The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

Abstract

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the $\beta$-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

1. Introduction

Nonlinear partial differential equations (NLPDEs) are frequently used in science and engineering to model a variety of nonlinear problems that can occur in real-life applications [1]. These equations, for instance, can be applied to the modeling of fluid dynamical issues, wave propagation in corrugated media, the study of earthquakes and seismic waves, and the modeling of optical fibers, among other things. The field of fluid dynamics is still important even if it is an older one that received a lot of attention. As an extension of differential equations (DEs) [2] of integer order, there are fractional order differential equations. Models of NLPDEs from physics and mathematics serve as essentials in theoretical sciences. Numerous practical fields, including meteorology, oceanography, and the aerospace industry, depend on a grasp of these NLPDEs.

Fractional differential equations (FDEs) [3] are becoming more and more prevalent today in a variety of disciplines, including dynamic systems and mathematics. Leibniz and L’Hôpital introduced the first idea for FDEs in 1695. In mathematical models incorporating FDEs, the nonlocal behavior of the fractional order derivatives provides the memory feature. Several researchers have been attracted to the flexibility of fractional theory and the numerous interesting features of fractional calculus (FC) [4,5,6,7,8,9]. New definitions of fractional derivatives were introduced by Caputo–Hadamard [10], Katugampola [11], Weyl [12], Riemann–Liouville [13], and Erdélyi–Kober [14], which enabled fractional calculus to deal with challenging natural phenomena. Throughout the past few decades, Caputo derivative [15] has been one of the most frequently employed fractional derivatives (FDs) in numerous research.

The mathematical model for small-amplitude long wave propagation in nonlinear dispersive media is described by the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation [16]. The BBM equation has been proven to be preferable to the Korteweg–De Vries (KdV) [17] equation. The wave-breaking models are essential to the BBM-Burger equation and the KdV equation. The KdV equation was driven by water waves, and it was utilized in many other physical systems as a model for long waves. Solitary wave solutions of the BBM-Burger equation [18] reflect the dynamics of waves in the medium and are essential to many fields such as physics and dispersive systems [19]. In this study, the BBM-Burger equation is considered for analytical solutions in the sense of $\beta$-derivative, CD, and M-TD. In $\beta$-derivative, the proposed model has the following form

$$D_{\beta ,t}^{\sigma }w-D_{\beta ,t}^{\sigma }{w}_{zz}+w_z+{\left(\frac{w^2}{2}\right)}_z=0,$$

(1)

where $D_{\beta ,t}^{\sigma }$ is $\beta$-derivative and $\sigma$ is fractional parameter.

In M-TD, the proposed model has the following form

$$D_{j,t}^{\sigma ,\beta }w-D_{j,t}^{\sigma ,\beta }{w}_{zz}+w_z+{\left(\frac{w^2}{2}\right)}_z=0,$$

(2)

where $D_{j,t}^{\sigma ,\beta }$ is M-TD and $\sigma$ and $\beta$ are fractional parameters.

In CD, the proposed model has the following form

$$D_{c,t}^{\sigma }w-D_{c,t}^{\sigma }{w}_{zz}+w_z+{\left(\frac{w^2}{2}\right)}_z=0,$$

(3)

where $D_{c,t}^{\sigma }$ is CD and $\sigma$ is fractional parameter.

The following is the BBM-Burger equation when $\beta$ = 0 and $\sigma$ = 1 are used.

$$w_t-w_{zzt}+w_z+{\left(\frac{w^2}{2}\right)}_z=0.$$

(4)

Atangana was the one who first introduced the fractional “$\beta$-derivative” [20,21]. The recently introduced derivatives, which are used to depict various medical situations, meet a number of requirements that were previously thought to be limits for the fractional derivatives. Basic and satisfying most of the criteria for the classical integral derivative, the conformable fractional derivative definition includes linearity, Rolle’s theorem, mean value theorem [22], product rule, quotient rule, power rule, and chain rule. The M-TD [23,24], which uses a Mittag–Leffler function with one parameter that satisfies several properties of integer-order calculus, was introduced in 2017 by Sousa and Oliveira. M-TD also satisfies the fundamental differential calculus mathematical principles, which stimulates additional research utilizing these newly formed notions. On conformable and M-TD models in the field of ocean engineering, there are some recent studies in the literature. The goal of these investigations is to find soliton solutions for the models with local derivatives. A novel solution for various FDEs is provided by the conformable fractional derivative, which aims to expand the ordinary derivative while satisfying some natural properties.

Numerous methods, including the extended ($\frac{G^{\prime }}{G^2}$) expansion method [25], the multiple exp-function method [26], the M-lump solution [27], Sardar-subequation technique [28], the Jacobi elliptic function method (JEFM) [29], Painleve analysis, and many others, can be used to find the solitary wave solutions of NLPDEs. However, in this work, the MAEM and RB sub-ODE method [30] have been utilized for finding efficient and effective traveling wave solutions. The RB sub-ODE technique is originally created to produce precise traveling wave solutions, solitary wave solutions, and peaked wave solutions for NLPDEs. Backlund transformation is applied to the RB equation [31]. NLPDE [32,33] may be converted into a set of algebraic equations using the RB equation and a traveling wave transformation.

An approach for creating precise differential equation solutions is the MAEM [34]. The auxiliary equation approach has been expanded in this way. It offers a simple method for handling the solutions of nonlinear evolution equations. This effective method has been used to achieve findings that are pleasing and aid in the investigation of answers to numerous issues that are appearing in applied mathematics and physics. Although there are many different types of traveling wave solutions that may be built using exact solution techniques, approximation solution approaches are also useful when studying evolution equations. This study was inspired by recent developments in fractional nonlinear evolution equations’ traveling wave solutions. Recently, it has been discovered that a variety of exact solution techniques are useful in creating potential wave behaviors that correspond to the physical system defined by a specific evolution equation. Reading the published research papers [21,24,35,36,37,38] also inspired the authors. In this paper, we use two effective analytical approaches to derive the precise soliton solutions of the BBM-Burger equation while taking into account three natural extensions of the classical derivative, namely the $beta$-derivative, M-TD, and CD. Additionally, using the Wolfram Mathematica 12 software, we provide several 2D and 3D graphical representations of the analytical soliton solutions of the BBM-Burger equation and investigate the impact of the fractional parameters employed in $beta$-derivative, M-TD, and CD.

This paper is organized as follows: Basic definitions and their properties are explained in Section 2. Section 3 represents the mathematical interpretation of the BBM-Burger equation. Section 4 and Section 5 conduct the algorithmic steps of the RB-sub ODE method and MAEM and apply them to the proposed model. In addition to computing, graphs are used to show how the result can be physically explained in Section 6. In Section 7, there are some concluding remarks to wrap up the work.

2. Preliminaries

The definitions of derivatives along with their fundamental properties are discussed in this section.

2.1. $\beta$-Derivative and Its Properties

Definition 1. 

The β-derivative is another kind of conformable derivative that can be defined, as [20]

$$D_{\beta ,t}^{\sigma }w\left(t\right)=\underset{\epsilon \to 0}{lim}\frac{w(t+\epsilon {(t+\frac{1}{\mathrm{\Gamma }\left(\sigma \right)})}^{1-\sigma })-w\left(t\right)}{\epsilon },0<\sigma \le 1.$$

The $\beta$-derivative has the following properties.

• The $\beta$-derivative is a linear operator; $D_{\beta ,t}^{\sigma }(cd\left(z\right)+rq\left(z\right))=c{D}_{\beta ,t}^{\sigma }d\left(z\right)+r{D}_{\beta ,t}^{\sigma }q\left(z\right)$, $\forall c,r\in \Re$.
• It satisfies the product rule; $D_{\beta ,t}^{\sigma }(d\left(z\right)\ast q\left(z\right))=q\left(z\right)D_{\beta ,t}^{\sigma }d\left(z\right)+d\left(z\right)D_{\beta ,t}^{\sigma }q\left(z\right)$.
• It satisfies the quotient rule; $D_{\beta ,t}^{\sigma }\left\{\frac{d\left(z\right)}{q\left(z\right)}\right\}=\frac{q\left(z\right)D_{\beta ,t}^{\sigma }d\left(z\right)-d\left(z\right)D_{\beta ,t}^{\sigma }q\left(z\right)}{q^2\left(z\right)}$.
• The $\beta$-derivative of a constant is zero; $D_{\beta ,t}^{\sigma }\left(c\right)=0$,for any constant c.

2.2. M-Truncated Derivative and Its Properties

This section defines an M-TD and presents several results that are surprisingly similar to those of classical calculus. Sousa et al. [39] recently presented the M-TD, which is a natural extension of the ordinary derivative. This derivative does not have the shortcomings of the preceding ones. The M-TD is also known as a conformable fractional derivative [23,40]. The M-TD can readily satisfy some features of classical calculus, including the quotient rule, product rule, linearity, chain rule, and function composition rule. The M-TD, which makes use of a one-parameter Mittag–Leffler function, also satisfies the requirements of integer-order calculus.

Definition 2. 

The M-TD for the function $w:\left. [0,\infty \right)\to$ R of order $\sigma \in \left(0,1\right)$ is defined, as [35]

$$D_{j,t}^{\sigma ,\beta }w\left(t\right)=\frac{\underset{\epsilon \to 0}{lim}w\left(t{E}_j^{\beta }\left(\epsilon t^{-\sigma }\right)\right)-w\left(t\right),}{\epsilon }$$

fort > 0. Where $E_j^{\beta }\left(.\right),\beta >0$ is a truncated Mittag–Leffler function of one parameter defined, as:

$$E_j^{\beta }\left(t\right)=\sum _{k=0}^j\frac{t^k}{\mathrm{\Gamma }\left(\beta k+1\right)}.$$

The M-TD has the following properties.

• The M-TD is a linear operator; $D_{j,z}^{\sigma ,\beta }(cd\left(z\right)+rq\left(z\right))=c{D}_{j,z}^{\sigma ,\beta }d\left(z\right)+r{D}_{j,z}^{\sigma ,\beta }q\left(z\right)$,$\forall c,r\in \Re$.
• It satisfies the product rule; $D_{j,z}^{\sigma ,\beta }(d\left(z\right)\ast q\left(z\right))=q\left(z\right)D_{j,z}^{\sigma ,\beta }d\left(z\right)+d\left(z\right)D_{j,z}^{\sigma ,\beta }q\left(z\right)$.
• It satisfies the quotient rule; $D_{j,z}^{\sigma ,\beta }\left\{\frac{d\left(z\right)}{q\left(z\right)}\right\}=\frac{q\left(z\right)D_{j,z}^{\sigma ,\beta }d\left(z\right)-d\left(z\right)D_{j,z}^{\sigma ,\beta }q\left(z\right)}{q^2\left(z\right)}$.
• The M-TD for a differentiable function $q\left(z\right)$ is defined, as:

  $$D_{j,z}^{\sigma ,\beta }q\left(z\right)=\frac{z^{1-\sigma }}{\mathrm{\Gamma }(\beta +1)}\frac{dq}{dz}.$$

2.3. Conformable Derivative

Definition 3. 

The conformable derivative of order σ for a function $w:\left. [0,\infty \right)\to \Re$ is written as:

$$D_{c,t}^{\sigma }w\left(t\right)=\underset{\epsilon \to 0}{lim}\frac{w(t+\epsilon {\left(t\right)}^{1-\sigma })-w\left(t\right)}{\epsilon },\forall t>0.$$

If w has σ-differentiability in any interval $(0,a)$ with a > 0, then

$$D_c^{\sigma }\left(w\left(0\right)\right)=\underset{t\to 0^{+}}{lim}{D}_c^{\sigma }\left(w\left(t\right)\right),$$

whenever the limit of the right-hand side exists.

Further, properties and theorems related to CD are discussed in [37].

3. Mathematical Interpretation of the Proposed Model

To obtain soliton solutions for Equation (4), the following transformations have been employed.

$$W(z,t)=w\left(\eta \right).$$

(5)

Three definitions are provided for the traveling wave parameter $\eta$.

For $\beta$-derivative, $\eta$ has the following form

$$\eta =Kz+\frac{R}{\sigma }{\left(t+\frac{1}{\mathrm{\Gamma }\left(\sigma \right)}\right)}^{\sigma }.$$

(6)

For M-TD, $\eta$ has the following form

$$\eta =Kz+\frac{R\mathrm{\Gamma }\left(\beta +1\right)}{\sigma }{t}^{\sigma }.$$

(7)

For CD, $\eta$ has the following form

$$\eta =Kz+\frac{R}{\sigma }{t}^{\sigma },$$

(8)

where $K,R$ are arbitrary constants with $K,R$≠ 0. Utilizing the transformation Equation (5) together with Equations (6)–(8), the obtained ordinary differential equation is

$$(K+R)w^{\prime }-K^{2}R{w}^{‴}+w{w}^{\prime }K=0.$$

This, when integrated with the integration constant set to zero, yields

$$(K+R)w-K^{2}R{w}^{″}+\frac{w^2}{2}K=0,$$

(9)

where $w^{\prime }=\frac{dw}{d\eta }$.

4. Application of RB Sub-ODE Method

According to RB sub-ODE method [30], the solution for Equation (9) is

$$w^{\prime }=H_1{w}^{2-L}+F_{1}w+G_1{w}^L,$$

(10)

where the constants $H_1,F_1,G_1$, and L will be found later.

Substituting Equation (10) into Equation (9), we have

$$\begin{array}{c}\hfill -3K^{2}R{F}_1{H}_{1}w{\left(\eta \right)}^2+K^{2}LR{F}_1{H}_{1}w{\left(\eta \right)}^2-2K^{2}R{H}_1^{2}w{\left(\eta \right)}^{3-L}+K^{2}LR{H}_1^{2}w{\left(\eta \right)}^{3-L}\\ \hfill -K^{2}R{F}_1{G}_{1}w{\left(\eta \right)}^{2L}-K^{2}LR{F}_1{G}_{1}w{\left(\eta \right)}^{2L}+Kw{\left(\eta \right)}^{1+L}+Rw{\left(\eta \right)}^{1+L}-K^{2}R{F}_1^{2}w{\left(\eta \right)}^{1+L}\\ \hfill -2K^{2}R{G}_1{H}_{1}w{\left(\eta \right)}^{1+L}+\frac{1}{2}Kw{\left(\eta \right)}^{2+L}-K^{2}LR{G}_1^{2}w{\left(\eta \right)}^{-1+3L}=0.\end{array}$$

(11)

Setting $L=0$ in the above equation, we obtain

$$\begin{array}{c}\hfill -K^{2}R{F}_1{G}_1+Kw\left(\eta \right)+Rw\left(\eta \right)-K^{2}R{F}_1^{2}w\left(\eta \right)-2K^{2}R{G}_1{H}_{1}w\left(\eta \right)+\frac{1}{2}Kw{\left(\eta \right)}^2\\ \hfill -3K^{2}R{F}_1{H}_{1}w{\left(\eta \right)}^2-2K^{2}R{H}_1^{2}w{\left(\eta \right)}^3.\end{array}$$

(12)

Adjusting each coefficient of $w^i(i=0,1,2,3)$ to zero, we have

$$\begin{array}{c}\hfill -K^{2}R{F}_1{G}_1=0,\\ \hfill K+R-K^{2}R{F}_1^2-2K^{2}R{G}_1{H}_1=0,\\ \hfill \frac{K}{2}-3K^{2}R{F}_1{H}_1=0,\\ \hfill -2K^{2}R{H}_1^2=0.\end{array}$$

(13)

Equation (13) produces the following results when it is solved.

$$H_1=\mp \frac{1}{6\sqrt{R}\sqrt{K+R}},F_1=\mp \frac{\sqrt{K+R}}{K\sqrt{R}},G_1=0.$$

Case 1.

when $L\ne 1,F_1\ne 0\mathrm{and}{G}_1=0,$ the algebraic solution can be obtained.

$$w\left(\eta \right)={\left(-\frac{H_1}{F_1}+P{\mathrm{e}}^{F_1\left(L-1\right)\eta }\right)}^{\frac{1}{L-1}},$$

(14)

$$W_{1,1}(z,t)=\frac{1}{{\left({\mathrm{e}}^{\frac{3\sqrt{K+R}\eta }{K\sqrt{R}}}P-\frac{K}{6\left(K+R\right)}\right)}^{1/3}}.$$

Case 2.

when $L\ne 1,H_1\ne 0\mathrm{and}{F_1}^2-4H_1{G}_1<0,$ the solitary periodic solutions can be obtained.

$$w\left(\eta \right)={\left(-\frac{F_1}{2H_1}+\frac{\sqrt{4H_1{G}_1-{\left(F_1\right)}^2}}{2H_1}tan\left(\frac{\left(1-L\right)\sqrt{4H_1{G}_1-{\left(F_1\right)}^2}}{2}\left(\eta +P\right)\right)\right)}^{\frac{1}{1-L}},$$

(15)

$$W_{1,2}(z,t)={\left(-\frac{3\left(K+R\right)}{K}-3\sqrt{R}\sqrt{K+R}\sqrt{-\frac{K+R}{K^{2}R}}tan\left(\frac{3}{2}\sqrt{-\frac{K+R}{K^{2}R}}\left(P+\eta \right)\right)\right)}^{1/3}.$$

and

$$w\left(\eta \right)={\left(-\frac{F_1}{2H_1}-\frac{\sqrt{4H_1{G}_1-{\left(F_1\right)}^2}}{2H_1}cot\left(\frac{\left(1-L\right)\sqrt{4H_1{G}_1-{\left(F_1\right)}^2}}{2}\left(\eta +P\right)\right)\right)}^{\frac{1}{1-L}},$$

(16)

$$W_{1,3}(z,t)={\left(-\frac{3\left(K+R\right)}{K}-3\sqrt{R}\sqrt{K+R}\sqrt{-\frac{K+R}{K^{2}R}}cot\left(\frac{3}{2}\sqrt{-\frac{K+R}{K^{2}R}}\left(P+\eta \right)\right)\right)}^{1/3}.$$

Case 3.

when $L\ne 1,H_1\ne 0\mathrm{and}{F_1}^2-4H_1{G}_1>0,$ these dark optical and solitary optical soliton solutions are found, respectively.

$$w\left(\eta \right)={\left(-\frac{F_1}{2H_1}-\frac{\sqrt{-4H_1{G}_1+{\left(F_1\right)}^2}}{2H_1}coth\left(\frac{\left(1-L\right)\sqrt{-4H_1{G}_1+{\left(F_1\right)}^2}}{2}\left(\eta +P\right)\right)\right)}^{\frac{1}{1-L}},$$

(17)

$$W_{1,4}(z,t)={\left(-\frac{3\left(K+R\right)}{K}-3\sqrt{R}\sqrt{K+R}\sqrt{\frac{K+R}{K^{2}R}}coth\left(\frac{3}{2}\sqrt{\frac{K+R}{K^{2}R}}\left(P+\eta \right)\right)\right)}^{1/3}.$$

and

$$w\left(\eta \right)={\left(-\frac{F_1}{2H_1}+\frac{\sqrt{-4H_1{G}_1+{\left(F_1\right)}^2}}{2H_1}tanh\left(\frac{\left(1-L\right)\sqrt{-4H_1{G}_1+{\left(F_1\right)}^2}}{2}\left(\eta +P\right)\right)\right)}^{\frac{1}{1-L}},$$

(18)

$$W_{1,5}(z,t)={\left(-\frac{3\left(K+R\right)}{K}-3\sqrt{R}\sqrt{K+R}\sqrt{\frac{K+R}{K^{2}R}}tanh\left(\frac{3}{2}\sqrt{\frac{K+R}{K^{2}R}}\left(P+\eta \right)\right)\right)}^{1/3}.$$

Case 4.

when $L\ne 1,H_1\ne 0\mathrm{and}{F_1}^2-4H_1{G}_1=0,$ the following algebraic solution is found.

$$w\left(\eta \right)={\left(-\frac{F_1}{2H_1}+\frac{1}{H_1\left(L-1\right)\left(\eta +P\right)}\right)}^{\frac{1}{1-L}},$$

(19)

$$W_{1,6}(z,t)={\left(-\frac{3\left(K+R\right)}{K}+\frac{2\sqrt{R}\sqrt{K+R}}{P+\eta }\right)}^{1/3}.$$

Case 5.

when $L\ne 1,H_1\ne 0\mathrm{and}{G}_1=0,$ the following solution is obtained.

$$w\left(\eta \right)={\left(H_1\left(L-1\right)\left(\eta +P\right)\right)}^{\frac{1}{L-1}},$$

(20)

$$W_{1,7}(z,t)=\frac{{\left(\frac{3}{2}\right)}^{1/4}}{{\left(\frac{P+\eta }{\sqrt{R}\sqrt{K+R}}\right)}^{1/4}}.$$

where P is an arbitrary constant.

5. Utilizing the MAEM

For obtaining the solutions, the MAEM [34] provides the general solution in the form

$$w\left(\eta \right)=H_0+\sum _{k=1}^m\left. [H_K{\left({\varphi }^h\right)}^k+F_k{\left({\varphi }^h\right)}^{-k}\right],$$

(21)

where $H_0$, $H_k$’s and $F_k$’s are unknown constants. The auxiliary equation defines the function $h\left(\eta \right)$.

$$h^{\prime }\left(\eta \right)=\frac{s+m{\varphi }^{-h}+n{\varphi }^h}{ln\varphi },$$

(22)

for arbitrary constant values of $s,m\mathrm{and}n$$(\varphi >0,\varphi \ne 1)$.

Cases for the Equation (22) are discussed below.

• If $s^2-4mn<0\mathrm{and}n\ne 0$, then, ${\varphi }^{h\left(\eta \right)}=\frac{-s+\sqrt{4mn-s^2}tan\left(\frac{\sqrt{4mn-s^2}\eta }{2}\right)}{2n},$ or ${\varphi }^{h\left(\eta \right)}=-\frac{s+\sqrt{4mn-s^2}cot\left(\frac{\sqrt{4mn-s^2}\eta }{2}\right)}{2n}.$
• If $s^2-4mn>0\mathrm{and}n\ne 0$, then, ${\varphi }^{h\left(\eta \right)}=-\frac{s+\sqrt{s^2-4mn}tanh\left(\frac{\sqrt{s^2-4mn}\eta }{2}\right)}{2n},$ or ${\varphi }^{h\left(\eta \right)}=-\frac{s+\sqrt{s^2-4mn}coth\left(\frac{\sqrt{s^2-4mn}\eta }{2}\right)}{2n}.$
• If $s^2-4mn=0\mathrm{and}n\ne 0$, then, ${\varphi }^{h\left(\eta \right)}=-\frac{2+s\eta }{2n\eta }$.

The highest order derivative $w^{″}$ and the highest order nonlinear term $w^2$ in Equation (9) are balanced according to the homogeneous balance principle, yields that $m=2$, gives

$$w\left(\eta \right)=H_0+H_1{\varphi }^h+F_1{\varphi }^{-h}+H_2{\varphi }^{2h}+F_2{\varphi }^{-2h}.$$

(23)

The following set of algebraic equations is obtained by equating each coefficient of ${\varphi }^{h\left(\eta \right)}$ to zero:

$$\begin{array}{c}{{\varphi }^{h\left(\eta \right)}}^{-4}:\frac{1}{2}K{F}_2\left(-12K{m}^{2}R+F_2\right)=0,\\ {{\varphi }^{h\left(\eta \right)}}^{-3}:K\left(-10KmRs{F}_2+F_1\left(-2K{m}^{2}R+F_2\right)\right)=0,\\ {{\varphi }^{h\left(\eta \right)}}^{-2}:-3K^{2}mRs{F}_1+\frac{K{F}_1^2}{2}+F_2\left(K+R-4K^{2}R\left(2mn+s^2\right)+K{H}_0\right)=0,\\ {{\varphi }^{h\left(\eta \right)}}^{-1}:F_1\left(K+R-K^{2}R\left(2mn+s^2\right)+K{H}_0\right)+K{F}_2\left(-6KnRs+H_1\right)=0,\\ {{\varphi }^{h\left(\eta \right)}}^0:-K^{2}nRs{F}_1-2K^2{n}^{2}R{F}_2+K{H}_0+R{H}_0+\frac{K{H}_0^2}{2}-K^{2}mRs{H}_1+K{F}_1{H}_1-2K^2{m}^{2}R{H}_2+K{F}_2{H}_2=0,\\ {{\varphi }^{h\left(\eta \right)}}^1:\left(K+R-K^{2}R\left(2mn+s^2\right)+K{H}_0\right)H_1+K\left(-6KmRs+F_1\right)H_2=0,\\ {{\varphi }^{h\left(\eta \right)}}^2:-3K^{2}nRs{H}_1+\frac{K{H}_1^2}{2}+\left(K+R-4K^{2}R\left(2mn+s^2\right)+K{H}_0\right)H_2=0,\\ {{\varphi }^{h\left(\eta \right)}}^3:K\left(-10KnRs{H}_2+H_1\left(-2K{n}^{2}R+H_2\right)\right)=0,\\ {{\varphi }^{h\left(\eta \right)}}^4:K\left(-10KnRs{H}_2+H_1\left(-2K{n}^{2}R+H_2\right)\right)=0.\end{array}$$

Solving the above equations, yields, the following families.

Family 1:

when $H_0=\frac{-K-R+8K^{2}mnR+K^{2}R{s}^2}{K},H_1=12KnRs,H_2=12K{n}^{2}R,F_1=0,F_2=0.$

The following cases have occurred.

• For $s^2-4mn<0\mathrm{and}n\ne 0$, the trigonometric solutions are found.

  $$W_{1,1}\left(z,t\right)=-1-\frac{R}{K}+8KmnR-2KR{s}^2+3KR\left(4mn-s^2\right)tan{\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)}^2,$$

  or

  $$W_{1,2}\left(z,t\right)=-1-\frac{R}{K}+KR\left(4mn-s^2\right)\left(-1+3csc{\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)}^2\right).$$
• For $s^2-4mn>0\mathrm{and}n\ne 0$, the hyperbolic solutions are obtained.

  $$W_{1,3}\left(z,t\right)=-1-\frac{R}{K}+8KmnR-2KR{s}^2+3KR\left(-4mn+s^2\right)tanh{\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}^2,$$

  or

  $$W_{1,4}\left(z,t\right)=-1-\frac{R}{K}+8KmnR-2KR{s}^2+3KR\left(-4mn+s^2\right)coth{\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}^2.$$

Family 2:

When $H_0=\frac{-K-R+8K^{2}mnR+K^{2}R{s}^2}{K},H_1=0,H_2=0,F_1=12KmRs,F_2=12K{m}^{2}R.$

The following cases are obtained.

• For $s^2-4mn<0\mathrm{and}n\ne 0$, the following trigonometric solutions resulted.

  $$W_{2,1}\left(z,t\right)=-1-\frac{R}{K}+KR\left(s^2+8mn\left(1+\frac{3\left(2mn-s^2+s\sqrt{4mn-s^2}tan\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)\right)}{{\left(s-\sqrt{4mn-s^2}tan\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)\right)}^2}\right)\right),$$

  or

  $$W_{2,2}\left(z,t\right)=-1-\frac{R}{K}+KR\left(s^2+8mn\left(\begin{array}{c}1+\frac{6mn}{{\left(s+\sqrt{4mn-s^2}cot\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)\right)}^2}\hfill \\ -\frac{3s}{s+\sqrt{4mn-s^2}cot\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)}\hfill \end{array}\right)\right).$$
• For $s^2-4mn>0\mathrm{and}n\ne 0$, the following hyperbolic solutions are found.

  $$W_{2,3}\left(z,t\right)=-1-\frac{R}{K}+KR\left(s^2+8mn\left(\begin{array}{c}1+\frac{6mn}{{\left(s+\sqrt{-4mn+s^2}tanh\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)\right)}^2}\hfill \\ -\frac{3s}{s+\sqrt{-4mn+s^2}tanh\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}\hfill \end{array}\right)\right),$$

  or

  $$W_{2,4}\left(z,t\right)=-1-\frac{R}{K}+KR\left(s^2+8mn\left(\begin{array}{c}1+\frac{6mn}{{\left(s+\sqrt{-4mn+s^2}coth\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)\right)}^2}\hfill \\ -\frac{3s}{s+\sqrt{-4mn+s^2}coth\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}\hfill \end{array}\right)\right).$$

Family 3:

When $H_0=\frac{-K-R+8K^{2}mnR+K^{2}R{s}^2}{K},H_1=12KnRs,H_2=12K{n}^{2}R,F_1=12KmRs,F_2=12K{m}^{2}R.$

The cases listed below have occurred.

• For $s^2-4mn<0\mathrm{and}n\ne 0$, the trigonometric solutions are found.

  $$\begin{array}{c}{W}_{3,1}\left(z,t\right)=-1-\frac{R}{K}-2KR{s}^2+3KR\left(4mn-s^2\right)tan{\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)}^2\hfill \\ \hfill +8KmnR\left(1+\frac{3\left(2mn-s^2+s\sqrt{4mn-s^2}tan\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)\right)}{{\left(s-\sqrt{4mn-s^2}tan\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)\right)}^2}\right),\end{array}$$

  or

  $$\begin{array}{c}{W}_{3,2}\left(z,t\right)=-1-\frac{R}{K}-2KR{s}^2+3KR\left(4mn-s^2\right)cot{\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)}^2\hfill \\ \hfill +8KmnR\left(\begin{array}{c}1+\frac{6mn}{{\left(s+\sqrt{4mn-s^2}cot\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)\right)}^2}\hfill \\ -\frac{3s}{s+\sqrt{4mn-s^2}cot\left(\frac{1}{2}\sqrt{4mn-s^2}\eta \right)}\hfill \end{array}\right).\end{array}$$
• For $s^2-4mn>0\mathrm{and}n\ne 0$, the following hyperbolic solutions are obtained.

  $$\begin{array}{c}{W}_{3,3}\left(z,t\right)=-1-\frac{R}{K}-2KR{s}^2+3KR\left(-4mn+s^2\right)tanh{\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}^2\hfill \\ \hfill +8KmnR\left(\begin{array}{c}1+\frac{6mn}{{\left(s+\sqrt{-4mn+s^2}tanh\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)\right)}^2}\hfill \\ -\frac{3s}{s+\sqrt{-4mn+s^2}tanh\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}\hfill \end{array}\right),\end{array}$$

  or

  $$\begin{array}{c}{W}_{3,4}\left(z,t\right)=-1-\frac{R}{K}-2KR{s}^2+3KR\left(-4mn+s^2\right)coth{\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}^2\hfill \\ \hfill +8KmnR\left(\begin{array}{c}1+\frac{6mn}{{\left(s+\sqrt{-4mn+s^2}coth\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)\right)}^2}\hfill \\ -\frac{3s}{s+\sqrt{-4mn+s^2}coth\left(\frac{1}{2}\sqrt{-4mn+s^2}\eta \right)}\hfill \end{array}\right).\end{array}$$

6. Graphical Illustration

This section provides a graphical representation of the BBM-Burger equation solutions that have been found. Concerning the 2D and 3D graphs of $W_{1,1}(z,t)$, $W_{1,3}(z,t)$, $W_{1,4}(z,t)$, respectively, provided the periodic and single wave solutions for the values $\sigma =0.5,1$, P $=1.7,5.5$, K $=14.5,15.5$, R $=-0.5$, $\beta =0.35$, within the interval $-5\le z\le 5,0\le t\le 2$ for 3D graph and t $=1$ for 2D plots, as shown in Figure 1, Figure 2 and Figure 3 by RB sub-ODE method. Figure 4 and Figure 5 represent the Kink and Pulse shape soliton 3D solutions of $W_{1,5}(z,t)$ and $W_{1,6}(z,t)$ for the values $\sigma =0.5$, P $=-1.7,5.5$, K $=14.5,10.5$, R $=14.5,0.5$, $\beta =0.35$ and 2D plots at t $=1$ for the same values in the interval $-5\le z\le 5,0\le t\le 2$ by RB sub-ODE method.

The solutions for the trigonometric and hyperbolic functions in $W_{1,1}(z,t)$ and $W_{1,3}(z,t)$ are that we obtain the anti-bell-shaped solitons and dark soliton solutions, respectively, by choosing the values $\sigma =0.5$, m $=1$, K $=2$, R $=1$, $\beta =0.35$, s $=0.1$, n $=1$, within the range $-10\le z\le 10,-10\le t\le 10$, and t $=1$ for 2D surfaces in Figure 6 and Figure 7 by MAEM. The dark soliton solution, in which the intensity profile of the soliton displays a dip in a uniform backdrop, this hole-soliton, often referred to as a dark soliton, causes a transient reduction in wave amplitude. Solutions for the Family 2 in $W_{2,1}(z,t)$ and $W_{2,3}(z,t)$, by MAEM provides the bright and bell-shaped soliton solutions by taking the values $\sigma =0.5$, m $=1$, K $=-2.5$, R $=1$, $\beta =0.35$, s $=0.1$, n $=1$, within the range $-10\le z\le 10,-10\le t\le 10$, and t $=1$ for 2D surfaces in Figure 8 and Figure 9.

In Figure 10 and Figure 11, the trigonometric and hyperbolic solutions of $W_{3,1}(z,t)$ and $W_{3,3}(z,t)$, we receive the M-shape and W-shape soliton solutions, respectively, by choosing the values $\sigma =0.5$, m $=1$, K $=-1$, R $=0.5$, $\beta =0.35$, s $=0.1$, n $=1$, within the domain $-10\le y\le 10,0\le t\le 5$, and t $=1$ for 2D surfaces by MAEM. In applied sciences, particularly in dispersive systems, the retrieved solutions are important for describing a variety of natural phenomena.

The construction of dark and bright solitons can be seen in Figure 1. The RB sub-ODE technique provides solutions in the algebraic form in Figure 1, the periodic form in Figure 2, and the hyperbolic form in Figure 3. The MAEM also gives trigonometric in Figure 6, Figure 8 and Figure 10 and hyperbolic in Figure 7, Figure 9 and Figure 11 solutions. The bell-shaped soliton in Figure 9, W-shaped soliton in Figure 10 and M-shaped soliton in Figure 11 are also achieved in this work. These graphs demonstrate the dynamical and dispersive behavior of the solitary wave solutions with a suitable choice of parameters. It can be noticed that the wave profile slightly varies when the fractional parameter’s value is changed without changing the form of the curve. A very useful comparison among the different fractional derivatives, including $\beta$, conformable, and M-TD’s, is shown in two-dimensional line graphs.

7. Conclusions

In this study, the RB sub-ODE approach and the MAEM were used to solve the nonlinear BBM-Burger problem, yielding novel, accurate, and analytical solitary wave solutions. These methods provided remarkable solutions that can be operated consistently and simply. Using these conventional and computerized techniques, we could comprehend difficult nonlinear differential equations in a range of scientific domains. The solitons and other traveling wave solutions of the governing model could be found by applying the definitions of derivatives with fractional parameters, i.e., $\beta$-derivative, CD, and M-TD, to each function. Trigonometric and hyperbolic function solutions could be found in the extracted solutions. The outcomes indicated that the MAEM and RB sub-ODE approaches might be used as helpful mathematical tools for extracting the various solitary wave solutions with different differential operators. The current work can be modified in the future to include more evolution equation kinds with various nonlinearities. This paper has studied the comparison of three derivatives. The analysis says that by changing the values of fractional parameters, an effect on wave solutions is observed but M-TD is considered more valuable because, by changing its parameter values, a smooth wave has been observed. This transitive is very effective and useful. The reason for smooth waves is the Mittag–Leffler function of one parameter, which is why better results are obtained in comparison with other derivatives. They can be used by the researcher in the next phases as well. Future research on the BBM-Burger equation may explore the fractional impacts on the solutions of the governing system using, the fractional local derivative, the Atangana–Baleanu derivative, and other recently proposed definitions of fractional derivatives. We can also consider the BBM-Burger equation with stochastic terms. This study confirms that the RB sub-ODE approach and MAEM are effective and useful mathematical methods and are applicable to investigating other fractional NLEEs in science and engineering.