the RVT theorem, the number of input variables, $X_i,$ must equal the number of output variables, $Y_i.$ If they are not equal, we must introduce fictitious random outputs to apply the theorem (Hussein and Selim, 2019).

Remark 2:

According to the previous theorem, we can find the marginal density distribution of any transformed random variable, say $Y_1,$ through the following integral: (3) $$f_{Y_1}\left(y_1\right)={\int }_{D_{y_2}}^{}{f}_{X_1,X_2}\left(s_1{(y}_1,y_2),s_2{(y}_1,y_2)\right)\left|J\right|d{y}_2,$$(3) where $D_{y_2}\subset \mathbb{R}$ is the domain over which the RV, $Y_2,$ is defined.

Proposition 1:

(RVT technique: general reciprocal for sum of two RVs)

Let $\mathit{X}={{(X}_1,X_2)}^T$ be a two-dimensional random vector with joint probability density function $f_{X_1,X_2}(x_1,x_2).$ Then the PDF of the single output $Z=A+\frac{D}{X_1+X_2}$ is given by: (4a) $$f_Z\left(z\right)=\underset{D_{x_2}}{\overset{}{\int }}\frac{\left|D\right|}{{(z-A)}^2}{f}_{X_1,X_2}(\frac{D}{z-A}-x_2,x_2)d{x}_2,$$(4a) or (4b) $$f_Z\left(z\right)=\underset{{\mathit{D}}_{{\mathit{x}}_1}}{\overset{}{\int }}\frac{\left|D\right|}{{(z-A)}^2}{f}_{X_1,X_2}(x_1,\frac{D}{z-A}-x_1)d{x}_1.$$(4b)

Proof:

Using Theorem 1 with introducing a fictitious random output $Y_2=X_2$ (according to Remark 1), the random vector $\mathrm{ }\mathit{Y}=r(\mathit{X})$ will take the form: (5) $$\mathit{Y}=\left(\begin{array}{c}{Y}_1=Z\\ Y_2=X_2\end{array}\right)=\left(\begin{array}{c}{r_1(X}_1,X_2)\\ r_2{(X}_1,X_2)\end{array}\right)=\left(\begin{array}{c}A+\frac{D}{X_1+X_2}\\ X_2\end{array}\right),$$(5) with inverse transformation $\mathit{X}=\mathit{s}\left(\mathit{Y}\right),$ takes the form: (6) $$\mathit{X}=\left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)=\left(\begin{array}{c}{s_1(Y}_1,Y_2)\\ s_2{(Y}_1,Y_2)\end{array}\right)=\left(\begin{array}{c}\frac{D}{Y_1-A}-Y_2\\ Y_2\end{array}\right).$$(6)

The Jacobian of transformation, Eq. (6)(6) $$\mathit{X}=\left(\begin{array}{c}{X}_1\\ X_2\end{array}\right)=\left(\begin{array}{c}{s_1(Y}_1,Y_2)\\ s_2{(Y}_1,Y_2)\end{array}\right)=\left(\begin{array}{c}\frac{D}{Y_1-A}-Y_2\\ Y_2\end{array}\right).$$(6) , is simplified to (7) $$J=-\frac{D}{{(y_1-A)}^2}.$$(7)

Consequently, from Eq. (2)(2) $$f_{Y_1,Y_2}(y_1,y_2)=f_{X_1,X_2}\left(s_1{(y}_1,y_2),s_2{(y}_1,y_2)\right)\left|J\right|.$$(2) , the joint PDF of the random vector $\mathit{Y}$ is given by: (8) $$f_{Y_1,Y_2}\left(y_1,y_2\right)=\frac{\left|\mathit{D}\right|}{{{(\mathit{y}}_1-\mathit{A})}^2}{f}_{X_1,X_2}\left(\frac{D}{y_1-A}-y_2,y_2\right)$$(8)

In view of Remark 2 and Eq. (3)(3) $$f_{Y_1}\left(y_1\right)={\int }_{D_{y_2}}^{}{f}_{X_1,X_2}\left(s_1{(y}_1,y_2),s_2{(y}_1,y_2)\right)\left|J\right|d{y}_2,$$(3) , the PDF of the RV, $Y_1$ will take the form: (9) $$f_{Y_1}\left(y_1\right)={\int }_{D_{y_2}}^{}\frac{\left|\mathit{D}\right|}{{{(\mathit{y}}_1-\mathit{A})}^2}{f}_{X_1,X_2}\left(\frac{D}{y_1-A}-y_2,y_2\right)d{y}_2,$$(9)

Or equivalently, (10) $$f_Z\left(z\right)={\int }_{D_{x_2}}^{}\frac{\left|\mathit{D}\right|}{{(\mathit{z}-\mathit{A})}^2}{f}_{X_1,X_2}\left(\frac{D}{z-A}-x_2,x_2\right)d{x}_2.$$(10)

Note: if we introduce the fictitious variable as $Y_2$ = $X_1,$ and follow the same procedures, we will get formula in Eq. (4b)(4b) $$f_Z\left(z\right)=\underset{{\mathit{D}}_{{\mathit{x}}_1}}{\overset{}{\int }}\frac{\left|D\right|}{{(z-A)}^2}{f}_{X_1,X_2}(x_1,\frac{D}{z-A}-x_1)d{x}_1.$$(4b) .

3. The probabilistic solution of the GSWW equation

In this section, the aim is to find the complete probabilistic solutions of the stochastic GSWW equation described by Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) where the two dispersion coefficients $\alpha \text{and} \beta$ are assumed continuous RVs. This can be achieved by computing the PDF of the dependent variable $\varphi \left(x,t\right).$ As clarified in Appendix A, the modified extended tanh-function (METF) method is applied to solve Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) in deterministic scenario. All acquired travelling wave solutions are in the form of trigonometric (periodic), hyperbolic (solitary), and rational function solutions. The evaluated solutions will be considered, in the following analysis, as the random transformations for the RVT technique to get the PDFs of the corresponding random solutions.

Let’s consider the formulas of the solutions, ${\varphi }_i\left(x,t\right)=a_0+\frac{H_i(x,t)}{\alpha +\beta },$ i = 1,2,…,5, given in Appendix A by Eqs. (A-20) to (A-24), as the input-output relations for the RVT technique, where the RVs, $\alpha$ and $\beta$ are the inputs and the solution SP, ${\varphi }_i\left(x,t\right),i=1,2,\dots .,5,$ is the single system output. To compute the PDF of the solution ${\varphi }_i(x,t),$ proposition 1 will be used with $Z={\varphi }_i,$ $\mathit{X}={{(X}_1,X_2)}^T={(\alpha ,\beta )}^T,$ $A=a_0$ and ${D=H}_i(x,t).$

Therefore, using Eq. (4), the PDF of the stochastic solution ${\varphi }_i(x,t)$ is: (11) $$f_{{\varphi }_i}\left({\varphi }_i\right)={\int }_{D_{\beta }}\frac{\left|H_i\right|}{({\varphi }_i-a_0{)}^2}{f}_{\alpha ,\beta }(-\beta +\frac{H_i}{{\varphi }_i-a_0},\beta )d\beta ,i=1,2,\dots ,5.$$(11)

Using the PDFs, any statistical moment of the solution SPs can be calculated via the expression (12) $$\mathbb{E}\left({\varphi }_i^m\right)={\int }_{D_{{\varphi }_i}}{\varphi }_i^m{f}_{{\varphi }_i}\left({\varphi }_i\right)d{\varphi }_i,i=1,2,\dots ,5.$$(12)

It is evident that the PDF of the solution SP is constructed whenever the solution SP, ${\varphi }_i\left(x,t\right),$ can be computed as a closed form in the input random variables. In addition, this closed form relation must be reversible such that every input random quantity can be computed explicitly as a function of the solution process, ${\varphi }_i\left(x,t\right),$ and all-other random quantities. This is necessary to compute the Jacobian of the inverse transformations and apply the RVT theory. Hence, the application of this technique depends on the type of input random variables and its relationship with the solution process in the closed form solution.

3.1. Applications

The above mathematical treatment of the problem is valid regardless the probability distribution of the random coefficients. To illustrate our theoretical results, the variabilities of the random dispersive coefficients, $\alpha$ and $\beta ,$ are assumed to be general uniform or exponential variabilities. Considering the coefficients are dependent random variables, which is the more general situation, I will suggest that the joint PDF of the input RVs. α. and $\beta$ is constructed using a copula transformation (Nelsen, 1999). In the following, these details will be declared.

Application 1: General exponential variabilities for $\alpha \mathit{ }\mathit{\text{and}}\mathit{ }\beta$

Let us assume that $\alpha$ and $\beta$ are general dependent exponential RVs with parameters, ${\tau }_1$ and ${\tau }_2$ respectively. These RVs are transformed by the Farlie-Gordon-Morgenstern (FGM) copula (Nelsen, 1999) to define a two-dimensional random vector ${(\alpha ,\beta )}^T$ with the following joint PDF: (13) $$f_{\mathit{\alpha },\mathit{\beta }}(\alpha ,\beta )=\{\begin{array}{l}\frac{2}{3}{\tau }_1{\tau }_2{\mathrm{e}}^{-2(\alpha {\tau }_1+\beta {\tau }_2)}(2-{\mathrm{e}}^{\alpha {\tau }_1}-{\mathrm{e}}^{\beta {\tau }_2}+2{\mathrm{e}}^{\alpha {\tau }_1+\beta {\tau }_2}),\alpha \ge 0\mathrm{ }\mathit{\text{and}}\mathrm{ }\beta \ge 0\\ \\ 0,\mathit{\text{otherwise}}.\end{array}$$(13)

 Substituting from Eq. (13)(13) $$f_{\mathit{\alpha },\mathit{\beta }}(\alpha ,\beta )=\{\begin{array}{l}\frac{2}{3}{\tau }_1{\tau }_2{\mathrm{e}}^{-2(\alpha {\tau }_1+\beta {\tau }_2)}(2-{\mathrm{e}}^{\alpha {\tau }_1}-{\mathrm{e}}^{\beta {\tau }_2}+2{\mathrm{e}}^{\alpha {\tau }_1+\beta {\tau }_2}),\alpha \ge 0\mathrm{ }\mathit{\text{and}}\mathrm{ }\beta \ge 0\\ \\ 0,\mathit{\text{otherwise}}.\end{array}$$(13) into Eq. (11)(11) $$f_{{\varphi }_i}\left({\varphi }_i\right)={\int }_{D_{\beta }}\frac{\left|H_i\right|}{({\varphi }_i-a_0{)}^2}{f}_{\alpha ,\beta }(-\beta +\frac{H_i}{{\varphi }_i-a_0},\beta )d\beta ,i=1,2,\dots ,5.$$(11) , a general closed form for the PDFs of all solution SPs is obtained in the following form: (14) $$f_{{\varphi }_i}({\varphi }_i;x,t)=\{\begin{array}{c}{G}_i({\varphi }_i;x,t),\frac{H_i(x,t)}{{\varphi }_i-a_0}\ge 0\mathrm{ }\\ \\ 0, \mathit{\text{otherwise}}\end{array},i=1,2,\dots ,5,$$(14) where (15) $$G_i\left({\varphi }_i;x,t\right)=\frac{2{\tau }_1{\mathrm{\tau }}_2\mathrm{ }\left|{\mathit{\text{Jac}}}_i\right|}{3(-2{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+2{\mathrm{\tau }}_2)}\times \{{{\tau }_1\left({\tau }_1-2{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{{\mathrm{\tau }}_2\left(2{\tau }_1-{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}-\left(3{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+2{{\mathrm{\tau }}_2}^2\right){\mathrm{e}}^{\frac{-{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{\left(2{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+3{{\mathrm{\tau }}_2}^2\right)\mathrm{e}}^{\frac{-{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}\},$$(15)

${\mathit{\text{Jac}}}_i=\frac{-H_i\left(x,t\right)}{{({\varphi }_i-a_0)}^2}$ and $H_i\left(x,t\right)$ are defined in Appendix A.

It is clear that the domain of ${\varphi }_i$ is sensitive to the form of the deterministic functions, $H_i\left(x,t\right),$ and its related variables and parameters. Hence, the type of deterministic solution of the problem will affect this domain. Hereinafter, the PDF corresponding to every type of solution will be presented more specifically.

Case 1:

The PDF of the rational solution

The rational solution ${\varphi }_1(x,t)$ is given in Eq. (A-20) and $H_1(x,t)=\frac{12-\kappa (x-\lambda t{)}^2}{(x-\lambda t)},$ $\kappa =\frac{1-\lambda }{\lambda }.$ In this case, the PDF associated to ${\varphi }_1(x,t),$ will take the following specific general form: (16) $$f_{{\varphi }_1}({\varphi }_1;x,t)=\left\{\begin{array}{c}\left.\begin{array}{l}U(x-\lambda t)G_1({\varphi }_1;x,t),\lambda >1\\ {\left[U\right(x-x_1)-U(x-\lambda t)+U(x-x_2)]G}_1({\varphi }_1;x,t), \lambda <1\end{array}\right\},{\varphi }_1\ge a_0\mathrm{ }\\ \\ \left.\begin{array}{l}U(\lambda t-x)G_1({\varphi }_1;x,t),\lambda >1\\ {\left[U\right(x_1-x)+U(x-\lambda t)-U(x-x_2)]G}_1({\varphi }_1;x,t), \lambda <1\end{array}\right\},{\varphi }_1{\le a}_0\end{array}\right.,$$(16) where $G_1\left({\varphi }_1;x,t\right)$ is defined according to Eq. (15)(15) $$G_i\left({\varphi }_i;x,t\right)=\frac{2{\tau }_1{\mathrm{\tau }}_2\mathrm{ }\left|{\mathit{\text{Jac}}}_i\right|}{3(-2{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+2{\mathrm{\tau }}_2)}\times \{{{\tau }_1\left({\tau }_1-2{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{{\mathrm{\tau }}_2\left(2{\tau }_1-{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}-\left(3{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+2{{\mathrm{\tau }}_2}^2\right){\mathrm{e}}^{\frac{-{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{\left(2{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+3{{\mathrm{\tau }}_2}^2\right)\mathrm{e}}^{\frac{-{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}\},$$(15) for i = 1 and (17) $$U(x-x_j)=\{\begin{array}{c}1, x\ge x_j\\ 0, x<x_j\end{array},x_1=\lambda t-\sqrt{\frac{12}{k}}\text{and} x_2=\lambda t+\sqrt{\frac{12}{k}}.$$(17)

Case 2:

The PDF of the Solitary solutions

The PDF of the solitary stochastic solution, ${\varphi }_2(x,t),$ as given in Eq. (A-21), is given by: (18) $$f_{{\varphi }_2}({\varphi }_2;t,x)=\{\begin{array}{c}U(x-\lambda t)G_2({\varphi }_2;x,t),{\varphi }_2\ge a_0\mathrm{ }\\ \\ U(\lambda t-x)G_2({\varphi }_2;x,t),{\varphi }_2\le a_0\mathrm{ }\end{array},\lambda >1,$$(18) where $\mathrm{ }{G}_2\left({\varphi }_2;x,t\right)$ is defined according to Eq. (15)(15) $$G_i\left({\varphi }_i;x,t\right)=\frac{2{\tau }_1{\mathrm{\tau }}_2\mathrm{ }\left|{\mathit{\text{Jac}}}_i\right|}{3(-2{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+2{\mathrm{\tau }}_2)}\times \{{{\tau }_1\left({\tau }_1-2{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{{\mathrm{\tau }}_2\left(2{\tau }_1-{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}-\left(3{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+2{{\mathrm{\tau }}_2}^2\right){\mathrm{e}}^{\frac{-{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{\left(2{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+3{{\mathrm{\tau }}_2}^2\right)\mathrm{e}}^{\frac{-{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}\},$$(15) , with $H_2(x,t)=6\sqrt{-\kappa }\mathit{\text{tanh}}\hspace{0.17em}\left[\frac{1}{2}\sqrt{-\kappa }(x-\lambda t)\right]$ and i = 2.

Similarly, for the solution, ${\varphi }_3(x,t),$ as given in Eq. (A-22), the PDF is given by: (19) $$f_{{\varphi }_3}({\varphi }_3;x,t)=\{\begin{array}{c}U(x-\lambda t)G_3({\varphi }_3;x,t),{\varphi }_3\ge a_0\mathrm{ }\\ \\ U(\lambda t-x)G_3({\varphi }_3;x,t),{\varphi }_3\le a_0\mathrm{ }\end{array},\lambda >1$$(19) where $\mathrm{ }{G}_3\left({\varphi }_3;x,t\right)$ is defined according to Eq. (15)(15) $$G_i\left({\varphi }_i;x,t\right)=\frac{2{\tau }_1{\mathrm{\tau }}_2\mathrm{ }\left|{\mathit{\text{Jac}}}_i\right|}{3(-2{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+2{\mathrm{\tau }}_2)}\times \{{{\tau }_1\left({\tau }_1-2{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{{\mathrm{\tau }}_2\left(2{\tau }_1-{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}-\left(3{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+2{{\mathrm{\tau }}_2}^2\right){\mathrm{e}}^{\frac{-{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{\left(2{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+3{{\mathrm{\tau }}_2}^2\right)\mathrm{e}}^{\frac{-{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}\},$$(15) , with $H_3(x,t)=6\sqrt{-\kappa }\mathit{\text{coth}}\left[\frac{1}{2}\sqrt{-\kappa }(x-\lambda t)\right],$ and i = 3.

Case 3:

The PDF of the periodic solutions

The PDF of the periodic stochastic solution, ${\varphi }_4(x,t),$ as given in Eq. (A-23), is given by: (20) $$f_{{\varphi }_4}({\varphi }_4;x,t)=\{\begin{array}{c}\sum _n\left[U\right(x-x_{2n})-{U(x-x_{3n})]G}_4({\varphi }_4;x,t),\mathrm{ }{\varphi }_4\ge a_0\mathrm{ }\\ \\ \sum _n\left[U\right(x-x_{1n})-{U(x-x_{2n})]G}_4({\varphi }_4;x,t),\mathrm{ }{\varphi }_4\le a_0\mathrm{ }\end{array},\lambda <1,$$(20) where $\mathrm{ }{G}_4\left({\varphi }_4;x,t\right)$ is defined according to Eq. (15)(15) $$G_i\left({\varphi }_i;x,t\right)=\frac{2{\tau }_1{\mathrm{\tau }}_2\mathrm{ }\left|{\mathit{\text{Jac}}}_i\right|}{3(-2{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+2{\mathrm{\tau }}_2)}\times \{{{\tau }_1\left({\tau }_1-2{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{{\mathrm{\tau }}_2\left(2{\tau }_1-{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}-\left(3{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+2{{\mathrm{\tau }}_2}^2\right){\mathrm{e}}^{\frac{-{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{\left(2{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+3{{\mathrm{\tau }}_2}^2\right)\mathrm{e}}^{\frac{-{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}\},$$(15) , with $H_4(x,t)=-6\sqrt{\kappa }\mathit{\text{tan}}\hspace{0.17em}\left[\frac{1}{2}\sqrt{\kappa }(x-\lambda t)\right]$ and i = 4, $x_{1n}=\lambda t+\frac{2\left(n-1\right)\pi }{\sqrt{k}}, x_{2n}=\lambda t\mathrm{\hspace{0.17em}}+\mathrm{\hspace{0.17em}}\frac{\left(2n-1\right)\pi }{\sqrt{k}},{\mathrm{ }x}_{3n}=\lambda t+\frac{2n\pi }{\sqrt{k}},n=0,\pm 1,\pm 2,\dots \dots .$

Similarly, for the last solution, ${\varphi }_5(x,t),$ as given in Eq. (A-24), one can get the following PDF: (21) $$f_{{\varphi }_5}({\varphi }_5;x,t)=\{\begin{array}{c}\sum _n\left[U\right(x-x_{1n})-{U(x-x_{2n})]G}_5({\varphi }_5;x,t),\mathrm{ }{\varphi }_5\ge a_0\mathrm{ }\\ \\ \sum _n\left[U\right(x-x_{2n})-{U(x-x_{3n})]G}_5({\varphi }_5;x,t),\mathrm{ }{\varphi }_5\le a_0\mathrm{ }\end{array}, \lambda <1,$$(21) where $\mathrm{ }{G}_5\left({\varphi }_5;x,t\right)$ is defined according to Eq. (15)(15) $$G_i\left({\varphi }_i;x,t\right)=\frac{2{\tau }_1{\mathrm{\tau }}_2\mathrm{ }\left|{\mathit{\text{Jac}}}_i\right|}{3(-2{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+{\mathrm{\tau }}_2)(-{\tau }_1+2{\mathrm{\tau }}_2)}\times \{{{\tau }_1\left({\tau }_1-2{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{{\mathrm{\tau }}_2\left(2{\tau }_1-{\mathrm{\tau }}_2\right)\mathrm{e}}^{\frac{-2{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}-\left(3{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+2{{\mathrm{\tau }}_2}^2\right){\mathrm{e}}^{\frac{-{{\tau }_{2}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}+{\left(2{{\tau }_1}^2-7{\tau }_1{\mathrm{\tau }}_2+3{{\mathrm{\tau }}_2}^2\right)\mathrm{e}}^{\frac{-{{\tau }_{1}H}_i\left(x,t\right)}{{\varphi }_i-a_0}}\},$$(15) , with $H_5(x,t)=6\sqrt{\kappa }\mathit{\text{cot}}\left[\frac{1}{2}\sqrt{\kappa }(x-\lambda t)\right],$ and i = 5.

Application 2: General positive Uniform variabilities for $\alpha \mathit{ }\mathit{\text{and}}\mathit{ }\beta$

Consider $\alpha$ and $\beta$ are general dependent uniform RVs defined on the intervals (0, a) and (0, b) respectively and $a<b.$ As in application 1, a transformation using the Farlie-Gordon-Morgenstern copula (Nelsen, 1999) is applied to define a two-dimensional RV ${(\alpha ,\beta )}^T$ with the following joint PDF: (22) $$\begin{array}{cc}{f}_{\mathit{\alpha },\mathit{\beta }}(\alpha ,\beta )=\{\begin{array}{l}\frac{2}{3\alpha \beta }(2-\frac{\alpha }{a}-\frac{\beta }{b}+\frac{2\alpha \beta }{\mathit{\text{ab}}}),\hspace{1em}0<\alpha <a\mathrm{ }\mathit{\text{and}}\mathrm{ }0<\beta <b\\ \\ 0,\mathit{\text{otherwise.}}\end{array}& \end{array}$$(22)

Similarly, as in application 1, the general closed form for the PDFs of all solution SPs is obtained in the following form: (23) $$f_{{\varphi }_i}({\varphi }_i;x,t)=\{\begin{array}{l}\begin{array}{c}{F}_{i1}({\varphi }_i;x,t),a_0+\frac{H_i(x,t)}{a+b}\le {\varphi }_i<a_0+\frac{H_i(x,t)}{b}\\ F_{i2}({\varphi }_i;x,t),a_0+\frac{H_i(x,t)}{b}\le {\varphi }_i<a_0+\frac{H_i(x,t)}{a}\end{array}\\ F_{i3}({\varphi }_i;x,t),a_0+\frac{H_i(x,t)}{\mathrm{a}}\le {\varphi }_i\\ 0,\mathit{\text{otherwise}}\end{array},i=1,2,\dots ,5,$$(23) where (26) $$F_{i1}\left({\varphi }_i;x,t\right)=-\frac{\left|{\mathit{\text{Jac}}}_i\right|}{9a^2{b}^2{\left(a_0-{\varphi }_i\right)}^3}(a^3{\left(a_0-{\varphi }_i\right)}^3-9a^{2}b{\left(a_0-{\varphi }_i\right)}^3+\left(a_{0}b-2H_i\left(x,t\right)-b{\varphi }_i\right){\left(a_{0}b+H_i\left(x,t\right)-b{\varphi }_i\right)}^2-3a\left(a_0-{\varphi }_i\right)(3{a_0}^2{b}^2+4a_{0}b{H}_i\left(x,t\right)+{H_i\left(x,t\right)}^2-6a_0{b}^2{\varphi }_i‐4b{H}_i\left(x,t\right){\varphi }_i+3b^2{{\varphi }_i}^2)),$$(26) (27) $$F_{i2}({\varphi }_i;x,t)=\frac{(9b-a)\left|{\mathit{\text{Jac}}}_i\right|}{9b^2},$$(27) (28) $$F_{i3}\left({\varphi }_i;x,t\right)=-\frac{\left|{\mathit{\text{Jac}}}_i\right|H_i\left(x,t\right)}{9a^2{b}^2{\left(a_0-{\varphi }_i\right)}^3}(3a\left(a_0-{\varphi }_i\right)\left(4a_{0}b+H_i\left(x,t\right)-4b{\varphi }_i\right)+H_i\left(x,t\right)\left(3a_{0}b+2H_i\left(x,t\right)-3b{\varphi }_i\right))$$(28) ${\mathit{\text{Jac}}}_i=\frac{-H_i\left(x,t\right)}{{({\varphi }_i-a_0)}^2}$ and $H_i\left(x,t\right)$ are defined in Appendix A.

4. Numerical results

In realizing the above work, let us assign numerical values for the distributions’ parameters used in the two previous applications. Specifically, ${\tau }_1=1$ and ${\tau }_2=3$ in application 1 and $a=2$ and $b=3$ in application 2. In each application the arbitrary constant $a_0=0.$ Now, all solutions can be represented graphically along the spatial dimension, $x,$ at specific wave speed, $\mathrm{ }\lambda ,$ and time, $t.$ The numerical results are presented for both applications. Specifically, some results corresponding to exponentially distributed dispersion coefficients, $\alpha$ and $\beta ,$ are illustrated through Figures 1–3 and for uniformly distributed ones, are illustrated through Figures 4–6.

Figure 1. Variations of: (a) PDF of the rational solution ${\varphi }_1(x,t)$ along the spatial dimension, $x,$ at wave speed, $\mathrm{ }\lambda =1.4$ and $t=3.0;$ (b) PDF of the rational solution ${\varphi }_1(x,t)$ with wave speed, $\mathrm{ }\lambda ,$ at $x=10$ and $t=3;$ (c) Mean rational solution, $\mathbb{E}\left[{\varphi }_1\right],$ with wave speed, $\mathrm{ }\lambda ,$ at $x=10$ and $t=3.0,$ (Exponential distributions).

Figure 2. Variations of: (a) PDF of the solitary solution ${\varphi }_3(x,t)$ along the spatial dimension, $x,$ at wave speed, $\mathrm{ }\lambda =1.5$ and $t=3.0;$ (b) PDF of the solitary solution ${\varphi }_3(x,t)$ with wave speed, λ, at $x=10$ and $t=3.0;$ (c) Mean solitary solution, $\mathbb{E}\left[{\varphi }_3\right],$ with wave speed, $\mathrm{ }\lambda ,$ at $x=10$ and $t=3.0,$ (Exponential distributions).

Figure 3. Variations of: (a) PDF of the periodic solution ${\varphi }_4(x,t)$ along the spatial dimension, $x,$ at wave speed, $\mathrm{ }\lambda =0.4$ and $t=3.0,$ taking $n=0,1,2;$ (b) PDF of the periodic solution, ${\varphi }_4(x,t)$ with wave speed, λ, at $x=0.8$ and $t=3.0;$ (c) Mean periodic solution, $\mathbb{E}\left[{\varphi }_4\right],$ with wave speed, $\mathrm{ }\lambda ,$ at $x=0.8,$ $t=3.0$ and $n=0,$ (Exponential distributions).

Figure 4. Variations of: (a) PDF of the rational solution, ${\varphi }_1(x,t)$ along the spatial dimension, $x,$ at wave speed, $\mathrm{ }\lambda =1.4$ and $t=3.0;$ (b) PDF of the rational solution ${\varphi }_1(x,t)$ with wave speed, $\mathrm{ }\lambda ,$ at $x=10$ and $t=3;$ (c) Mean rational solution, $\mathbb{E}\left[{\varphi }_1\right],$ with wave speed, $\mathrm{ }\lambda ,$ at $x=10$ and $t=3.0,$ (Uniform distributions).

Figure 5. Variations of: (a) PDF of the solitary solution, ${\varphi }_3(x,t)$ along the spatial dimension, $x,$ at wave speed, $\mathrm{ }\lambda =1.5$ and $t=3.0;$ (b) PDF of the solitary solution ${\varphi }_3(x,t)$ with wave speed, λ, at $x=10$ and $t=3.0;$ (c) Mean solitary solution, $\mathbb{E}\left[{\varphi }_3\right],$ with wave speed, $\mathrm{ }\lambda ,$ at $x=10$ and $t=3.0,$ (Uniform distributions).

Figure 6. Variations of: (a) PDF of the periodic solution, ${\varphi }_4(x,t)$ along the spatial dimension, $x,$ at wave speed, $\mathrm{ }\lambda =0.4$ and $t=3.0,$ taking $n=0,1,2;$ (b) PDF of the periodic solution ${\varphi }_4(x,t)$ with wave speed, λ, at $x=0.8$ and $t=3.0;$ (c) Mean periodic solution, $\mathbb{E}\left[{\varphi }_4\right],$ with wave speed, $\mathrm{ }\lambda ,$ at $x=0.8,$ $t=3.0$ and $n=0,$ (Uniform distributions).

In application 1, Figures 1a, 2a, and 3a show the 3D plot of the PDF corresponding to the rational stochastic solution ${\varphi }_1(x,t),$ the solitary stochastic solution, ${\varphi }_3(x,t),$ and the periodic stochastic solution, ${\varphi }_4(x,t).$ They show the behavior of the PDFs with the spatial dimension, x, at a given wave speed, $\lambda ,$ and time, t. The effect of wave speed on the PDFs behavior is clarified through Figures 1b, 2b and 3b which are plotted for different $\lambda$ at fixed spatial dimension, x and time, t. It is noticeable that the mean values of the selected solutions’ SPs are increasing as wave speed, $\mathrm{ }\lambda ,$ increases. These behaviors can be noted more clearly through Figures 1c, 2c and 3c in which the behavior of the means of the solutions’ SPs, $\mathbb{E}\left[{\varphi }_1\right],\mathbb{E}\left[{\varphi }_3\right]$ and $\mathbb{E}\left[{\varphi }_4\right],$ that are calculated using Eq. (12)(12) $$\mathbb{E}\left({\varphi }_i^m\right)={\int }_{D_{{\varphi }_i}}{\varphi }_i^m{f}_{{\varphi }_i}\left({\varphi }_i\right)d{\varphi }_i,i=1,2,\dots ,5.$$(12) with $m=1,$ are elucidated. From these figures, one can note the considerable increase of the mean vertical displacement of the surface due to the increase of the wave speed which is physically acceptable.

For the sack of comparison, the corresponding solutions for application 2 are presented through Figures 4–6. It is evident that the behaviors of the solutions in application 2 are the same as corresponding ones in applications 1. This assures the reliability of the obtained mathematical findings.

5. Conclusions

In this paper, the stochastic shallow water wave equation with dual random dispersion coefficients has been investigated via implementing a sound stochastic technique, named RVT, combined with the modified extended tanh-function (METF) Method. Firstly, this nonlinear partial differential equation has been solved deterministically using the METF method. Then, the RVT technique has been constructively applied to get the full probabilistic exact solution by deducing the PDF of the solution SP in a general closed form.

The source of generality is that the dispersion coefficients were assumed to be non-equal RVs. Additionally, the proposed mathematical treatment of the problem permitted these random coefficients to be dependent RVs with arbitrary probability distributions. Further, using the obtained general PDFs, any statistical moments of the solution SPs could be derived through Eq. (12)(12) $$\mathbb{E}\left({\varphi }_i^m\right)={\int }_{D_{{\varphi }_i}}{\varphi }_i^m{f}_{{\varphi }_i}\left({\varphi }_i\right)d{\varphi }_i,i=1,2,\dots ,5.$$(12) .

To verify the proposed technique of solution, two applications were presented using different variabilities for the dispersion coefficients. The graphical results for the two applications are physically acceptable and are of the same behavior for corresponding solutions that makes the mathematical findings dependable.

The contribution of this work is that the proposed randomized model is more general and realistic than the previously introduced in (Hussein and Selim, 2015). Moreover, the implemented technique gives new full probabilistic solutions, represented by the solutions’ PDFs, of the considered problem in closed forms. In addition, the symbolic analysis and established analytical solutions affirm that the current technique is effective, robust, reliable and straightforward.

To my knowledge, this contribution was not presented previously for this problem. Therefore, this is a stride for future solving of other nonlinear differential equations with random coefficients and/or with random force using the proposed methodology. Alternatively, solving the same model using RVT technique combined with other deterministic techniques, like Jacobi elliptic function and generalized exponential rational function method, to derive other new stochastic wave solutions.

Data availability statement

Data sharing is not applicable to this article since no data sets were taken from outside sources.

Disclosure statement

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Notes on contributors

A. Hussein

A. Hussein has been awarded the Ph.D. degree in Engineering Mathematics, Faculty of Engineering, Cairo Univ. Egypt, 2005. His research interests are in the areas: Methods for solving stochastic Differential Equations (Spectral and Transformation approaches), Analytical methods for solving Nonlinear evolution equations, Radiative transfer, and transport phenomena in stochastic media. His current Job: Associate Prof. of Engineering Mathematics, Applied College, Umm Al-Qura University, Saudi Arabia. He has published research articles in reputable international journals of mathematical and engineering sciences.

Appendix A

A.1 The modified extended-tanh function (METF) method

The METF method, as introduced by El-Wakil et al. (2005) and El-Wakil and Abdo (2007), can be summarized in the following:

Consider the following (1 + 1) PDE: (A-1) $$\mathrm{\Gamma }(\varphi ,{\varphi }_x,{\varphi }_t,{\varphi }_{\mathit{\text{tx}}},{\varphi }_{\mathit{\text{xx}},}\dots )=0.$$(A-1)

Assuming a traveling wave solution $\varphi (x,t)=\mathrm{\Phi }(\xi ),$ where $\xi =x-\lambda t,$ Eq. (A-1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) is converted into the following ordinary differential equation (ODE): (A-2) $$F(\mathrm{\Phi },{\mathrm{\Phi }}^{\prime },{\mathrm{\Phi }}^{″},{\mathrm{\Phi }}^{‴},\dots )=0,$$(A-2) with $’:=d/d\xi .$ Integrating of Eq. (A-2) will reduce its order, considering the integration constant to be zero. After that, the solution may be represented by the formula: (A-3) $$\mathrm{\Phi }\left(\xi \right)=a_0+{\sum }_{i=1}^M\left[a_i{\omega }^i\left(\xi \right)+b_i{\omega }^{-i}\left(\xi \right)\right].$$(A-3)

Balancing the nonlinear term with the highest-order linear term in Eq. (A-2), the integer M can be determined. Additionally, $\omega (\xi )$ is the solution of the next Reccati differential equation (A-4) $${\omega }^{\prime }=l+{\omega }^2,$$(A-4) where $l$ is a constant to be evaluated.

Depending on the value of $l,$ Eq. (A-4) has the following solutions: (A-5) $$\omega \left(\xi \right)=-\frac{1}{\xi } \mathrm{\text{for}}\mathrm{ }l=0,$$(A-5) (A-6) $$\omega \left(\xi \right)=\{\begin{array}{c}-\sqrt{-l}\mathit{\text{\hspace{0.5em}tanh}}\hspace{0.17em}[\sqrt{-l}\xi ]\\ -\sqrt{-l}\mathit{\text{\hspace{0.5em}coth}}[\sqrt{-l}\xi ]\end{array}\text{for}l<0,$$(A-6) and (A-7) $$\omega \left(\xi \right)=\{\begin{array}{c}\sqrt{l}\mathit{\text{tan}}\hspace{0.17em}[\sqrt{l}\xi ]\\ -\sqrt{l}\mathit{\text{cot}}[\sqrt{l}\xi ]\end{array}\text{for}l>0.$$(A-7)

Inserting Eqs. (A-3) and (A-4) into the ODE (A-2) leads to a system of algebraic equations in the unknowns $a_0,a_i,b_i,l$ and $\lambda .$ By solving this system, the solutions of Eq. (A-1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) can be obtained.

A.2 The deterministic solution of the GSWW equation

The METF method is applied to obtain a traveling wave solution of Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) as follows:

Under the mapping, $\xi =x-\lambda t;\lambda$ is the wave speed, Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) is transformed to the next ODE: (A-8) $$\lambda \left[{\mathrm{\Phi }}^{″″}+\left(\alpha +\beta \right){\mathrm{\Phi }}^{\prime }{\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}^{″}\right]+{\mathrm{\Phi }}^{″}=0$$(A-8)

Balancing ${\mathrm{\Phi }}^{″″}$ with ${\mathrm{\Phi }}^{\prime }{\mathrm{\Phi }}^{″},$ leads to $M=1.$ Then, according to Eq. (A-3), (A-9) $$\mathrm{\Phi }\left(\xi \right)=a_0+a_1\omega +b_1{\omega }^{-1}.$$(A-9)

Merging Eqs. (A-4) and (A-9) into Eq. (A-8) gives an algebraic equation in powers of $\mathrm{ }\omega .$ Equating the coefficients of ${\omega }^i$ and ${\omega }^{-i}(i=0,\hspace{0.17em}1,\hspace{0.17em}2,\dots ,5)\hspace{0.17em}$to zero leads to the next system of equations in the parameters $a_1,b_1,l$ and $\lambda :$ (A-10) $$l{a}_1(1-\lambda +8l\lambda +(\alpha +\beta )l\lambda a_1-(\alpha +\beta \left)\lambda b_1\right)=0,$$(A-10) (A-11) $$l{b}_1(1-\lambda +8l\lambda +(\alpha +\beta )l\lambda a_1-(\alpha +\beta \left)\lambda b_1\right)=0,$$(A-11) (A-12) $$a_1(1-\lambda +20l\lambda +2(\alpha +\beta )l\lambda a_1-(\alpha +\beta \left)\lambda b_1\right)=0,$$(A-12) (A-13) $$l{b}_1(1-\lambda +20l\lambda +(\alpha +\beta )l\lambda a_1-2(\alpha +\beta \left)\lambda b_1\right)=0,$$(A-13) (A-14) $$\lambda a_1(12+(\alpha +\beta \left)a_1\right)=0,$$(A-14) (A-15) $$l\lambda b_1(12l-(\alpha +\beta \left)b_1\right)=0.$$(A-15)

Using Mathematica package, the system of algebraic equations is solved, and the next sets of solutions are obtained. (A-16) $$l=0,a_1=-\frac{12}{(\alpha +\beta )},b_1=\frac{1-\lambda }{(\alpha +\beta )\lambda }.$$(A-16) (A-17) $$l=\frac{1-\lambda }{4\lambda },a_1=-\frac{12}{(\alpha +\beta )},b_1=0.$$(A-17) (A-18) $$l=\frac{1-\lambda }{4\lambda },a_1=0,b_1=\frac{3(1-\lambda )}{(\alpha +\beta )\lambda }.$$(A-18) (A-19) $$l=\frac{1-\lambda }{16\lambda },a_1=-\frac{12}{(\alpha +\beta )},b_1=\frac{3(1-\lambda )}{4(\alpha +\beta )\lambda }.$$(A-19)

Depending on Eqs. (A-16), the deterministic solution of Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) is the next rational form: (A-20) $${\varphi }_1(x,t)=a_0+\frac{H_1(x,t)}{(\alpha +\beta )}, x\ne \lambda t,$$(A-20) where $H_1(x,t)=\frac{12-\kappa (x-\lambda t{)}^2}{(x-\lambda t)},$ $\kappa =\frac{1-\lambda }{\lambda }$ and $a_0$ is an arbitrary constant.

In view of Eqs. (A-17), and for $\kappa <0$ (or $\lambda >1$), the solution of Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) will take the subsequent solitary wave forms: (A-21) $${\varphi }_2(x,t)=a_0+\frac{H_2(x,t)}{(\alpha +\beta )},$$(A-21) (A-22) $${\varphi }_3(x,t)=a_0+\frac{H_3(x,t)}{(\alpha +\beta )},$$(A-22) where

$H_2(x,t)=6\sqrt{-\kappa }\mathit{\text{tanh}}\hspace{0.17em}\left[\frac{1}{2}\sqrt{-\kappa }(x-\lambda t)\right]\text{and}{H}_3(x,t)=6\sqrt{-\kappa }\mathit{\text{coth}}\left[\frac{1}{2}\sqrt{-\kappa }(x-\lambda t)\right].$

For $\kappa >0$ (or $\lambda <1$), one can get the following periodic solutions: (A-23) $${\varphi }_4(x,t)=a_0+\frac{H_4(x,t)}{(\alpha +\beta )},$$(A-23) (A-24) $${\varphi }_5(x,t)=a_0+\frac{H_5(x,t)}{(\alpha +\beta )},$$(A-24) where $$H_4(x,t)=-6\sqrt{\kappa }\mathrm{\text{tan}}\hspace{0.17em}\left[\frac{1}{2}\sqrt{\kappa }(x-\lambda t)\right]\text{and}{H}_5(x,t)=6\sqrt{\kappa }\mathit{\text{cot}}\left[\frac{1}{2}\sqrt{\kappa }(x-\lambda t)\right].$$

Using Eq. (A-18), identical solution to that is given by Eq. (A-21) will be obtained. Utilizing Eq. (A-19), the solution corresponding to $\kappa <0$ (or $\lambda >1$) is: $${\varphi }_6(x,t)=a_0+\frac{3}{(\alpha +\beta )}\sqrt{-\kappa }\left\{\mathit{\text{coth}}\left[\frac{1}{4}\sqrt{-\kappa }(x-\lambda t)\right]+\mathit{\text{tanth}}\left[\frac{1}{4}\sqrt{-\kappa }(x-\lambda t)\right]\right\}.$$

Or equivalently, (A-25) $${\varphi }_6(x,t)=a_0+\frac{6\sqrt{-\kappa }}{(\alpha +\beta )}\mathit{\text{coth}}\left[\frac{1}{2}\sqrt{-\kappa }(x-\lambda t)\right],$$(A-25) which is identical to $\mathrm{ }{\varphi }_3\left(x,t\right).$

Again, depending on Eq. (A-19), the solution of Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) corresponding to $\kappa >0$ (or $\lambda <1$) takes the form: $${\varphi }_7(x,t)=a_0+\frac{3}{(\alpha +\beta )}\sqrt{\kappa }\left\{\mathit{\text{cot}}\left[\frac{1}{4}\sqrt{\kappa }(x-\lambda t)\right]-\mathrm{\text{tan}}\hspace{0.17em}\left[\frac{1}{4}\sqrt{\kappa }(x-\lambda t)\right]\right\}.$$

Or equivalently, (A-26) $${\varphi }_7(x,t)=a_0+\frac{6\sqrt{\kappa }}{(\alpha +\beta )}\mathit{\text{cot}}\left[\frac{1}{2}\sqrt{\kappa }(x-\lambda t)\right],$$(A-26) which is identical to ${\varphi }_5\left(x,t\right).$

Therefore, only Eqs. (A-20)–(A-24) represent the independent deterministic solutions of Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) . All solutions are verified by substituting them back into Eq. (1)(1) $${\varphi }_{\mathit{\text{xxxt}}}+\alpha {\varphi }_x{\varphi }_{\mathit{\text{xt}}}+\beta {\varphi }_t{\varphi }_{\mathit{\text{xx}}}-{\varphi }_{\mathit{\text{xt}}}-{\varphi }_{\mathit{\text{xx}}}=0,$$(1) .