Construction of the Closed Form Wave Solutions for TFSMCH and (1 1) Dimensional TFDMBBM Equations via the EMSE Technique⁺

Abstract

The purpose of this study is to investigate a series of novel exact closed form traveling wave solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE technique. The considered FONLEEs are used to delineate the characteristic of diffusion in the creation of shapes in liquid beads arising in plasma physics and fluid flow and to estimate the external long waves in nonlinear dispersive media. These equations are also used to characterize various types of waves, such as hydromagnetic waves, acoustic waves, and acoustic gravity waves. Here, we utilize the Caputo-type fractional order derivative to fractionalize the considered FONLEEs. Some trigonometric and hyperbolic trigonometric functions have been used to represent the obtained closed form traveling wave solutions. Furthermore, here, we reveal that the EMSE technique is a suitable, significant, and dominant mathematical tool for finding the exact traveling wave solutions for various FONLEEs in mathematical physics. We draw some 3D, 2D, and contour plots to describe the various wave behaviors and analyze the physical consequence of the attained solutions. Finally, we make a numerical comparison of our obtained solutions and other analogous solutions obtained using various techniques.

1. Introduction

The research on the ECFTWSs of NLEEs has significant applications in various nonlinear fields of sciences, such as physics, chemistry, biology, solid-state physics, the proliferation of shallow water wave mechanics, engineering, etc. (for instance, see [1,2,3,4,5,6]). On the other hand, it is well established that FOD-based models have more advantages than the corresponding IOD-based models. There are more degrees of freedom in FOD-based models. FOD provides an exceptional mechanism to describe the memory and hereditary properties of different ingredients and procedures. Moreover, FOD is a dominating tool for controlling several fractional order NLEEs, which are very important in the above-mentioned nonlinear fields of sciences (for example, see the monographs in [7,8,9] and the references therein). Therefore, in this article, we fractionalize the SMCH equation [10,11] and DMBBM equation [12,13,14], applying partial Caputo FOD [15], and obtain the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation, as follows:

$${}_0{}^{C}D_t^{\alpha }u+2k{u}_x+\beta u^2{u}_x-u_{xxt}=0$$

(1)

and

$${}_0{}^{C}D_t^{\alpha }u+u_x-\gamma u^2{u}_x+u_{xxx}=0.$$

(2)

In the last few decades, various techniques have been applied to solve different NLEEs as well as their fractional versions (for example, see [5,6,7,8,10,11,12,13,14,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39] and the references therein). In this research article, we formulate a more general, straightforward, and powerful technique known as the EMSE technique. Furthermore, so far, we know that there is a gap regarding the use of ECFTWSs to formulate the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation by using EMSE like method, and to fill up this gap, here, we use the EMSE technique to obtain some new ECFTWSs of the considered TFSMCH equation and (1 + 1) dimensional TFDMBBM equation. The main objective of this paper is to determine some new, suitable, and acceptable ECFTWSs to formulate the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE method. The specific objectives of this article are as follows:

➢

The formulation of a more general, straightforward, and powerful technique to determine the ECFTWSs of several FONLEEs of mathematical physics.

➢

Involving the Caputo-type fractional order derivative to fractionalize the well-known SMCH equation and (1 + 1) dimensional DMBBM equation.

➢

Investigating a series of novel ECFTWSs for the TFSMCH equation and the (1 + 1) dimensional TFDMBBM equation via the EMSE method.

➢

Analyzing the physical consequence of the attained solutions for certain values of the involved parameters using 3D, 2D, and contour figures.

The next sections of this paper are described as follows:

A brief discussion on PFOD is given in Section 2. Section 3 introduces the TFSMCH equation and the (1 + 1) dimensional TFDMBBM equation. Then, in Section 4, we describe the EMSE method. We determine some new ECFTWSs for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation using the EMSE method. In Section 6, we display some 3D, 2D, and contour figures of the obtained solutions and explain their corresponding physical consequences. In Section 7, we provide a comparative study of our newly obtained solutions and the previously obtained analogous solutions for the considered FONLEEs. Furthermore, in Section 7, we provide a brief discussion on our obtained results. Finally, we conclude this article in Section 8.

2. Notes on Partial Fractional Order Derivatives

The PFOD has been used in various higher order spatial dimensions like the anomalous diffusion phenomenon in ${\mathbb{R}}^2$ (for example, see the monograph in [15]). Among different definitions of PFOD, the most frequently used definitions are partial Riemann–Liouville FOD and partial Caputo FOD (for instance, see [15,40,41,42,43]).

Definition 1.

(see [15,40,41,42,43]). Let $\alpha \in \left[n-1, n\right], n\in \mathbb{N}$ and $u\in L^1\left(J, {\mathbb{R}}^n\right)$. The Riemann–Liouville partial fractional integral of $u\left(x, t\right)$ of order $\alpha$ regarding $t$ is as follows:

$$I_t^{\alpha }u\left(x, t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_0^t{\left(t-T\right)}^{\alpha -1}u\left(x,T\right)dT, t>0$$

(3)

Here, $\mathrm{\Gamma }(\alpha )$ represents Euler’s Gamma function of $\alpha$, and the given integral exists.

Definition 2.

(see [15,40,41,42,43]). Let $\alpha \in \left[n-1, n\right], n\in \mathbb{N}$ and $u\in L^1\left(J, {\mathbb{R}}^n\right)$. The Riemann–Liouville PFOD of $u\left(x, t\right)$ of order $\alpha$ regarding $t$ is as follows:

$${}_0{}^{RL}D_t^{\alpha }u\left(x, t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\partial }^n}{\partial t^n}}u(x, t)& \mathrm{f}\mathrm{o}\mathrm{r} \alpha =n,\\ {\displaystyle \frac{{\partial }^n}{\partial t^n}}\left(I_t^{n-\alpha }u(x, t)\right)& for \alpha \in \left[n-1, n\right), n\in \mathbb{N}.\end{array}\right.$$

(4)

Here, $I_t^{\alpha }u\left(x, t\right)$ denotes the Riemann–Liouville partial fractional integral of $u\left(x, t\right)$ of order $\alpha$.

Definition 3.

(see [15,40,41,42,43]). Let $\alpha \in \left[n-1, n\right], n\in \mathbb{N}$ and $u\in L^1\left(J, {\mathbb{R}}^n\right)$. The partial Caputo FOD of $u(x, t)$ of order $\alpha$ regarding $t$ is as follows:

$${}_0{}^{C}D_t^{\alpha }u\left(x, t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\partial }^n}{\partial t^n}}u(x, t)& \mathrm{f}\mathrm{o}\mathrm{r} \alpha =n,\\ I_t^{n-\alpha }\left({\displaystyle \frac{{\partial }^n}{\partial t^n}}u(x, t)\right)& for \alpha \in \left[n-1, n\right), n\in \mathbb{N}.\end{array}\right.$$

(5)

Here, $I_t^{\alpha }u\left(x, t\right)$ denotes the Riemann–Liouville partial fractional integral of $u\left(x, t\right)$ of order $\alpha$, and $\mathrm{\Gamma }(\alpha )$ is the Euler’s Gamma function of $\alpha$.

3. Governing Equations

3.1. TFSMCH Equation

In fluid dynamics, the CH equation is the following NLPDE:

$$u_t+3u{u}_x+2k{u}_x-u_{xxt}=2u_x{u}_{xx}-u{u}_{xxx}$$

(6)

where u represents the velocity of fluids in a horizontal direction and k represents a constant of critical shallow water wave speed. In 1993, this CH equation was established by Reberto Camassa and Darry Holm [44] as a bi-Hamiltonian model for waves in shallow water. Then, in 2006, Wazwaz [10] studied a set of significant CH equations and named it a modified $\alpha$-equation, which takes the following form:

$$u_t-u_{xxt}+(\alpha +1)u^2{u}_x-\alpha u_x{u}_{xx}-u{u}_{xxx}=0, \forall \alpha \in \mathbb{N}.$$

(7)

Taking $\alpha =2$ in (7), we yield the MCH equation in the following form:

$$u_t-u_{xxt}+3u^2{u}_x-2u_x{u}_{xx}-u{u}_{xxx}=0$$

(8)

In 2012, Irshad et al. [45] simplified the MCH Equation (8) and obtained the following SMCH equation, including $k,\beta$ (parameters):

$$u_t+{2ku}_x-u_{xxt}+\beta u^2{u}_x=0, \forall \beta >0.$$

(9)

The SMCH Equation (9) is a very significant NLPDE, which can define several wave incidents in ocean engineering.

For the generalization of the SMCH equation, in this article we involve the partial Caputo FOD given by Definition 3 in (9) and obtain the TFSMCH Equation (1) involving parameters $k \mathrm{and} \beta$.

3.2. (1 + 1) Dimensional TFDMBBM Equation

The BBM equation has been recognized for its various important implementations. It represents a long-wave model in nonlinear dispersive systems. It is an important NLPDE, which was used to regularize the following long-wave equation:

$$u_t+u_x+u^n{u}_x+u_{xxt}=0,$$

(10)

where $u\left(x, t\right)$ maps from $\mathbb{R}\times \mathbb{R}$ to $\mathbb{R}$ and the subscripts denote the order of partial derivatives with regard to t (time) and x (position coordinate). In 1972, Benjamin, Bona, and Mahony [46] developed the BBM equation as an advancement of the famous Korteweg–De vries (KdV) equation, as follows:

$$u_t+u_x+u{u}_x+u_{xxx}=0$$

(11)

For $n=2$, Equation (10) is identified as the MBBM equation, and the dispersive form of the MBBM equation is known as the (1 + 1) dimensional DMBBM equation, which is written as follows:

$$u_t+u_x-\gamma u^2{u}_x+u_{xxx}=0$$

(12)

Equation (12) was developed to approximate the surface long waves in nonlinear dispersive media. Furthermore, the (1 + 1) dimensional DMBBM equation was also used to analyze various types of waves, such as hydromagnetic waves in cold plasma, acoustic gravity waves in compressible fluids, etc. (for instance, see [13,14]). In this paper, we fractionalize the (1 + 1) dimensional DMBBM Equation (12) using the partial Caputo FOD and obtain the (1 + 1) dimensional TFDMBBM Equation (2) involving parameter $\gamma$.

4. Extended Modified Simple Equation Technique

Here, we discuss the well-known MSE technique in an extended form for selecting the solvability of several FONLEEs; we named it as the EMSE technique.

Suppose the following NLEEs:

$$\mathcal{P}\left(u,{}_0{}^{C}D_t^{\alpha }u,{}_0{}^{C}D_t^{2\alpha }u,{}_0{}^{C}D_t^{\alpha }{u}_x,{}_0{}^{C}D_t^{2\alpha }{u}_x,u_x,u_{xx},u_{xxx},\cdots \right)=0, 0\le \alpha \le 1,t>0,$$

(13)

where $x, t$are independent variables, ${}_0{}^{C}D_t^{\alpha }u,{}_0{}^{C}D_t^{2\alpha }u,{}_0{}^{C}D_t^{\alpha }{u}_x,{}_0{}^{C}D_t^{2\alpha }{u}_x, \dots$ are partial Caputo fractional order derivatives (FODs) of various orders of the unknown function $u=u\left(x,t\right)$ and its partial derivative, and $\mathcal{P}$ represents a polynomial of $u\left(x,t\right)$ and its partial Caputo FODs of various orders, along with the nonlinear terms. Here, the partial integer order derivatives are as follows:

$$u_t={\displaystyle \frac{\partial u}{\partial t}},u_x={\displaystyle \frac{\partial u}{\partial x}},u_{tt}={\displaystyle \frac{{\partial }^{2}u}{\partial t^2}},u_{xx}={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}$$

To solve the NLEE (13) in our EMSE technique, we apply five main steps, which are given below:

• Step 1: First, we convert the real variables $x$ and $t$ to a traveling wave variable $\xi$ by using the following formula:

  $$\xi =x\pm w{\displaystyle \frac{t^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}};u\left(x,t\right)=F\left(\xi \right),$$

  (14)

  where $w$ is the traveling wave velocity.

Then, we transform the partial Caputo fractional differential Equation (13) to an ordinary differential Equation (ODE), applying (14) and we obtain the following:

$$\mathcal{P}\left(u, \pm w{u}^{\prime }, k{u}^{\prime }, {k^{2}u}^{″}, w^2{u}^{″}\dots \right)=0,$$

(15)

where $\mathcal{P}$ is a polynomial of $u\left(\xi \right)$ and its various order ordinary derivatives with regard to $\xi$.

• Step 2: In this step, we suppose that the ECFTWSs of Equation (15) are as follows:

  $$F\left(\xi \right)={\sum }_{j=0}^N{a}_j{\left\{{\displaystyle \frac{s^{\prime }\left(\xi \right)}{s\left(\xi \right)}}\right\}}^j,$$

  (16)

  where $N$ is the positive integer, $a_j$ are unknown constants, with $a_N\ne 0$, and $s\left(\xi \right)$ is an unknown function $s\left(\xi \right),$ with $s^{\prime }\left(\xi \right)={\displaystyle \frac{ds}{d\xi }}\ne 0$. Here, we determine the values of $a_j$ and $s\left(\xi \right)$.
• Step 3: In this step, we obtain the value of $N$ using a homogeneous balancing of the highest-order derivatives and nonlinear terms of Equation (15).

Furthermore, if $deg\left[F\left(\xi \right)\right]=N$, then we can obtain the degree of any other expressions using the following formula:

$$deg\left[{\displaystyle \frac{d^{m}F\left(\xi \right)}{d{\xi }^m}}\right]=N+m \mathrm{and} deg\left[F^m{\left\{{\displaystyle \frac{d^{q}F\left(\xi \right)}{d{\xi }^q}}\right\}}^p\right]=mN+p\left(N+q\right)$$

(17)

• Step 4: Here, first we substitute Equation (16) in Equation (15). Then, we compute $F^{\prime }, F^{″}, F^{‴}$and approximate $s\left(\xi \right)$. After that, we obtain a polynomial of ${\left({\displaystyle \frac{1}{s\left(\xi \right)}}\right)}^j, j=0,1,2\dots$. From the subsequent polynomial, all the coefficients of ${\left({\displaystyle \frac{1}{s\left(\xi \right)}}\right)}^j$ are set to zero. Then, we obtain a set of algebraic and differential equations with $a_j$, $s\left(\xi \right),$ and other necessary parameters.
• Step 5: This step is the final step of the EMSE technique. In this step, we use the obtained solutions for Step 4 to determine $s\left(\xi \right)$ and the values of further required parameters.

5. Construction of Solutions

Here, we explore various novel ECFTWSs for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation by executing the EMSE technique.

5.1. Solutions for TFSMCH Equation

In this subsection, we apply the EMSE technique to evaluate some new ECFTWSs and solitary wave solutions for the TFSMCH equation.

The TFSMCH Equation (1) is as follows:

$${}_0{}^{C}D_t^{\alpha }u+2k{u}_x+\beta u^2{u}_x-u_{xxt}=0$$

(18)

where $\beta >0$ and $k\in \mathbb{R}$ are parameters.

Applying the traveling wave transformation $u\left(x, t\right)=u\left(\xi \right),\xi =x-{\displaystyle \frac{w t^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}$ according to (14), we convert Equation (18) into the following nonlinear ODE (NLODE):

$${-w u}^{\prime }+2k{u}^{\prime }+{w u}^{‴}+\beta u^2{u}^{\prime }=0,$$

(19)

Integrating both sides of (19) with respect to $\xi$ and considering the integrating constant as zero, we yield the following:

$$\left(2k-w\right)u+{w u}^{″}+{\displaystyle \frac{\beta }{3}}{u}^3=0,$$

(20)

Applying the homogeneous balancing of the highest-order derivative $u^{″}$ and the nonlinear term of the highest-order $u^3$ arising in (20), we obtain $N=1$. Hence, the solution for (20) is as follows:

$$u=a_0+a_1\left({\displaystyle \frac{s^{\prime }\left(\xi \right)}{s\left(\xi \right)}}\right),$$

(21)

where $a_0$ and $a_1$ are unknown constants, with $a_1\ne 0,$ and $s\left(\xi \right)$ is an unknown function.

Subsequently, we first determine $u,u^3$, and $u^{″}$ from Equation (21) and then put them in Equation (20) and equate the coefficients of $s^0,s^{-1},s^{-2},s^{-3}$ to zero to obtain the following:

$$2k{a}_0-w{a}_0+{\displaystyle \frac{1}{3}}\beta {a_0}^3=0$$

(22)

$$2k{a}_1{s}^{\prime }-w{a}_1{s}^{\prime }+\beta {a_0}^2{a}_1{s}^{\prime }+w{a}_1{s}^{‴}=0$$

(23)

$$\beta a_0{{a_1}^2\left(s^{\prime }\right)}^2-3w{a}_1{s}^{\prime }{s}^{″}=0$$

(24)

$$2w{a}_1{\left(s^{\prime }\right)}^3+{\displaystyle \frac{1}{3}}\beta {{a_1}^3\left(s^{\prime }\right)}^3=0.$$

(25)

Since ${ a}_1\ne 0$, from (22) and (25), we yield the following:

$$a_0=0,a_0=\pm {\displaystyle \frac{\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}$$

and

$$a_1=\pm {\displaystyle \frac{i\sqrt{6w}}{\sqrt{\beta }}}.$$

Consequently, using (23) and (24), we obtain the following:

$$s\left(\xi \right)=c_3+{\displaystyle \frac{{\beta }^2{a_1}^2{{a_0}^2{c}_1 e}^{\frac{3\left(w-2k-\beta {a_0}^2\right) \xi }{\beta a_0{a}_1}}}{{9\left(w-2k-\beta {a_0}^2\right)}^2}}$$

(26)

where, $\xi =x-{\displaystyle \frac{w{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}$.

• Case 1: When$a_0=0$, the solution is neglected.
• Case 2: When $a_0={\displaystyle \frac{\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}},$ $a_1=\pm {\displaystyle \frac{i\sqrt{6w}}{\sqrt{\beta }}}$, and $s\left(\xi \right)={\mathrm{c}}_3+{\displaystyle \frac{{\mathrm{c}}_1{{\beta }^2{a}_0^2{a}_1^{2}e}^{\frac{3(w-2k-\beta a_0^2)\xi }{\beta a_0{a}_1}}}{9{(\mathrm{w}-2k-\beta a_0^2)}^2}}$, the solution for Equation (21) is as follows:

  $$u(\xi )={\displaystyle \frac{\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}+{\displaystyle \frac{\sqrt{3}{c}_1{we}^{\pm \frac{i\sqrt{2}\sqrt{w-2k} \xi }{\sqrt{w}}}}{\sqrt{w-2k}\sqrt{\beta }(c_3+\frac{c_1{we}^{\pm \frac{i\sqrt{2}\sqrt{w-2k} \xi }{\sqrt{w}}}}{4k-2w})}}$$

  (27)

The solution given in (27) is an expedient and a more universal traveling wave solution, which could not be found in former analogous research. For the various values of $c_1$ and $c_3$, we may obtain the various solutions using the EMSE technique, but for brevity, here, we could not display all the other solutions.

For $w>2k$, if we choose $c_1=-{\displaystyle \frac{2\left(2k-w\right)c_3}{w}}$, using the symbolic computational software MATHEMATICA 12, we convert the solution in (27) to the following trigonometric form:

$$u\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}cot\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right)$$

(28)

Also, for $w>2k,$ if we choose ${ c}_1={\displaystyle \frac{2\left(2k-w\right)c_3}{w}}$, the above solution in (27) reduces to the following soliton form:

$$u\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}tan\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right)$$

(29)

The solutions obtained in (28) and (29) represent the periodic wave solutions.

Again, for $w<2k$, Equations (28) and (29) yield the following periodic wave solutions, respectively:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}coth\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right)$$

(30)

and

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}tanh\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right)$$

(31)

The solutions obtained in Equations (30) and (31) represent the singular periodic wave solutions.

• Case 3: When $a_0=-{\displaystyle \frac{\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}},$ $a_1=\pm {\displaystyle \frac{i\sqrt{6w}}{\sqrt{\beta }}},$ and $s\left(\xi \right)={\mathrm{c}}_3+{\displaystyle \frac{{\mathrm{c}}_1{{\beta }^2{a}_0^2{a}_1^{2}e}^{\frac{3(\mathrm{w}-2k-\beta a_0^2)\xi }{\beta a_0{a}_1}}}{9{(\mathrm{w}-2k-\beta a_0^2)}^2}}$, we obtain the following solution from Equation (21):

  $$u\left(\xi \right)=-{\displaystyle \frac{\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}+{\displaystyle \frac{2\sqrt{3}{c}_{1}w\sqrt{\left(w-2k\right)} e^{\pm \frac{i\sqrt{2}\sqrt{\left(w-2k\right)}\xi }{\sqrt{w}}}}{\sqrt{\beta }(c_{1}w{e}^{\pm \frac{i\sqrt{2}\sqrt{\left(w-2k\right)}\xi }{\sqrt{w}}}+4k{c}_3-2w{c}_3)}}$$

  (32)

The solution given in (32) is an expedient and a more universal traveling wave solution, which could not be found in former analogous research. For the various values of $c_1$ and $c_3$, we may obtain the various solutions using the EMSE technique, but for brevity, here, we could not display all the other solutions.

For $w>2k,$ if we choose $c_1=-{\displaystyle \frac{2\left(2k-w\right)c_3}{w}}$, using the symbolic computational software MATHEMATICA, we convert the solution in (27) to the following trigonometric form:

$$u\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}cot\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right)$$

(33)

Also, for $w>2k$, if we choose${ c}_1={\displaystyle \frac{2\left(2k-w\right)c_3}{w}}$, the above solution in (32) reduces to the following soliton form:

$$u\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}tan\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right)$$

(34)

The solutions obtained in (33) and (34) are well-known periodic wave solutions.

Furthermore, for $w<2k$, Equations (33) and (34) yield the following periodic wave solutions, respectively:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}coth\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right)$$

(35)

and

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}tanh\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right)$$

(36)

The solutions obtained in Equations (35) and (36) represent the singular periodic wave solutions.

If we summarize the solutions obtained in Equations (28), (29), (33) and (34) and the solutions given in Equations (30), (31), (35) and (36), we attain the following solutions:

$$u\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}cot\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right)$$

(37)

$$u\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}tan\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right)$$

(38)

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}coth\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right)$$

(39)

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}tanh\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right)$$

(40)

Applying Equation (14) in Equations (37), (38), (39) and (40), we obtain the following solutions for the TFSMCH Equation (18):

$$u_{\left(1, 2\right)}\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}cot\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right),$$

(41)

$$u_{\left(\mathrm{3,4}\right)}\left(\xi \right)=\pm {\displaystyle \frac{i\sqrt{3\left(w-2k\right)}}{\sqrt{\beta }}}tan\left({\displaystyle \frac{\sqrt{\left(w-2k\right)} \xi }{\sqrt{2w}}}\right),$$

(42)

$$u_{\left(5, 6\right)}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}coth\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right),$$

(43)

$$u_{\left(7, 8\right)}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3\left(2k-w\right)}}{\sqrt{\beta }}}tanh\left({\displaystyle \frac{\sqrt{\left(2k-w\right)} \xi }{\sqrt{2w}}}\right),$$

(44)

where $\xi =x-{\displaystyle \frac{w{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$.

The solutions obtained in Equations (41), (42), (43) and (44) are new, significant, and more universal for the TFSMCH Equation (18).

Consequently, with respect to its application, the attained ECFTWSs of the TFSMCH Equation (18) may play a noteworthy role in explaining the gravity water waves, water wave mechanics, turbulent motion, and fluid flow.

Remark 1.

For the solutions given in Equations (41), (42), (43) and (44), it can be observed that$w\ne 2k$.

5.2. Solutions for (1 + 1) Dimensional TFDMBBM Equation

Here, we apply the EMSE technique to evaluate some novel ECFTWSs and solitary wave solutions for the (1 + 1) dimensional TFDMBBM equation.

The (1 + 1) dimensional TFDMBBM Equation (2) is as follows:

$${}_0{}^{C}D_t^{\alpha }u+u_x-\gamma u^2{u}_x+u_{xxx}=0$$

(45)

where $\gamma$ is a constant, with $\gamma \ne 0$.

Applying the traveling wave transformation $u\left(x, t\right)=u\left(\xi \right),\xi =x-{\displaystyle \frac{w t^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}$ according to (14), we convert Equation (45) into the following NLODE:

$${\left(1-w\right) u}^{\prime }-\gamma u^2{u}^{\prime }+u^{‴}=0$$

(46)

Integrating both sides of (46) regarding to $\xi$ and considering the integrating constant as zero, we obtain the following:

$$\left(1-w\right)u-{\displaystyle \frac{1}{3}}\gamma u^3+u^{″}=0$$

(47)

Applying the homogeneous balancing of the highest-order derivative $u^{″}$ and the nonlinear term of the highest-order $u^3$ arising in (47), we obtain $N=1$. Hence, the solution for (47) is as follows:

$$u=b_0+b_1\left({\displaystyle \frac{r^{\prime }\left(\xi \right)}{r\left(\xi \right)}}\right)$$

(48)

where $b_0$ and $b_1$ are unknown constants, with $b_1\ne 0$, and $r\left(\xi \right)$ is an unknown function.

Subsequently, we first determine $u,u^3$, and $u^{″}$ from Equation (48) and then put them in Equation (47) and equate the coefficients of $r^0, r^{-1},r^{-2},r^{-3}$ to zero to obtain the following:

$$\left(1-w\right)b_0-{\displaystyle \frac{1}{3}} \gamma b_0^3=0$$

(49)

$$\left(1-w\right)r^{\prime }-\gamma b_0^2 r^{\prime }+r^{‴}=0$$

(50)

$$\gamma b_0 b_1{ r}^{\prime }+3r^{″}=0$$

(51)

$$\left(6b_1-\gamma b_1^3\right){\left(r\prime \right)}^3=0$$

(52)

Solving Equations (49) and (52), we obtain the following:

$$b_0=0, { b}_0=\pm {\displaystyle \frac{\sqrt{3 (1-w)}}{\sqrt{\gamma }}}$$

and

$$b_1=\pm {\displaystyle \frac{\sqrt{6}}{\sqrt{\gamma }}},$$

since

$$b_1\ne 0.$$

Consequently, by solving Equations (50) and (51), we obtain the following:

$$r\left(\xi \right)=c_2+{\displaystyle \frac{{\gamma }^2{b_0}^2{b_1}^2{c}_1{e}^{\frac{3(1-w-\gamma {b_0}^2)\xi }{\gamma b_0{b}_1}}}{9{(1-w-\gamma {b_0}^2)}^2}},$$

where $\xi =x-{\displaystyle \frac{w{t}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}}$.

• Case 1: When $b_0=0$, the solution is neglected.
• Case 2: When $b_0={\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}$,$b_1=\pm {\displaystyle \frac{\sqrt{6}}{\sqrt{\gamma }}}$, and, consequently, $r(\xi )=c_2+{\displaystyle \frac{{c_{1}e}^{\pm \sqrt{\left(2-2w\right)}\xi }}{2-2w}}$, we obtain the solution in (48), as follows:

  $$u(\xi )={\displaystyle \frac{\sqrt{3}\sqrt{1-w}}{\sqrt{\gamma }}}-{\displaystyle \frac{\sqrt{6}{c}_1{e}^{\pm \sqrt{\left(2-2w\right)}\xi }}{\sqrt{2-2w}\sqrt{\gamma }\left(c_2+\frac{c_1{e}^{\pm \sqrt{\left(2-2w\right)}\xi }}{2(1-w)}\right)}}$$

  (53)

The solution given in (53) is an expedient and a more universal traveling wave solution, which could not be found in former analogous research. For the various values of $c_1$ and $c_2$, we may obtain the various solutions using the EMSE technique, but for brevity, here, we could not display all the other solutions.

Considering $(1-w)>0$ and $c_1=-2c_2(1-w)$and using the symbolic computational software MATHEMATICA, we convert the solution given in (53) to the following trigonometric form:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}coth\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right)$$

(54)

Again, considering $(1-w)>0$ and $c_1=2c_2(1-w)$ from Equation (53), we yield the following:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}tanh\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right)$$

(55)

The solutions obtained in Equations (54) and (55) are well-known singular periodic wave solutions.

Furthermore, for $\left(1-w\right)<0,$ Equations (54) and (55) yield the following periodic solutions:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}cot\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(56)

and

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}tan\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(57)

The solutions obtained in (56) and (57) represent the periodic wave solutions.

• Case 3: If we put $b_0=-{\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}$,${ b}_1=\pm {\displaystyle \frac{\sqrt{6}}{\sqrt{\gamma }}}$, and, consequently, $r(\xi )=c_2+{\displaystyle \frac{c_1{e}^{\pm \sqrt{\left(2-2w\right)}\xi }}{2-2w}}$, the solution (48) is as follows:

  $$u(\xi )=-{\displaystyle \frac{\sqrt{3}\sqrt{1-w}}{\sqrt{\gamma }}}+{\displaystyle \frac{\sqrt{3}{c}_1{e}^{\sqrt{\left(2-2w\right)}\xi }}{\sqrt{1-w}\sqrt{\gamma }\left(c_2+\frac{{c_{1}e}^{\sqrt{\left(2-2w\right)}\xi }}{2(1-w)}\right)}}$$

  (58)

The solution given in (58) is an expedient and a more universal traveling wave solution, which could not be found in former analogous research. For the various values of $c_1$ and $c_2$, we may obtain the various solutions using the EMSE technique, but for brevity, here, we could not show all the other solutions.

Considering $(1-w)>0$ and $c_1=-2c_2(1-w)$and using the symbolic computational software MATHEMATICA, we convert the solution given in (58) to the following trigonometric form:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}coth\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right)$$

(59)

Again, considering $(1-w)>0$ and $c_1=2c_2(1-w)$ from Equation (59), we yield the following:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}tanh\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right)$$

(60)

The solutions obtained in Equations (59) and (60) are well-known singular periodic wave solutions.

Furthermore, for $\left(1-w\right)<0$, Equations (59) and (60) yield the following periodic solutions:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}cot\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(61)

and

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}tan\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(62)

The solutions obtained in (61) and (62) are well-known periodic wave solutions.

If we summarize the solutions obtained in Equations (54), (55), (59) and (60) and the solutions obtained in Equations (56), (57), (61) and (62), we attain the following subsequent solutions:

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}coth\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right)$$

(63)

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}tanh\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right)$$

(64)

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}cot\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(65)

$$u\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}tan\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(66)

Applying Equation (14) in Equations (63), (64), (65) and (66), we obtain the following solutions:

$$u_{(9, 10)}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}coth\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right),$$

(67)

$$u_{(11, 12)}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(1-w)}}{\sqrt{\gamma }}}tanh\left({\displaystyle \frac{\sqrt{\left(1-w\right)} \xi }{\sqrt{2}}}\right),$$

(68)

$$u_{(13, 14)}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}cot\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right),$$

(69)

$$u_{(15, 16)}\left(\xi \right)=\pm {\displaystyle \frac{\sqrt{3(w-1)}}{\sqrt{\gamma }}}tan\left({\displaystyle \frac{\sqrt{\left(w-1\right)} \xi }{\sqrt{2}}}\right)$$

(70)

where $\xi =x-{\displaystyle \frac{w{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$.

The solutions obtained in Equations (67), (68), (69) and (70) are new, significant, and more general to the (1 + 1) dimensional TFDMBBM Equation (45).

Furthermore, with respect to its application, the attained ECFTWSs of TFSMCH Equation (45) may play a noteworthy role in analyzing various types of waves, such as hydromagnetic waves in cold plasma, acoustic gravity waves in compressible fluids, etc.

Remark 2.

For the solutions given in Equations (67), (68), (69) and (70), it can be observed that$w\ne 1$.

6. Pictorial Demonstration and Their Physical Importance

In this section, we provide the pictorial demonstration and physical importance of the attained solutions. Here, we use the symbolic computational software MATHEMATICA to draw 3D, 2D, and contour plots of the attained solutions, which help us describe the physical sketches more clearly.

6.1. Pictorial Demonstrations of Attained Solutions for TFSMCH Equation

The sketches for the attained solutions in Equations (41), (42), (43) and (44) for various values of the parameters are shown below (Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10):

6.2. Pictorial Demonstrations of Attained Solutions for (1 + 1) Dimensional TFDMBBM Equation

The sketches of figures of the obtained solutions in Equations (67), (68), (69) and (70) for various values of the parameters are shown below (Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18).

6.3. Physical Importance of Attained Solutions

This subsection describes the physical importance of the obtained solutions. Here, we gained sixteen new solutions, where (41), (42), (69) and (70) represent four positive solutions and four negative solutions in trigonometric form and (43), (44), (67) and (68) represent four positive solutions and four negative solutions in hyperbolic trigonometric form. Based on Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 and Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18, we claim that the attained solutions will be effective for testing their physical importance, if we know the relevant inner processes of physical occurrences.

Several shapes have been yielded for the solutions for Equations (41), (42), (43) and (44) within the interval of $0\le x, t\le 20$ and for the solutions for Equations (67), (68), (69) and (70) within the interval of $-4\le x\le 4, 0\le t\le 5$ for the different parameter values. For the set of values $w=5, k=1.5, \beta =1, \alpha =0.5; w=5, k=1.5, \beta =1, \alpha =0.65; w=5, k=1.5,\beta =1, \alpha =0.95$ and $w=5, k=1.5, \beta =1, \alpha =0.5;$ $w=15, k=1.5, \beta =1, \alpha =0.5;$ $w=5, k=1.5, \beta =1, \alpha =0.95$, Figure 1, Figure 2 and Figure 3 and Figure 4, Figure 5 and Figure 6 represent the solutions for Equations (41) and (42), respectively, which are periodic traveling wave solutions for the TFSMCH equation. Similarly, the periodic traveling wave solutions for the (1 + 1) dimensional TFDMBBM equation given by (69) and (70) are shown in Figure 15 and Figure 16 and in Figure 17 and Figure 18 for the following set of values: $w=2.8, \gamma =2.0, \alpha =0.5$; $w=2, \gamma =3.0, \alpha =0.5$and $w=2.56, \gamma =3, \alpha =0.5$; $w=2, \gamma =3, \alpha =0.5$. Consequently, Figure 7 and Figure 8 show the solitary wave solution for the TFSMCH equation, specified by (43) for the following set of values: $w=1, k=1.5, \beta =1, \alpha =0.5$ and $w=2, k=1.5, \beta =1, \alpha =0.5$. Figure 9 and Figure 10 represent the soliton and solitonic solutions for the TFSMCH equation given by (44) for the following set of values:$w=1, k=1.5, \beta =1, \alpha =0.5$ and $w=1, k=1.5, \beta =1, \alpha =0.95$. Consequently, Figure 11 and Figure 12 express the solitary wave solution for the (1 + 1) dimensional TFDMBBM equation given by (67) for the following set of values: $w=0.75, \gamma =1, and \alpha =0.96$ and $w=0.67, \gamma =1, and \alpha =0.5$. Moreover, Figure 13 and Figure 14 represent the kink-type wave solution for the (1 + 1) dimensional TFDMBBM equation given by (68) for the following set of values: $w=0.05, \gamma =1.0, \alpha =0.5$ and $w=0.20, \gamma =1.0, \alpha =0.5$.

Furthermore, we observed that the shapes of the plotted figures in case of the solutions given by (41) to (44) for the TFSMCH equation are not singular periodic waves for $w\le 15,$ $k=1.5, \mathsf{\beta }=1$ and within the interval of $0\le x, t\le 20$. For the values of the parameters in the interval $0\le x, t\le 10$, the solutions may be singular periodic waves. For brevity, here, we could not show the other plotted figures of the solutions given in (41) to (44) considering$w>15$. Similarly, we observed that the shapes of the plotted figures in case of the solutions given by (67) to (70) for the (1 + 1) dimensional TFDMBBM equation are not singular periodic waves for $w<3$ and within the intervals $-4\le x\le 4 \mathrm{a}\mathrm{n}\mathrm{d} 0\le t\le 5$. For different values of parameters within the intervals $-4\le x\le 4 \mathrm{a}\mathrm{n}\mathrm{d} 0\le t\le 5$, the figures’ shapes may be singular periodic waves. For brevity, here, we could not show the other figures of the solutions given by (67) to (70) considering$w\ge 3$.

7. Comparison of Results and Analysis

7.1. Comparison of Obtained Results

In this section, we compare our obtained solutions for the TFSMCH equation and (1 + 1) dimensional TFDMBBM equation by using the EMSE technique with the other analogous results obtained by various techniques (

Table 1. Comparison of the attained solutions and solutions of Ghaffar et al. [24], Alam and Ali [47], and Liu et al. [11] for the TFSMCH equation.

The Attained Solutions	Ghaffar et al.’s Solutions	Alam and Ali’s and Liu et al.’s Solutions
$\mathrm{If} {\mathit{m}}_{\mathbf{1}}={\displaystyle \frac{\sqrt{\mathbf{3}(\mathbf{2}\mathit{k}-\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{w}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{5,6})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{t}\mathit{h}\left({\displaystyle \frac{\sqrt{\left(\mathbf{2}\mathit{k}-\mathit{w}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathit{w}}}}\right)$	If $m_1={\displaystyle \frac{(\mathrm{V}-2k)a_1}{V b_1}}\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}=1$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{19}(\eta )=m_{1}coth\left({\displaystyle \frac{\eta }{2}}\right)$	If $m_1=0, d=0, \sqrt{\mathsf{\Omega }}=1, \mathrm{A}=1$ and $\mathsf{\Phi }=x-\mathrm{v}\mathrm{t}$, then the solution is $u_1(\mathsf{\Phi })=m_{1}coth\left({\displaystyle \frac{\mathsf{\Phi }}{2}}\right)$
$\mathrm{If} {\mathit{m}}_{\mathbf{2}}=-{\displaystyle \frac{\sqrt{\mathbf{3}(\mathbf{2}\mathit{k}-\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{w}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{5,6})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{2}}\mathit{c}\mathit{o}\mathit{t}\mathit{h}\left({\displaystyle \frac{\sqrt{\left(\mathbf{2}\mathit{k}-\mathit{w}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathit{w}}}}\right)$	If $m_2={\displaystyle \frac{(\mathrm{V}-2k)a_1}{V b_1}}\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}=1$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{19}(\eta )=m_{2}coth\left({\displaystyle \frac{\eta }{2}}\right)$	If $m_2=0, d=0, \sqrt{\mathsf{\Omega }}=1, \mathrm{A}=2$ and $\mathsf{\Phi }=x-\mathrm{v}\mathrm{t}$, then the solution is $u_6(\mathsf{\Phi })=m_{2}coth\left({\displaystyle \frac{\mathsf{\Phi }}{2}}\right)$
$\mathrm{If} {\mathit{m}}_{\mathbf{1}}={\displaystyle \frac{\mathit{i}\sqrt{\mathbf{3}(-\mathbf{2}\mathit{k}+\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{v}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{1,2})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{t}\left({\displaystyle \frac{\sqrt{\left(\mathbf{w}-\mathbf{2}\mathit{k}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathbf{w}}}}\right)$	If $\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}={\displaystyle \frac{1}{\sqrt{2}}}$, $m_1={\displaystyle \frac{a_0(V{\lambda }^2+2V+4k)}{2V b_1}}$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{10}(\eta )=m_{1}cot({\displaystyle \frac{-\eta }{2}})$	If $m_2=0, d=0, B=0, t\sqrt{\mathsf{\Omega }}=1, \mathrm{A}=1$ and $\mathsf{\Phi }=x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_3(\mathsf{\Phi })=m_{1}cot\left({\displaystyle \frac{-\mathsf{\Phi }}{2}}\right)$
$\mathrm{If} {\mathit{m}}_{\mathbf{2}}={\displaystyle \frac{\mathit{i}\sqrt{\mathbf{3}(-\mathbf{2}\mathit{k}+\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{v}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{1,2})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{2}}\mathit{c}\mathit{o}\mathit{t}\left({\displaystyle \frac{\sqrt{\left(\mathbf{w}-\mathbf{2}\mathit{k}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathbf{w}}}}\right)$	If $\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}={\displaystyle \frac{1}{\sqrt{2}}}$, $m_2={\displaystyle \frac{a_0(V{\lambda }^2+2V+4k)}{2V b_1}}$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{10}(\eta )=m_{2}cot({\displaystyle \frac{-\eta }{2}})$	If $m_1={\displaystyle \frac{1}{2}}, d=0, t\sqrt{\mathsf{\Omega }}=1, \mathrm{A}=2$ and $\mathsf{\Phi }=x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_8(\mathsf{\Phi })=m_{1}cot\left({\displaystyle \frac{-\mathsf{\Phi }}{2}}\right)$
$\mathrm{If} {\mathit{m}}_{\mathbf{1}}=-{\displaystyle \frac{\sqrt{\mathbf{3}(\mathbf{2}\mathit{k}-\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{v}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{5,6})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{t}\mathit{h}\left({\displaystyle \frac{\sqrt{\left(\mathbf{2}\mathit{k}-\mathit{w}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathbf{w}}}}\right)$	If $\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}=1$, $m_2={\displaystyle \frac{(V+2k)a_1}{V{b}_1}}$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{19}(\eta )=m_{2}coth\left({\displaystyle \frac{-\eta }{2}}\right)$	If $m_1={\displaystyle \frac{1}{2}}, d=0, \sqrt{\mathsf{\Delta }}=1, \mathrm{A}=2$ and $\mathsf{\Phi }=x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_7(\mathsf{\Phi })=m_{2}coth\left({\displaystyle \frac{-\mathsf{\Phi }}{2}}\right)$
$\mathrm{If} {\mathit{m}}_{\mathbf{1}}={\displaystyle \frac{\sqrt{\mathbf{3}(\mathbf{2}\mathit{k}-\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{v}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{5,6})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{t}\mathit{h}\left({\displaystyle \frac{\sqrt{\left(\mathbf{2}\mathit{k}-\mathit{w}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathbf{w}}}}\right)$	If $\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}=1$, $a_1=1, m_2={\displaystyle \frac{(V+2k)}{V{b}_1}}={\displaystyle \frac{2}{5}}$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{20}(\eta )={\displaystyle \frac{2}{5}}coth\left({\displaystyle \frac{-\eta }{2}}\right)$	If $m_1={\displaystyle \frac{1}{2}}, d=0, \sqrt{\mathsf{\Delta }}=1, \mathrm{A}=2$ and $\mathsf{\Phi }=x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_6(\mathsf{\Phi })={\displaystyle \frac{2}{5}}coth\left({\displaystyle \frac{-\mathsf{\Phi }}{2}}\right)$
$\mathrm{If} {\mathit{m}}_{\mathbf{1}}={\displaystyle \frac{-\mathit{i}\sqrt{\mathbf{3}(-\mathbf{2}\mathit{k}+\mathit{w})}}{\sqrt{\mathit{\beta }}}}$$\mathrm{and} \mathit{\xi }=\mathit{x}-{\displaystyle \frac{\mathit{v}{\mathit{t}}^{\mathit{\alpha }}}{\mathbf{\Gamma }\left(\mathit{\alpha }+\mathbf{1}\right)}}$ , $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{1,2})}\left(\mathit{\xi }\right)={\mathit{m}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{t}\left({\displaystyle \frac{\sqrt{\left(\mathbf{w}-\mathbf{2}\mathit{k}\right)}\mathit{\xi }}{\sqrt{\mathbf{2}\mathbf{w}}}}\right)$	If $\sqrt{{\displaystyle \frac{-(2V-4k)}{V}}}={\displaystyle \frac{1}{\sqrt{2}}}$,$m_1={\displaystyle \frac{a_1(V-2k)}{2V{b}_1}}=1$ and $\eta =x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_{20}(\eta )=cot\left({\displaystyle \frac{-\eta }{2}}\right)$	If $m_1={\displaystyle \frac{1}{2}}, m_2=0,d=0, \sqrt{\mathsf{\Delta }}=1, \mathrm{A}=2$ and $\mathsf{\Phi }=x-{\displaystyle \frac{v{t}^{\alpha }}{\Gamma \left(\alpha +1\right)}}$, then the solution is $u_8(\mathsf{\Phi })=cot\left({\displaystyle \frac{-\mathsf{\Phi }}{2}}\right)$

,

Table 2. Comparison of our obtained solutions and the solutions of Ghaffar et al. [24], Mohyud-Din et al. [48], and Najafi et al. [12] for TFSMCH equations.

Our Obtained Solutions	Ghaffar et al.’s Solutions	Mohyud-Din et al.’s and Najafi et al.’s Solutions
$\mathrm{If} \mathit{w}=\mathbf{3}, \mathit{k}=\mathbf{1}, \mathit{\beta }=\mathbf{1},$ $\mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{0},$$\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{1,2})}\left(\mathit{\xi }\right)=-\mathbf{2.311870540}\sqrt{\mathbf{3}}$	If $\lambda =1, \mathsf{\mu }=-1, k=1,{\mathrm{a}}_1=2\sqrt{3}, b_0=1, x=0,t=0,$ then the solution is $u_{(1)}\left(\eta \right)=-2\sqrt{3}$.	If $k=1, \beta =-1, b_1={\displaystyle \frac{1}{2}}. \eta =0, c_1=1$, then the solution is $u_4\left(\eta \right)$ $=-2\sqrt{3}$.
$\mathrm{If} \mathit{w}=\mathbf{6}\sqrt{\mathbf{2}}, \mathit{k}=\mathbf{1}, \mathit{\beta }=\mathbf{2.65},$ $\mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{0}, \mathit{t}=\mathbf{1}$, then the solution is ${\mathit{u}}_{(\mathbf{3,4})}\left(\mathit{\xi }\right)=\mathbf{1.033337426}$	If $\lambda =\sqrt{2}, \mathsf{\mu }=-1, k=1, {\mathrm{a}}_1=3\sqrt{6}, b_0=\sqrt{2}, b_1=1, V=6\sqrt{2}, x=0, t=1,$ then the solution is $u_{(3)}\left(\eta \right)=$ ${\displaystyle \frac{3\sqrt{3}}{5}}$.	If $\lambda =1, k=1, \beta =-1, b_0=1, \eta =0, c_1=1.0,\mathsf{\mu }=1.0,$ we obtain $u_4\left(\eta \right)$ $=$ ${\displaystyle \frac{3\sqrt{3}}{5}}$.
$\mathrm{If} \mathit{w}=\mathbf{6}\sqrt{\mathbf{2}}, \mathit{k}=\mathbf{1}, \mathit{\beta }=\mathbf{2.65},$ $\mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{0}, \mathit{t}=\mathbf{1}$, then the solution is ${\mathit{u}}_{(\mathbf{3,4})}\left(\mathit{\xi }\right)=\mathbf{1.033337426}$	If $\lambda =\sqrt{2}, \mathsf{\mu }=-1, k=1, {\mathrm{a}}_1=3\sqrt{6}, b_0=\sqrt{2}, b_1=1, V=6\sqrt{2}, x=0, t=1,$ then the solution is $u_{(3)}\left(\eta \right)=$ ${\displaystyle \frac{3\sqrt{3}}{5}}$.	If $\lambda =1, k=1, \beta =-1, b_0=1, \eta =0, c_1=1.0, \mathsf{\mu }=1.0,$ we obtain $u_9\left(\eta \right)$ $=$ ${\displaystyle \frac{3\sqrt{3}}{5}}$.
$\mathrm{If} \mathit{w}=\mathbf{6}, \mathit{k}=\mathbf{2.5}, \mathit{\beta }={\displaystyle \frac{\mathbf{1}}{\mathbf{6}}},$ $\mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{0}, \mathit{t}=\mathbf{1}$, $\mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{5,6})}\left(\mathit{\xi }\right)=\mathbf{0.6990667209}\sqrt{\mathbf{6}}$	If $k={\displaystyle \frac{5}{2}}, {\mathrm{a}}_1=1, b_1=-{\displaystyle \frac{1}{6}}, V=6, x=0, t=0,$ then the solution is $u_{(7)}\left(\eta \right)=$ $\sqrt{6}$.	If $\lambda =\sqrt{-1}, k=1, \beta =1, b_0=0, \eta =0, c_1=1,\mathsf{\mu }=1,$ then the solution is $u_6\left(\eta \right)$ $=\sqrt{6}$.

,

Table 3. Comparison of the attained solutions to Nurul et al.’s [32] solutions for TFSMCH equations.

The Attained Solutions	Nurul et al.’s [32] Solutions
$\mathrm{If} \mathit{w}=\mathbf{5}, \mathit{k}=\mathbf{1.5}, \mathit{\beta }=\mathbf{1}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{1}$$, \mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{1,2})}\left(\mathit{\xi }\right)=\pm \mathbf{1.3544913}$	If $w=5, k=1.5, \beta =1, x=1, t=1$, then the solution is $u_{(\mathrm{1,2})}\left(\xi \right)=\pm 0.54276099$
$\mathrm{If} \mathit{w}=\mathbf{5}, \mathit{k}=\mathbf{1.5}, \mathit{\beta }=\mathbf{1}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{1}$$, \mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{3,4})}\left(\mathit{\xi }\right)=\pm \mathbf{4.4297070}$	If $w=5, k=1.5, \beta =1, x=1, t=1$, then the solution is $u_{(\mathrm{3,4})}\left(\xi \right)=\pm 0.306282$
$\mathrm{If} \mathit{w}=\mathbf{1}, \mathit{k}=\mathbf{1.5}, \mathit{\beta }=\mathbf{1}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{2}, \mathit{t}=\mathbf{1}$$, \mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{5,6})}\left(\mathit{\xi }\right)=\pm \mathbf{3.48832}$	If $w=1, k=1.5, \beta =1, x=2, t=1$, then the solution is $u_{(\mathrm{5,6})}\left(\xi \right)=\pm 3.6388$
$\mathrm{If} \mathit{w}=\mathbf{1}, \mathit{k}=\mathbf{1.5}, \mathit{\beta }=\mathbf{1}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{2}, \mathit{t}=\mathbf{1}$$, \mathrm{then} \mathrm{the} \mathrm{solution} \mathrm{is} {\mathit{u}}_{(\mathbf{7,8})}\left(\mathit{\xi }\right)=\pm \mathbf{1.72002}$	If $w=1, k=1.5, \beta =1, x=2, t=1$, then the solution is $u_{(\mathrm{7,8})}\left(\xi \right)=\pm 1.64889$

and

Table 4. Comparison of the attained solutions to Khan et al.’s [49] solutions for the (1 + 1) dimensional TFDMBBM equation.

The Attained Solutions	Khan et al.’s [49] Solutions
If $\mathit{w}=\mathbf{0.20},\mathit{\gamma }=\mathbf{1}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{1}$, then the solution is ${\mathit{u}}_{\left(\mathbf{9},\mathbf{10}\right)}\left(\mathit{\xi }\right)=\pm \mathbf{3.4123295}$	If $w=0.20, \alpha =1, x=1, t=1$, we obtain ${\mathrm{u}}_{3,4}\left(\xi \right)=\pm 3.3187879$
If $\mathit{w}=\mathbf{0.20}, \mathit{\gamma }=\mathbf{1}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{1},$ then the solution is ${\mathbf{u}}_{\left(\mathbf{11},\mathbf{12}\right)}\left(\mathit{\xi }\right)=\pm \mathbf{0.70333184}$	If $w=0.20, \alpha =1, x=1, t=1,$ then the solution is ${\mathrm{u}}_{\mathrm{1,2}}\left(\xi \right)=\pm 0.72315557$
If $\mathit{w}=\mathbf{2}, \mathit{\gamma }=\mathbf{3}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{1}$, then the solution is ${\mathbf{u}}_{\left(\mathbf{13},\mathbf{14}\right)}\left(\mathit{\xi }\right)=\pm \mathbf{0.81220004}$	If $w=2, \alpha =3, x=1, t=1$, then the solution is ${\mathrm{u}}_{5,6}\left(\xi \right)=\pm 0.85451043$
If $\mathit{w}=\mathbf{2}, \mathit{\gamma }=\mathbf{3}, \mathit{\alpha }=\mathbf{0.5}, \mathit{x}=\mathbf{1}, \mathit{t}=\mathbf{1}$, then the solution is ${\mathbf{u}}_{\left(\mathbf{15},\mathbf{16}\right)}\left(\mathit{\xi }\right)=\pm \mathbf{1.2312237}$	If $w=2, \alpha =3, x=1, t=1$, then the solution is ${\mathrm{u}}_{\mathrm{7,8}}\left(\xi \right)=\pm 1.1702607$

).

7.2. Analysis of Obtained Results

Many authors have studied the integer order version of TFSMCH and (1 + 1) dimensional TFDMBBM equations applying various methods for establishing their exact traveling wave solutions. Alam and Ali [47] and Liu et al. [11] applied the generalized (G^//G)-expansion technique in the SMCH equation, Nurul et al. [32] used the MSE method in the SMCH equation, and Ghaffar et al. [24] used the extended rational $exp\left(\left({\displaystyle \frac{-{\psi }^{/}}{\psi }}\right)\left(\eta \right)\right)$ expansion method in the fractional SMCH equation. Mohyud-Din et al. [48] used the rational $exp\left(\phi \left(\eta \right)\right)$ expansion method in TFSMCH, and Najafi et al. [12] applied He’s semi-inverse technique in the CH equation and SMCH equation. Lu et al. [50] used the modified extended auxiliary equation mapping technique in the SMCH equation, and Khan et al. [49] used the MSE method in the (1 + 1) dimensional DMBBME. From the comparison in Table 1, we observed that Ghaffar et al. [24] obtained the solution $u_{19}\left(\eta \right)$, Alam and Ali [47] and Liu et al. [11] obtained the solutions$u_1\left(\eta \right) \mathrm{a}\mathrm{n}\mathrm{d} u_6(\eta )$, which are almost equivalent to our obtained solution $u_{(\mathrm{1,2})}\left(\xi \right)$ for the TFSMCH equation. In addition, our obtained solution $u_{(\mathrm{1,2})}\left(\xi \right)$ for the TFSMCH equation is equivalent to the solutions $u_{10}\left(\eta \right) \mathrm{a}\mathrm{n}\mathrm{d} u_{20}(\eta )$ of Ghaffar et al. [24] and the solutions $u_3\left(\eta \right) \mathrm{a}\mathrm{n}\mathrm{d} u_8(\eta )$ of Alam and Ali [47] and Liu et al. [11], respectively.

From the comparison in Table 2, we observed that our obtained solutions $u_{(\mathrm{1,2})}\left(\xi \right), u_{(\mathrm{3,4})}\left(\xi \right), \mathrm{a}\mathrm{n}\mathrm{d} u_{(\mathrm{5,6})}\left(\xi \right)$ for the TFSMCH equation are almost equivalent to the solutions $u_1\left(\eta \right), u_3\left(\eta \right), \mathrm{a}\mathrm{n}\mathrm{d} u_7(\eta )$ of Ghaffar et al. [24], respectively, as well as the respective solutions $u_4\left(\eta \right), u_9\left(\eta \right), \mathrm{a}\mathrm{n}\mathrm{d} u_6(\eta )$ of Mohyud-Din et al. [48] and Najafi et al. [12].

From the comparison in Table 3, we observed that our obtained solutions for the TFSMCH equation are almost equivalent to the solutions of Nurul et al. [32]. Finally, from the comparison in Table 4, we observed that our obtained solutions $u_{(\mathrm{9,10})}\left(\xi \right), u_{(\mathrm{11,12})}\left(\xi \right),$ $u_{(\mathrm{13,14})}\left(\xi \right)m \mathrm{a}\mathrm{n}\mathrm{d} u_{(\mathrm{15,16})}\left(\xi \right)$ for the (1 + 1) dimensional TFDMBBM equation are almost equivalent to the solutions $u_{3,4}\left(\xi \right), u_{1,2}\left(\xi \right), u_{5,6}\left(\xi \right), and u_{7,8}\left(\xi \right)$ of Khan et al. [49], respectively. Although the aforementioned discussion indicates that our obtained closed form wave solutions are equivalent to the existing closed form wave solutions, there is a gap in the order of the involed derivative. In the comparison in Table 1, Table 2, Table 3 and Table 4, we used $\alpha =0.5$ (order of the derivative) only, but if we use the value of $\alpha$ as $\alpha =0.1, 0.2, 0.3, 0.4, 0.5, \cdots , 1$, then, we can determine many types of distinct new ECFTWSs for the considered NLEEs, which have not been found in the existing literature, and this concept will increase the practical applications of our considered TFSMCH equation and (1 + 1) dimensional TFDMBBM equation. Moreover, here, we observed that the EMSE method obtained more suitable and accurate closed form wave solutions than the other analogous existing method. Therefore, we can claim that our obtained solution generates more accurate and suitable results than other existing parallel methods in the recent literature. Furthermore, it has been established that the dynamics of the fractional order models generated by TFSMCH and (1 + 1) dimensional TFDMBBM equations are more stable than their equivalent integer order models since the stability domain in the fractional order models is greater than the corresponding domain of the integer order models. In this paper, all the obtained results were verified through the computer programming language Maple.

8. Conclusions

In this study, we executed and symbolically accomplished many types of innovative and interesting precise ECFTWSs for the nonlinear TFSMCH and (1 + 1) dimensional TFDMBBM equations. Here, we determined which conditions are formed and how the behavior of waves varies as time is changed by the stated equations using a more general, straightforward, efficient, and powerful EMSE method, with the help of partial Caputo FOD. We obtained a path of ECFTWSs for the considered NLEEs, including kink type, periodic type, and soliton type, with a variety of free parameters. The effect of variation in a parameter is illustrated in the diagram, and we attempted to describe the position by giving the low and high range values. By balancing the parameters, we showed how the wave form changes. Our achieved solutions play an important role in characterizing dispersion in plasma physics arising from liquid drops, optical fibers, fluid flow, surface long waves in nonlinear media, hydromagnetic waves in cold plasma, acoustic waves in inharmonic crystals, and acoustic gravity waves in compressible fluids. The attained solutions in this paper are represented graphically using 3D, 2D, and contour plots and create a new dimension in mathematical physics. These established results confirm that the EMSE method is more general, straightforward, conformable, and effective. Furthermore, it may be applied to find the ECFTWSs for many other NLEES arising in mathematical physics and engineering.