Analytical solution to fractional differential equation arising in thermodynamics

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<big>'''Analytical solution to fractional differential equation arising in thermodynamics'''</big></div>

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Areej Almuneef <math display="inline">{}^{1,a}</math>, Ahmed Eissa Hagag <math display="inline">{}^{2,b,\ast }</math><br/> <math display="inline">{}^{1}</math>Department of Mathematical Science, College of Science, Princess Nourah bint Abdulrahman University,<br/>P.O. Box 105862, Riyadh 11656, Saudi Arabia<br/> <math display="inline">{}^{2}</math>Department of Basic Science, Faculty of Engineering, Sinai University, Ismailia, Egypt<br/> <math display="inline">{}^{1,a}</math>[mailto:aaalmuneef@pnu.edu.sa aaalmuneef@pnu.edu.sa]<br/> <math display="inline">{}^{2,b,\ast }</math>[mailto:E-mail:ahmed.shehata@su.edu.eg E-mail:ahmed.shehata@su.edu.eg]</div>-->
==Abstract==

The analysis of nonlinear events related to physical phenomena is a popular issue in the modern day. The essential purpose of this work is to discover a novel approximate solution to the fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation (BBMPB) utilizing the natural decomposition method (NDM) of fractional order. The suggested approach provides analytical solutions that are extremely near to the exact solution obviating the complexities associated with many other approaches. The expected issue’s uniqueness theorem and convergence analysis are explored using Banach’s fixed-point theory. The reliability and accuracy of the recommended method were tested using numerical simulations. The graphs and tables reflect the results. The comparison of the suggested scheme’s solution with the exact solutions demonstrates that the scheme is efficient, methodical, and extremely exact in tackling nonlinear complicated phenomena.

'''Keywords''': Fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation, convergence analysis, fractional natural decomposition method

==1. Introduction==

Differential equations (DEs) are becoming extremely important in industrial applications. These are necessary and stimulating since the majority of the operations are connected with rates of change, which are clearly shown by them. In particular, DEs provide concepts for analyzing occurrences and creating ideas in medicine, finance, engineering, economics, and other related fields of research [1-2]. The analysis and examination of these kinds of equations are based on the survey of the foundations that govern the majority of physical phenomena. Furthermore, the analysis of nonlinear systems using fractional operators is crucial for studying phenomena in everyday life. While illustrating real-world issues connected with complexity, the researchers investigated its characteristics in greater depth and discovered that each notion has its own boundaries. However, several scholars discovered numerous limits and flaws in classical calculus while researching problems involving memory or hereditary characteristics. Many researchers use the core concepts and accompanying principles of FC to illustrate their points of view on many types of nonlinear phenomena [3-5]. They later suggested additional operators defined using fractional order. Accordingly, many scholars are drawn to the notion of fractional calculus while examining various models [6-8].

The research on nonlinear analysis in relation to the everyday demands of living beings drew the attention of all scholars due to its importance in modernization. Finding the solution for the relevant system is as important as modeling with mathematical tools. In this way, there are various techniques accessible in the literature [9-11]. Furthermore, each algorithm has its own set of requirements as well as its own set of restrictions. On the other hand, scholars are developing new techniques by overcoming constraints such as large computations, low precision, complex procedures, calculating time, and so on. There are several strategies available in the literature, many of which are quite accurate. The Adomian decomposition technique is one of the approaches with excellent accuracy and dependability [12-13]. Researchers are always exploring and attempting to suggest new techniques by altering, fostering, combining, or upgrading current ones. In this way, the researchers proposed a new method by introducing natural transform (NT) to the ADM, which is called the natural decomposition method (NDM) in the classical order [14-16]. Then this method was generalized and presented in fractional order [17-20].

In many areas of mathematics and science, pseudo-parabolic equations are found, and the highest-order term in these equations has a one-time derivative. They have been utilized to study clay consolidation, thermodynamics, shear in second-order fluids, fluid flow in fissured rock, and propagation of long waves with tiny amplitudes, among other things [21-24]. The generalized Benjamin-Bona-Mahony-Burgers (BBMB) equation is a significant particular instance of pseudo-parabolic-type equations that can be written as follows:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{\omega }_{t}-{\omega }_{xxt}-\alpha {\omega }_{xx}+\gamma {\omega }_{x}+g{\left( \omega \right) }_{x}\,\, = \,\, 0,</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(1)
|}

where <math display="inline">\gamma</math>  denotes any genuine constant value, <math display="inline">\alpha</math>  denotes a positive constant, <math display="inline">\omega \left( x,t\right)</math>  is the horizontal fluid velocity, and <math display="inline">g\left( \omega \right)</math>  is a nonlinear <math display="inline">{C}^{2}</math> smooth function. Peregrine [25], and Benjamin et al. [26] suggested an alternative regularised long-wave equation if <math display="inline">\alpha =</math><math>0</math>, <math display="inline">\gamma =1,</math> and <math display="inline">g{\left( \omega \right) }_{x}=</math><math>\omega {\omega }_{x}</math> in Eq. (1), which is known as the Kortewegde Vries equation

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{\omega }_{t}+{\omega }_{xxt}+{\omega }_{x}+\omega {\omega }_{x}\,\, = \,\, 0.</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(2)
|}


If <math display="inline">g{\left( \omega \right) }_{x}=\theta \omega {\omega }_{x}+</math><math>\beta {\omega }_{xxx}</math> in Eq. (1), then the generic form of the BBMPB equation is thus obtained as follows

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{\omega }_{t}-{\omega }_{xxt}-\alpha {\omega }_{xx}+\gamma {\omega }_{x}+\theta \omega {\omega }_{x}+\beta {\omega }_{xxx}\,\, = \,\, 0.</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(3)
|}


If we put <math display="inline">\alpha =\beta =0</math> in Eq. (3), then this is the Benjamin-Bona-Mahony (BBM) equation in its general form

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{\omega }_{t}-{\omega }_{xxt}+\gamma {\omega }_{x}+\theta \omega {\omega }_{x}\,\, = \,\, 0,</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(4)
|}

where <math display="inline">\theta \not =0</math> and <math display="inline">\gamma</math>  are arbitrary constants. Eq. (4) contains various forms of BBM equations as seen in the research [27-30].

In this article, we use NDM to solve the fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation. In addition, the behavior of the results is described in terms of fractional order. The remainder of the work is organized as follows: in the next section, we explain the fundamental concepts of FC and NDM of fractional order, which are then used to obtain the needed results. Section 3 shows the basic solution method of the proposed technique using the Caputo fractional operator. Section 4 proves the proposed algorithm’s uniqueness theorem and convergence analysis. In Section 5, we obtain the solution to the fractional nonlinear BBMPB problem using the fundamental NDM. In addition, we give the numerical results and graphs for the found solution in the same section. Finally, we draw conclusions about the stored findings in terms of the considered technique and model.

==2. Preliminaries to FC==

===2.1 Definition I===

The fractional integral operator of Riemann-Liouville of a function <math display="inline">\omega \left( \theta \right) \in {C}_{\zeta },\zeta \geq -</math><math>1</math> is defined in Podlubny [3]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{J}_{a}^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{1}{\Gamma \left( \zeta +1\right) }\displaystyle\int_{a}^{\theta }\omega \left( \theta \right) {\left( d\theta \right) }^{\zeta }=\frac{1}{\Gamma \left( \zeta \right) }\displaystyle\int_{a}^{\theta }{\left( \theta -\upsilon \right) }^{\zeta -1}\omega \left( \upsilon \right) d\upsilon ,&\theta ,\zeta >0,\\\omega \left( \theta \right) ,&\zeta =0.\end{matrix}\right.</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(5)
|}

===2.2 Definition II ===
The Caputo fractional differential operator of order <math display="inline">\zeta >0</math> is defined in Podlubny [3]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{D}_{\ast }^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{{d}^{k}}{d{\theta }^{k}}\omega \left( \theta \right) ,&\zeta =k\displaystyle\in N,\\\frac{1}{\Gamma \left( k-\zeta \right) }\displaystyle\int_{0}^{\theta }{\left( \theta -\upsilon \right) }^{k-\zeta -1}{\omega }^{\left( k\right) }\left( \upsilon \right) d\upsilon ,&k-1<\zeta \leq k\in N.\end{matrix}\right.</math> 
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(6)
|}

===2.3 Definition III===

The Mittag-Leffler of a one-parameter function <math display="inline">{E}_{\zeta }\left( \theta \right)</math>  with <math display="inline">\zeta >0</math> is given in Mainardi [31]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{E}_{\zeta }\left( \theta \right) =\sum _{i=0}^{\infty }\frac{{\theta }^{i}}{\Gamma \left( i\zeta +1\right) }</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(7)
|}

===2.4 Definition IV===

The natural transform (NT) of <math display="inline">\omega \left( \theta \right)</math>, which is defined as

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{N}^{+}\left\{\omega (\theta ) \right\} \,\, = \,\, \int_{-\infty}^{\infty}{e}^{-s\theta}\omega (z\theta) d\theta,\quad s,z\in (-\infty,\infty).</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(8)
|}

===2.5 Definition V===

The effect of the natural transform on the Caputo operator is given in Loonker and Banerji [32]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{N}^{+}\left\{ {D}^{\zeta }\omega \left( \theta \right) \right\} \,\, = \,\, \frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left\{ \omega \left( \theta \right) \right\} -</math><math>\sum _{i=0}^{n-1}\frac{{z}^{i-\zeta }}{{s}^{i+1-\zeta }}{\left[ {D}^{i}\omega \left( \theta \right) \right] }_{\theta =0},n-</math><math>1<\zeta \leq n.</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(9)
|}

==3. Construction of fractional NDM==

We consider a general form of fractional nonlinear partial differential equation to demonstrate the underlying theory and solution technique of the suggested approach as

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{D}_{t}^{\zeta }\omega \left( x,t\right) +R\omega \left( x,t\right) +F\omega \left( x,t\right) \,\, = \,\, \hslash \left( x,t\right) ,n-1<\zeta \leq n,</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(10)
|}

with the initial condition

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\omega \left( x,0\right) \,\, = \,\, v\left( x\right) ,</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(11)
|}

where <math display="inline">{D}_{t}^{\zeta }=\frac{{\partial }^{\zeta }}{\partial {t}^{\zeta }}</math> denotes the Caputo operator of <math display="inline">\omega \left( x,t\right)</math>, <math display="inline">R</math> denotes the linear function, <math display="inline">F</math> denotes the non-linear function, and <math display="inline">\hslash \left( x,t\right)</math>  signifies the source term. Using the NT on Eq. (10), we get

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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|<math>{N}^{+}\left[D_{t}^{\zeta}\omega\left(x,t\right)\right]+\mathbb{N}^{+}\left[\mathcal{R}\omega(x,t)\right]+\mathbb{N}^{+}\left[\mathcal{F}\omega(x,t)\right] \,\, = \,\, \mathbb{N}^{+}\left[\hbar\left(x,t\right)\right].</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|
|}

Applying definition 5, we get

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{N}^{+}\left[ \omega \left( x,t\right) \right] \,\, = \,\,  \frac{{z}^{\zeta }}{{s}^{\zeta }}\sum _{i=0}^{n-1}\frac{{z}^{i-\zeta }}{{s}^{i+1-\zeta }}{\left[ {D}^{i}\omega \left( t\right) \right] }_{t=0}+\frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ \hslash \left( x,t\right) \right] -\frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ R\omega \left( x,t\right) +F\omega \left( x,t\right) \right].</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(12)
|}


Utilize the inverse NT on the above equation to obtain

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\omega \left( x,t\right) \,\, = \,\, H\left( x,t\right) +{N}^{-}\left\{ \frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ \hslash \left( x,t\right) -R\omega \left( x,t\right) -F\omega \left( x,t\right) \right] \right\} .</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(13)
|}

<math display="inline">H\left( x,t\right)</math>  identified using nonhomogeneous terms and the provided guess condition. The infinite series solution is given as

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\omega (x,t)=\sum_{n=0}^{\infty }{\omega}_{n}(x,t), \qquad F\omega (x,t) =\sum_{n=0}^{\infty }{A}_{n},</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(14)
|}

where <math display="inline">{A}_{n}</math> is signifies the nonlinear component of <math display="inline">F\omega \left( x,t\right)</math>, and we have

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\sum _{n=0}^{\infty }{\omega }_{n}\left( x,t\right) \,\, = \,\, H\left( x,t\right) +{N}^{-}\left\{ \frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ \hslash \left( x,t\right) -R\omega \left( x,t\right) -F\omega \left( x,t\right) \right] \right\} .</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(15)
|}


Lastly, the analytical solutions are provided in the form of

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\omega \left( x,t\right) \,\, = \,\, \sum _{n=0}^{\infty }{\omega }_{n}\left( x,t\right) .</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(16)
|}

<span id='convergence-analysis-of-ndm'></span>

==4. Convergence analysis of NDM==

The uniqueness and existence theorems are the instruments that lead one to infer that there is only one solution that satisfies a specific initial condition for a given problem.

===4.1 Theorem 1===

The solution provided with the aid of NDM for the BBMPM equation is unique wherever <math display="inline">\chi \in \left( 0,1\right)</math>, where

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\chi \,\, = \,\, \left\{ {\epsilon }^{3}+\alpha {\varphi }^{2}-\gamma \varphi -\theta \left( X+Y\right) -\beta {\varphi }^{3}\right\} \Xi </math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(17)
|}

===Proof===

The analytical solution determined for the BBMPM equation is given as

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\omega \left( x,t\right) =\sum _{j=0}^{\infty }{\omega }_{j}\left( x,t\right) {q}^{j},</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(18)
|}

where

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>{\omega }_{m+1}\left( x,t\right) ={N}^{-}\left\{ \frac{\phi \left( x\right) }{s}\right\} +{N}^{-}\Bigg\{ \left( \frac{{z}^{\zeta }}{{s}^{\zeta }}\right) {N}^{+}\Bigg[{\left( {\omega }_{m}\right) }_{xxt}+\alpha {\left( {\omega }_{m}\right) }_{xx}-\gamma {\left( {\omega }_{m}\right) }_{x}-</math><math>\theta \sum _{i=0}^{m}{\omega }_{i}{\left( {\omega }_{m-i}\right) }_{x}-\beta {\left( {\omega }_{m}\right) }_{3x}\Bigg]\Bigg\}.</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(19)
|}


Let <math display="inline">\omega</math>  and  be the two solutions for the BBMPM equation such that <math display="inline">\left| \omega \right| \leq X</math> and <math display="inline">\left| \right| \leq Y</math>, then usage of the equation above, we obtain

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\vert \omega -\vert \,\, = \,\, \vert {N}^{-}\left\{ \left( \frac{{z}^{\zeta }}{{s}^{\zeta }}\right) {N}^{+}\left[ \left( {\omega }_{xxt}-{}_{xxt}\right) +\alpha \left( {\omega }_{xx}-{}_{xx}\right) -\gamma \left( {\omega }_{x}-{}_{x}\right) -\theta \left( {\omega }^{2}-{}^{2}\right) -\beta \left( {\omega }_{3x}-{}_{3x}\right) \right] \right\} \vert .</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(20)
|}

Transform by using the convolution principle for NT, we obtain

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\begin{align}\vert \omega -\vert &=\,\,\int_{0}^{\phi }\left\{ \vert {\omega }_{2xt}-{}_{2xt}\vert +\alpha \vert {\omega }_{2x}-{}_{2x}\vert -\gamma \vert {\omega }_{x}-{}_{x}\vert -\theta \vert {\omega }^{2}-{}^{2}\vert -\beta \vert {\omega }_{3x}-{}_{3x}\vert \right\} \frac{{\left( \phi -\tau \right) }^{\zeta }}{\Gamma \left( \zeta +1\right) }d\tau .\\
&\leq \,\,\int_{0}^{\phi }\left\{ \frac{{\partial }^{3}}{\partial {x}^{2}\partial t}\vert \omega -\vert +\frac{\alpha {\partial }^{2}}{\partial {x}^{2}}\vert \omega -\vert -\frac{\gamma \partial }{\partial x}\vert \omega -\vert -\theta \vert \left( \omega -\right) \left( \omega +\right) \vert -\frac{\beta {\partial }^{3}}{\partial {x}^{3}}\vert \omega -\vert \right\} \frac{{\left( \phi -\tau \right) }^{\zeta }}{\Gamma \left( \zeta +1\right) }d\tau .\\
&\leq \,\,\left\{ \frac{{\partial }^{3}}{\partial {x}^{2}\partial t}\vert \omega -\vert +\frac{\alpha {\partial }^{2}}{\partial {x}^{2}}\vert \omega -\vert -\frac{\gamma \partial }{\partial x}\vert \omega -\vert -\theta \vert \omega -\vert \vert \omega +\vert -\frac{\beta {\partial }^{3}}{\partial {x}^{3}}\vert \omega -\vert \right\} \int_{0}^{\phi }\frac{{\left( \phi -\tau \right) }^{\zeta }}{\Gamma \left( \zeta +1\right) }d\tau .\\
&\leq \,\,\left\{ {\epsilon }^{3}\vert \omega -\vert +\alpha {\varphi }^{2}\vert \omega -\vert -\gamma \varphi \vert \omega -\vert -\theta \left( X+Y\right) \vert \omega -\vert -\beta {\varphi }^{3}\vert \omega -\vert \right\} \frac{{\phi }^{\zeta +1}}{\Gamma \left( \zeta +2\right) }, \end{align}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(21)
|}

where <math display="inline">{\varphi }^{n}=\frac{{\partial }^{n}}{\partial {x}^{n}},n=</math><math>1,2,3</math> and <math display="inline">{\epsilon }^{3}=\frac{{\partial }^{3}}{\partial {x}^{2}\partial t}</math>. To minimise the previous equation as follows, we can use the integral mean value [33]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center;margin:auto;width: 100%;"
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| <math>\begin{align}\vert \omega -\vert \,\, & \leq \,\, \left\{ {\epsilon }^{3}\vert \omega -\vert +\alpha {\varphi }^{2}\vert \omega -\vert -\gamma \varphi \vert \omega -\vert -\theta \left( X+Y\right) \vert \omega -\vert -\beta {\varphi }^{3}\vert \omega -\vert \right\} \Xi .\\ 
&\leq \,\,\chi \left| \omega -\right|,\end{align}  </math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(22)
|}

<math display="inline">\therefore \left( 1-\chi \right) \vert \omega -\vert \leq 0</math>. Since <math display="inline">0<\chi <1</math>, therefore <math display="inline">\vert \omega -</math><math>\vert =0</math>, which gives <math display="inline">\omega =</math>, where <math display="inline">\omega =</math><math>\frac{{\phi }^{\zeta +1}}{\Gamma \left( \zeta +2\right) }</math>. Hence, the analytical solution is unique.

===4.2 Theorem 2===

Presume that

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: center; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\left\| F\left( \omega \right) -F\left( s\right) \right\| \,\,\leq \,\, \chi \left\| \omega -s\right\| ,\forall \omega ,s\in B,\chi \in \left( 0,1\right) ,</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(23)
|}

where <math display="inline">B</math> is a Banach space with <math display="inline">F:B\rightarrow B</math>. The preceding theorem and the fixed-point principle of Banach [34] was used to infer that <math display="inline">F</math> has a fixed point. Furthermore, the analytical solution acquired utilizing the suggested procedure converges with a random election for <math display="inline">{\omega }_{0},{s}_{0}\in B</math> to a fixed point of <math display="inline">F</math> and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: center; margin:auto;width: 100%;" 
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| style="text-align: center;" |  <math>\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert \,\,\leq \,\, \frac{{\chi }^{\phi }}{1-\chi }\left\| {\omega }_{1}-{\omega }_{0}\right\| .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(24)
|}

===Proof===

Assume that <math display="inline">B</math> a Banach space <math display="inline">\left( C\left[ J\right] ,\Vert .\Vert \right)</math>  of all continuous functions. We can agree that <math display="inline">\left\{ {\omega }_{\mu }\right\}</math>  is a Cauchy sequence in the Banach space as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert \,\, &=\,\,\underbrace{\max}_{\phi \in J}\left| {\omega }_{\mu }-{\omega }_{\sigma }\right| \\
&=\,\, \underbrace{\max}_{\phi \in J}\left| {N}^{-}\left\{\left( \frac{{z}^{\zeta }}{{s}^{\zeta }}\right) {N}^{+}\left[\left( \frac{{\partial }^{3}{\omega }_{\mu -1}}{\partial {x}^{2}\partial t}-\frac{{\partial }^{3}{\omega }_{\sigma -1}}{\partial {x}^{2}\partial t}\right) +\alpha \left( \frac{{\partial }^{2}{\omega }_{\mu -1}}{\partial {x}^{2}}-\frac{{\partial }^{2}{\omega }_{\sigma -1}}{\partial {x}^{2}}\right) \right.\right.\right. \\
& \left.\left. \left. -\gamma \left( \frac{{\partial }^{2}{\omega }_{\mu -1}}{\partial x}-\frac{{\partial }^{2}{\omega }_{\sigma -1}}{\partial x}\right)-\theta \left( {\omega }_{\mu -1}^{2}-{\omega }_{\sigma -1}^{2}\right) -\beta \left( \frac{{\partial }^{3}{\omega }_{\mu -1}}{\partial {x}^{3}}-\frac{{\partial }^{3}{\omega }_{\sigma -1}}{\partial {x}^{3}}\right) \right]\right\} \right| \end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(25)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &\leq \,\, \underbrace{\max}_{\phi \in J}\left[ {N}^{-}\left\{ \left( \frac{{z}^{\zeta }}{{s}^{\zeta }}\right) {N}^{+}\left[ \left| \frac{{\partial }^{3}{\omega }_{\mu -1}}{\partial {x}^{2}\partial t}-\frac{{\partial }^{3}{\omega }_{\sigma -1}}{\partial {x}^{2}\partial t}\right| +\alpha \left| \frac{{\partial }^{2}{\omega }_{\mu -1}}{\partial {x}^{2}}-\frac{{\partial }^{2}{\omega }_{\sigma -1}}{\partial {x}^{2}}\right|  \right. \right. \right. \\
&\left. \left. \left. -\gamma \left| \frac{{\partial }^{2}{\omega }_{\mu -1}}{\partial x}-\frac{{\partial }^{2}{\omega }_{\sigma -1}}{\partial x}\right| -\theta \left| {\omega }_{\mu -1}-{\omega }_{\sigma -1}\right| \left| {\omega }_{\mu -1}+{\omega }_{\sigma -1}\right| -\beta \left| \frac{{\partial }^{3}{\omega }_{\mu -1}}{\partial {x}^{3}}-\frac{{\partial }^{3}{\omega }_{\sigma -1}}{\partial {x}^{3}}\right| \right] \right\} \right] .\end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(26)
|}

Transform by using the convolution principle for NT, we obtain

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}\left\| {\omega }_{\mu }-{\omega }_{\sigma }\right\| & \leq \,\,\underbrace{\max}_{\phi \in J}\left[ \left\{ {\epsilon }^{3}\left| {\omega }_{\mu -1}-{\omega }_{\sigma -1}\right| +\alpha {\varphi }^{2}\left| {\omega }_{\mu -1}-{\omega }_{\sigma -1}\right| -\gamma \varphi \left| {\omega }_{\mu -1}-{\omega }_{\sigma -1}\right| \right. \right. \\
&\left. \left. -\theta \left( X+Y\right) \left| {\omega }_{\mu -1}-{\omega }_{\sigma -1}\right| -\beta {\varphi }^{3}\left| {\omega }_{\mu -1}-{\omega }_{\sigma -1}\right| \right\} \int_{0}^{\phi }\frac{{\left( \phi -\tau \right) }^{\zeta }}{\Gamma \left( \zeta +1\right) }d\tau \right] .\end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(27)
|}


To minimise the previous equation as follows, we can use the integral mean value [33]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &\,\,\,\, \leq \,\,\,\, \underbrace{\max}_{\phi \in J}\left[ \left\{ {\epsilon }^{3}\vert {\omega }_{\mu -1}-{\omega }_{\sigma -1}\vert +\alpha {\varphi }^{2}\vert {\omega }_{\mu -1}-{\omega }_{\sigma -1}\vert -\gamma \varphi \vert {\omega }_{\mu -1}-{\omega }_{\sigma -1}\vert \right. \right. \\
&\left. \left. \qquad -\theta \left( X+Y\right) \vert {\omega }_{\mu -1}-{\omega }_{\sigma -1}\vert -\beta {\varphi }^{3}\vert {\omega }_{\mu -1}-{\omega }_{\sigma -1}\vert \right\} \Xi \right] \\
\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &\,\,\,\,\leq \,\,\,\, \chi \Vert {\omega }_{\mu -1}-{\omega }_{\sigma -1}\Vert .\end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(28)
|}

Subtracting <math display="inline">\mu</math>  by <math display="inline">\sigma +</math><math>1</math>, we obtain

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\left\| {\omega }_{\sigma +1}-{\omega }_{\sigma }\right\| \leq \chi \left\| {\omega }_{\sigma }-\right. </math><math>\left. {\omega }_{\sigma -1}\right\| \leq {\chi }^{2}\left\| {\omega }_{\sigma -1}-\right. </math><math>\left. {\omega }_{\sigma -2}\right\| \leq \cdots \leq {\chi }^{\sigma }\left\| {\omega }_{1}-\right. </math><math>\left. {\omega }_{0}\right\| .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(29)
|}

By using triangular inequality, we obtain

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &=\,\,\Vert {\omega }_{\sigma +1}+{\omega }_{\sigma +2}+{\omega }_{\mu }-{\omega }_{\sigma +1}-{\omega }_{\sigma +2}-{\omega }_{\sigma }\Vert \\
&=\,\,\Vert {\omega }_{\sigma +1}+{\omega }_{\sigma +2}+\cdots +{\omega }_{\mu }-{\omega }_{\mu -1}-\cdots -{\omega }_{\sigma +2}-{\omega }_{\sigma +1}-{\omega }_{\sigma }\Vert \\
&\leq \,\,\left\{ {\chi }^{\sigma }+{\chi }^{\sigma +1}+\cdots +{\chi }^{\mu -1}\right\} \Vert {\omega }_{1}-{\omega }_{0}\Vert \\
&\leq \,\,{\chi }^{\sigma }\left\{ 1+\chi +\cdots +{\chi }^{\mu -\sigma -1}\right\} \Vert {\omega }_{1}-{\omega }_{0}\Vert \\
&\leq \,\,{\chi }^{\sigma }\left\{ \frac{1-{\chi }^{\mu -\sigma -1}}{1-\chi }\right\} \Vert {\omega }_{1}-{\omega }_{0}\Vert .\end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(30)
|}

As <math display="inline">\chi \in \left( 0,1\right)</math>, so <math display="inline">1-</math><math>{\chi }^{\mu -\sigma -1}<1</math>, then we get

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert \,\,\leq\,\, \frac{{\chi }^{\sigma }}{1-\chi }\Vert {\omega }_{1}-{\omega }_{0}\Vert .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(31)
|}

Since <math display="inline">\Vert {\omega }_{1}-{\omega }_{0}\Vert <\infty</math>, we find that <math display="inline">\Vert {\omega }_{\mu }-</math><math>{\omega }_{\sigma }\Vert \rightarrow 0</math> when <math display="inline">\mu</math>  and <math display="inline">\sigma \rightarrow \infty</math>. This shows that the sequence <math display="inline">\left\{ {\omega }_{\mu }\right\}</math>  generated by NDM is a convergent Cauchy sequence.
<span id='solution-for-bbmpb-equation'></span>

==5. Solution for BBMPB equation==

To offer the solution to the relevant problem, we will use the fractional natural decomposition approach. We will provide four examples to demonstrate the dependability of the proposed method. In this part, we will look at the new fractional Benjamin Bona Mahony Peregrine Burgers equation, which is stated as follows

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}\frac{{\partial }^{\zeta }\omega \left( x,t\right) }{\partial {t}^{\zeta }}&\,\, = \,\,\frac{{\partial }^{3}\omega \left( x,t\right) }{\partial {x}^{2}\partial t}+\alpha \frac{{\partial }^{2}\omega \left( x,t\right) }{\partial {x}^{2}}-\gamma \frac{\partial \omega \left( x,t\right) }{\partial x}\\
&\,\quad -\theta \omega \left( x,t\right) \frac{\partial \omega \left( x,t\right) }{\partial x}-\beta \frac{{\partial }^{3}\omega \left( x,t\right) }{\partial {x}^{3}},\quad 0<\zeta \leq 1,\end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(32)
|}

in the operator form, with initial condition

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\omega \left( x,0\right) \,\, = \,\, \phi \left( x\right) .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(33)
|}

Using NT on Eq. (32), we may obtain

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>{N}^{+}\left[ {D}_{t}^{\zeta }\omega \left( x,t\right) \right] \,\,=\,\,{N}^{+}\left[ \frac{{\partial }^{3}\omega }{\partial {x}^{2}\partial t}+\alpha \frac{{\partial }^{2}\omega }{\partial {x}^{2}}-\gamma \frac{\partial \omega }{\partial x}-\theta \omega \frac{\partial \omega }{\partial x}-\beta \frac{{\partial }^{3}\omega }{\partial {x}^{3}}\right] .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(34)
|}

By using the natural transformation, we find that

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\frac{{s}^{\zeta }}{{z}^{\zeta }}{N}^{+}\left[ \omega \left( x,t\right) \right] -]n-1\sum \frac{{z}^{i-\zeta }}{{s}^{i+1-\zeta }}{\left[ {D}^{i}\omega \right] }_{t=0}\,\, =\,\, {N}^{+}\left[ \frac{{\partial }^{3}\omega }{\partial {x}^{2}\partial t}+\alpha \frac{{\partial }^{2}\omega }{\partial {x}^{2}}-\gamma \frac{\partial \omega }{\partial x}-\theta \omega \frac{\partial \omega }{\partial x}-\beta \frac{{\partial }^{3}\omega }{\partial {x}^{3}}\right] .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(35)
|}

By Eqs. (34) and (35), we find that

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>{N}^{+}\left[ \omega \left( x,t\right) \right] \,\, =\,\, \frac{\phi \left( x\right) }{s}+\frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ \frac{{\partial }^{3}\omega }{\partial {x}^{2}\partial t}+\alpha \frac{{\partial }^{2}\omega }{\partial {x}^{2}}-\gamma \frac{\partial \omega }{\partial x}-\theta \omega \frac{\partial \omega }{\partial x}-\beta \frac{{\partial }^{3}\omega }{\partial {x}^{3}}\right] .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(36)
|}

By using the inverse NT to above Eq.

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\omega \left( x,t\right) \,\, = \,\, \phi \left( x\right) +{N}^{-}\left\{ \frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ \frac{{\partial }^{3}\omega }{\partial {x}^{2}\partial t}+\alpha \frac{{\partial }^{2}\omega }{\partial {x}^{2}}-\gamma \frac{\partial \omega }{\partial x}-\theta \omega \frac{\partial \omega }{\partial x}-\beta \frac{{\partial }^{3}\omega }{\partial {x}^{3}}\right] \right\} .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(37)
|}

Suppose that the solution to the infinite series of <math display="inline">\omega \left( x,t\right) =</math><math>\sum _{n=0}^{\infty }{\omega }_{n}\left( x,t\right)</math>. Keep in mind <math display="inline">\omega {\omega }_{x}=</math><math>\sum _{n=0}^{\infty }{A}_{n}</math> is the Adomian polynomial and signify the nonlinear terms. Eq. (37) may be rewritten using this term as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\sum _{n=0}^{\infty }{\omega }_{n}\left( x,t\right) \,\, =\,\, \phi \left( x\right) +{N}^{-}\left\{ \frac{{z}^{\zeta }}{{s}^{\zeta }}{N}^{+}\left[ \sum _{n=0}^{\infty }{\omega }_{xxt}+\alpha \sum _{n=0}^{\infty }{\omega }_{xx}-\gamma \sum _{n=0}^{\infty }{\omega }_{x}-\theta \sum _{n=0}^{\infty }{A}_{n}-\beta \sum _{n=0}^{\infty }{\omega }_{xxx}\right] \right\} .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(38)
|}

===5.1 Application 1===

The initial condition for Eq. (32) take the following form [35]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\omega \left( x,0\right) \,\, =\,\, -\left( \frac{\beta +\gamma }{\theta }\right) +\frac{\beta +\gamma }{\theta }\tanh\left( \frac{-\beta -\gamma }{2\delta }x\right) ,</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(39)
|}

If <math display="inline">\alpha =\beta =\gamma =\theta =1</math>, we get

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>{\omega }_{1}\,\, =\,\, \frac{3{t}^{\zeta }\left( \cosh\left( 2x\right) -3\right) {sech}^{4}\left( x\right) }{\Gamma \left( \zeta +1\right) },</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(40)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}{\omega }_{2}\,\, & =\,\, \frac{3{t}^{2\zeta }{sech}^{7}\left( x\right) }{4\Gamma \left( 2\zeta +1\right) }\left\{ 3\left[ 546\sinh\left( x\right) -93\sinh\left( 3x\right) +\sinh\left( 5x\right) \right] +64\cosh\left( x\right) -32\cosh\left( 3x\right) \right\}\\ 
&\quad +\frac{6\zeta {t}^{2\zeta -1}}{\Gamma \left( 2\zeta +1\right) }\left( -34\cosh\left( 2x\right) +\cosh\left( 4x\right) +45\right) {sech}^{6}\left( x\right) ,\cdots \end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(41)
|}

The prior analytical solution leads to the following exact solution [35]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\omega \left( x,t\right) =-\left( \frac{\beta +\gamma }{\theta }\right) +\frac{\beta +\gamma }{\theta }tanh\left( \frac{-\beta -\gamma }{2\delta }\left( x+\right. \right. </math><math>\left. \left. \beta t\right) \right) .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(42)
|}

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and [[#tab-1|Table 1]] demonstrate that the exact solution of Eq. (32) has a generic type that is equivalent to the aforementioned analytical solutions. In order to comprehend the geometric behavior of our approximation to Eq. (32), shown in [[#img-1|Figure 1]], the exact solution in two and three dimensions is compared to the 2nd iteration of NTM. When <math display="inline">\zeta =</math><math>1</math>, <math display="inline">\zeta =0.95</math>, <math display="inline">\zeta =</math><math>0.90</math> and <math display="inline">\zeta =0.80</math>, the NTM solution and the exact solution were also compared.

<div class="center" style="font-size: 75%;">'''Table 1'''. Comparison of case 1 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math></div> 

<div id='tab-1'></div>
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;" 
|-style="text-align:center"
! <math display="inline">x</math> !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math> !!  <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\mathit{\boldsymbol{1}}\right)</math> !! Absolute error !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.95}}\right)</math>  !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.90}}\right)</math> 
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|2
|  style="text-align: center;vertical-align: bottom;"|-3.990109507
|  style="text-align: center;vertical-align: bottom;"|-3.564048679
|  style="text-align: center;vertical-align: bottom;"|4.260608E-01
|  style="text-align: center;vertical-align: bottom;"|-3.556574735
|  style="text-align: center;vertical-align: bottom;"|-3.549577917
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|4
|  style="text-align: center;vertical-align: bottom;"|-3.999818408
|  style="text-align: center;vertical-align: bottom;"|-3.990634473
|  style="text-align: center;vertical-align: bottom;"|9.183935E-03
|  style="text-align: center;vertical-align: bottom;"|-3.990469718
|  style="text-align: center;vertical-align: bottom;"|-3.990315480
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|6
|  style="text-align: center;vertical-align: bottom;"|-3.999996673
|  style="text-align: center;vertical-align: bottom;"|-3.999827971
|  style="text-align: center;vertical-align: bottom;"|1.687026E-04
|  style="text-align: center;vertical-align: bottom;"|-3.999824943
|  style="text-align: center;vertical-align: bottom;"|-3.999822109
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|8
|  style="text-align: center;vertical-align: bottom;"|-3.999999939
|  style="text-align: center;vertical-align: bottom;"|-3.999996849
|  style="text-align: center;vertical-align: bottom;"|3.090061E-06
|  style="text-align: center;vertical-align: bottom;"|-3.999996793
|  style="text-align: center;vertical-align: bottom;"|-3.999996741
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|10
|  style="text-align: center;vertical-align: bottom;"|-3.999999998
|  style="text-align: center;vertical-align: bottom;"|-3.999999942
|  style="text-align: center;vertical-align: bottom;"|5.659651E-08
|  style="text-align: center;vertical-align: bottom;"|-3.999999941
|  style="text-align: center;vertical-align: bottom;"|-3.999999940
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|12
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|-3.999999998
|  style="text-align: center;vertical-align: bottom;"|1.036601E-09
|  style="text-align: center;vertical-align: bottom;"|-3.999999998
|  style="text-align: center;vertical-align: bottom;"|-3.999999998
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|14
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|1.898592E-11
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|16
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|3.477218E-13
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|18
|  style="text-align: center;vertical-align: bottom;"|-4
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|6.217248E-15
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|  style="text-align: center;vertical-align: bottom;"|-3.999999999
|-style="text-align:center"
|  style="text-align: center;vertical-align: bottom;"|20
|  style="text-align: center;vertical-align: bottom;"|-4
|  style="text-align: center;vertical-align: bottom;"|-4
|  style="text-align: center;vertical-align: bottom;"|0
|  style="text-align: center;vertical-align: bottom;"|-4
|  style="text-align: center;vertical-align: bottom;"|-4
|}


<div id='img-1'></div>
{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;" 
|-style="background:white;"
|align="center" | 
{|style="margin: 0em auto 0.1em auto;width:auto;" 
|+
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:1ex.jpg|276x276px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[File:1app.jpg|258x258px]] 
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Exact solution of <math display="inline">\omega (x,t)</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) NDM solution of <math display="inline">\omega (x,t)</math>
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:1exapp.jpg|240x240px]]
|style="text-align: center;padding:10px;width: 59%;"| [[File:1alphaX.jpg|246x246px]] 
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) Absolute error at <math display="inline">t = 0.01</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(d) Comparison of results at <math display="inline">t = 0.01</math>
|-
| style="text-align: center;padding:10px;width: 49%;"|[[Image:Draft_Hagag_831060637-image5.jpeg|276px]] 
|  style="text-align: center;padding:10px;width: 50%;"|[[File:1alphaT.jpg|227x227px]] 
|-
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(e) Comparison between exact and various values of <math display="inline">\zeta</math>
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(f) Comparison between various values of <math display="inline">\zeta</math> at <math display="inline">x = 1</math>
|}
|-
| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 1'''. Periodic wave analytical (NDM) solutions <math display="inline">\omega (x,t)</math> of Eq. (32) with initial condition (39) and <math display="inline">\alpha = \beta = \gamma = \theta = 1</math>
|}

===5.2 Application 2===

The initial condition for Eq. (32) take the following form [35]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\omega \left( x,0\right) \,\, =\,\,-\left( \frac{\beta +\gamma }{\theta }\right) -\frac{\beta +\gamma }{\theta }coth\left( \frac{-\beta -\gamma }{2\delta }x\right) ,</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(43)
|}

If <math display="inline">\alpha =\beta =\gamma =\theta =1</math>, we get

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>{\omega }_{1}\,\, =\,\,-\frac{3{t}^{\zeta }\left( cosh\left( 2x\right) +3\right) {csch}^{4}\left( x\right) }{\Gamma \left( \zeta +1\right) },</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(44)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{align}{\omega }_{2}\,\, & =\,\,\frac{3{t}^{2\zeta }{csch}^{7}\left( x\right) }{4\Gamma \left( 2\zeta +1\right) }\left\{ 64\sinh\left( x\right) +32\sinh\left( 3x\right) +1638\cosh\left( x\right) +279\cosh\left( 3x\right) +3\cosh\left( 5x\right) \right\} \\
&\quad +\frac{3{t}^{2\zeta -1}}{\Gamma \left( 2\zeta \right) }\left( 34\cosh\left( 2x\right) +\cosh\left( 4x\right) +45\right) {csch}^{6}\left( x\right) ,\cdots \end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(45)
|}

The prior analytical solution leads to the following exact solution [35]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\omega \left( x,t\right) \,\, =\,\,-\left( \frac{\beta +\gamma }{\theta }\right) -\frac{\beta +\gamma }{\theta }coth\left( \frac{-\beta -\gamma }{2\delta }\left( x+\beta t\right) \right) .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(46)
|}

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and [[#tab-2|Table 2]] demonstrate that the exact solution of Eq. (32) has a generic type that is equivalent to the aforementioned analytical solutions. In order to comprehend the geometric behavior of our approximation to Eq. (32), shown in [[#img-2|Figure 2]] , the exact solution in two and three dimensions is compared to the 2nd iteration of NTM. When <math display="inline">\zeta =</math><math>1</math>, <math display="inline">\zeta =0.95</math>, <math display="inline">\zeta =</math><math>0.90</math> and <math display="inline">\zeta =0.80</math>, the NTM solution and the exact solution were also compared.

<div class="center" style="font-size: 75%;">'''Table 2'''. Comparison of case 2 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math></div>

<div id='tab-2'></div>
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;" 
|-style="text-align:center"
! <math display="inline">x</math> !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math> !!  <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\mathit{\boldsymbol{1}}\right)</math> !! Absolute error !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.95}}\right)</math>  !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.90}}\right)</math> 
|-style="text-align:center"
|  style="text-align: center;vertical-align: top;"|2
|  style="text-align: center;vertical-align: top;"|-4.009939646
|  style="text-align: center;vertical-align: top;"|-4.600112243
|  style="text-align: center;vertical-align: top;"|5.901725E-01
|  style="text-align: center;vertical-align: top;"|-4.610901692
|  style="text-align: center;vertical-align: top;"|-4.621002357
|-
|  style="text-align: center;vertical-align: top;"|4
|  style="text-align: center;vertical-align: top;"|-4.000181607
|  style="text-align: center;vertical-align: top;"|-4.009420444
|  style="text-align: center;vertical-align: top;"|9.238836E-03
|  style="text-align: center;vertical-align: top;"|-4.009586308
|  style="text-align: center;vertical-align: top;"|-4.009741583
|-
|  style="text-align: center;vertical-align: top;"|6
|  style="text-align: center;vertical-align: top;"|-4.000003326
|  style="text-align: center;vertical-align: top;"|-4.000172047
|  style="text-align: center;vertical-align: top;"|1.687210E-04
|  style="text-align: center;vertical-align: top;"|-4.000175075
|  style="text-align: center;vertical-align: top;"|-4.000177909
|-
|  style="text-align: center;vertical-align: top;"|8
|  style="text-align: center;vertical-align: top;"|-4.000000060
|  style="text-align: center;vertical-align: top;"|-4.000003150
|  style="text-align: center;vertical-align: top;"|3.090068E-06
|  style="text-align: center;vertical-align: top;"|-4.000003206
|  style="text-align: center;vertical-align: top;"|-4.000003258
|-
|  style="text-align: center;vertical-align: top;"|10
|  style="text-align: center;vertical-align: top;"|-4.000000001
|  style="text-align: center;vertical-align: top;"|-4.000000057
|  style="text-align: center;vertical-align: top;"|5.659651E-08
|  style="text-align: center;vertical-align: top;"|-4.000000058
|  style="text-align: center;vertical-align: top;"|-4.000000059
|-
|  style="text-align: center;vertical-align: top;"|12
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000001
|  style="text-align: center;vertical-align: top;"|1.036601E-09
|  style="text-align: center;vertical-align: top;"|-4.000000001
|  style="text-align: center;vertical-align: top;"|-4.000000001
|-
|  style="text-align: center;vertical-align: top;"|14
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|1.898598E-11
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|-
|  style="text-align: center;vertical-align: top;"|16
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|3.474607E-13
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|-
|  style="text-align: center;vertical-align: top;"|18
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|6.010944E-15
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|-
|  style="text-align: center;vertical-align: top;"|20
|  style="ext-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|1.019605E-16
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|}


<div id='img-2'></div>
{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;" 
|-style="background:white;"
|align="center" | 
{|style="margin: 0em auto 0.1em auto;width:auto;" 
|+
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:2ex.jpg|222x222px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image8.jpeg|216px]] 
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Exact solution of <math display="inline">\omega (x,t)</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) NDM solution of <math display="inline">\omega (x,t)</math>
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:2exapp.jpg|240x240px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[File:2ex+app.jpg|264x264px]] 
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) Absolute error at <math display="inline">t = 0.01</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(d) Comparison of results at <math display="inline">t = 0.01</math>
|-
| style="text-align: center;padding:10px;width: 49%;"|[[Image:Draft_Hagag_831060637-image11.jpeg|282px]]
|  style="text-align: center;padding:10px;width: 50%;"|[[File:2alphaT.jpg|252x252px]]
|-
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(e) Comparison between exact and various values of <math display="inline">\zeta</math>
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(f) Comparison between various values of <math display="inline">\zeta</math> at <math display="inline">x = 1</math>
|}
|-
| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 2'''. Periodic wave analytical (NDM) solutions <math display="inline">\omega (x,t)</math> of Eq. (32) with initial condition (42) and <math display="inline">\alpha = \beta = \gamma = \theta = 1</math>
|}

===5.3 Application 3===

The initial condition for Eq. (32) take the following form [35]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\omega \left( x,0\right) &=&-\left( \frac{\beta +\gamma }{\theta }\right) +\frac{\beta +\gamma }{2\theta }tanh\left( \frac{-\beta -\gamma }{4\delta }x\right) +\frac{\beta +\gamma }{2\theta }coth\left( \frac{-\beta -\gamma }{4\delta }x\right) ,\quad \quad \quad \quad \quad (47)\end{matrix}</math>
|}
</div>

If <math display="inline">\alpha =\beta =\gamma =\theta =1</math>, we get

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}{\omega }_{1}&=&-\frac{3{t}^{\alpha }\left( cosh\left( 2x\right) +3\right) {csch}^{4}\left( x\right) }{\Gamma \left( \alpha +1\right) },\end{matrix}\quad \quad \quad \quad \, \, (48)</math>
|}
</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}{\omega }_{2}&=&-\frac{3{t}^{2\zeta }{csch}^{7}\left( x\right) }{4\Gamma \left( 2\zeta +1\right) }\left\{ 64sinh\left( x\right) +32sinh\left( 3x\right) +1638cosh\left( x\right) +279cosh\left( 3x\right) +3cosh\left( 5x\right) \right\} \\&&-\frac{3{t}^{2\zeta -1}}{\Gamma \left( 2\zeta \right) }\left( 34cosh\left( 2x\right) +cosh\left( 4x\right) +45\right) {csch}^{6}\left( x\right) ,\cdots \end{matrix}\quad (49)</math>
|}
</div>

The prior analytical solution leads to the following exact solution [35]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\omega \left( x,t\right) &=&-\left( \frac{\beta +\gamma }{\theta }\right) +\frac{\beta +\gamma }{2\theta }tanh\left( \frac{-\beta -\gamma }{4\delta }\left( x+\beta t\right) \right) +\frac{\beta +\gamma }{2\theta }coth\left( \frac{-\beta -\gamma }{4\delta }\left( x+\beta t\right) \right) .\quad \quad \quad \, (50)\end{matrix}</math>
|}
</div>

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and [[#tab-3|Table 3]] demonstrate that the exact solution of Eq. (32) has a generic type that is equivalent to the aforementioned analytical solutions. In order to comprehend the geometric behavior of our approximation to Eq. (32), shown in [[#img-3|Figure 3]], the exact solution in two and three dimensions is compared to the 2nd iteration of NTM. When <math display="inline">\zeta =</math><math>1</math>, <math display="inline">\zeta =0.95</math>, <math display="inline">\zeta =</math><math>0.90</math> and <math display="inline">\zeta =0.80</math>, the NTM solution and the exact solution were also compared.

<div class="center" style="font-size: 75%;">'''Table 3'''. Comparison of case 3 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math></div>

<div id='tab-3'></div>
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;" 
|-style="text-align:center"
! <math display="inline">x</math> !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math> !!  <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\mathit{\boldsymbol{1}}\right)</math> !! Absolute error !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.95}}\right)</math>  !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.90}}\right)</math> 
|-style="text-align:center"
|  style="text-align: center;vertical-align: top;"|2
|  style="text-align: center;vertical-align: top;"|-4.009939646
|  style="text-align: center;vertical-align: top;"|-4.600112243
|  style="text-align: center;vertical-align: top;"|5.901725E-01
|  style="text-align: center;vertical-align: top;"|-4.610901692
|  style="text-align: center;vertical-align: top;"|-4.621002357
|-
|  style="text-align: center;vertical-align: top;"|4
|  style="text-align: center;vertical-align: top;"|-4.000181607
|  style="text-align: center;vertical-align: top;"|-4.009420444
|  style="text-align: center;vertical-align: top;"|9.238836E-03
|  style="text-align: center;vertical-align: top;"|-4.009586308
|  style="text-align: center;vertical-align: top;"|-4.009741583
|-
|  style="text-align: center;vertical-align: top;"|6
|  style="text-align: center;vertical-align: top;"|-4.000003326
|  style="text-align: center;vertical-align: top;"|-4.000172047
|  style="text-align: center;vertical-align: top;"|1.687210E-04
|  style="text-align: center;vertical-align: top;"|-4.000175075
|  style="text-align: center;vertical-align: top;"|-4.000177909
|-
|  style="text-align: center;vertical-align: top;"|8
|  style="text-align: center;vertical-align: top;"|-4.000000060
|  style="text-align: center;vertical-align: top;"|-4.000003150
|  style="text-align: center;vertical-align: top;"|3.090068E-06
|  style="text-align: center;vertical-align: top;"|-4.000003206
|  style="text-align: center;vertical-align: top;"|-4.000003258
|-
|  style="text-align: center;vertical-align: top;"|10
|  style="text-align: center;vertical-align: top;"|-4.000000001
|  style="text-align: center;vertical-align: top;"|-4.000000057
|  style="text-align: center;vertical-align: top;"|5.659651E-08
|  style="text-align: center;vertical-align: top;"|-4.000000058
|  style="text-align: center;vertical-align: top;"|-4.000000059
|-
|  style="text-align: center;vertical-align: top;"|12
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000001
|  style="text-align: center;vertical-align: top;"|1.036601E-09
|  style="text-align: center;vertical-align: top;"|-4.000000001
|  style="text-align: center;vertical-align: top;"|-4.000000001
|-
|  style="text-align: center;vertical-align: top;"|14
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|1.898619E-11
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|-
|  style="text-align: center;vertical-align: top;"|16
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|3.476832E-13
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|-
|  style="text-align: center;vertical-align: top;"|18
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|6.403864E-15
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|-
|  style="text-align: center;vertical-align: top;"|20
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|7.907517E-17
|  style="text-align: center;vertical-align: top;"|-4.000000000
|  style="text-align: center;vertical-align: top;"|-4.000000000
|}


<div id='img-3'></div>
{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;" 
|-style="background:white;"
|align="center" | 
{|style="margin: 0em auto 0.1em auto;width:auto;" 
|+
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:3ex.jpg|242x242px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image14.jpeg|236px]]
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Exact solution of <math display="inline">\omega (x,t)</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) NDM solution of <math display="inline">\omega (x,t)</math>
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:3exapp.jpg|224px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[Image:Draft_Hagag_831060637-image16.jpeg|218px]]
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) Absolute error at <math display="inline">t = 0.01</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(d) Comparison of results at <math display="inline">t = 0.01</math>
|-
| style="text-align: center;padding:10px;width: 49%;"|[[File:3alphaX.jpg|248px]]
|  style="text-align: center;padding:10px;width: 50%;"|[[File:3alphaT.jpg|242px]] 
|-
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(e) Comparison between exact and various values of <math display="inline">\zeta</math>
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(f) Comparison between various values of <math display="inline">\zeta</math> at <math display="inline">x = 1</math>
|}
|-
| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 3'''. Periodic wave analytical (NDM) solutions <math display="inline">\omega (x,t)</math> of Eq. (32) with initial condition (46) and <math display="inline">\alpha = \beta = \gamma = \theta = 1</math>
|}

===5.4 Application 4===

The initial condition for Eq. (32) take the following form [35]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\omega \left( x,0\right) \,\,=\,\, \frac{-10\beta \mu -10\gamma \mu +\delta }{20\theta \mu }-\frac{\left( 6\delta \mu \right) {\tanh}^{2}\left( \mu x\right) }{5\theta }-\frac{\left( 12\delta \mu \right) \tanh\left( \mu x\right) }{5\theta },</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(51)
|}

where <math display="inline">\mu =\frac{5\beta +5\gamma \sqrt{25{\beta }^{2}+50\beta \gamma +25{\gamma }^{2}-24{\delta }^{2}}}{24\delta }</math>. If <math display="inline">\alpha =</math><math>\beta =\gamma =\theta =1</math>, we get

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" |<math>\begin{align}{\omega }_{1}\,\, & =\,\,\frac{{t}^{\zeta }}{10800\Gamma \left( \zeta +1\right) }\Bigg\{ -3\left( 959\sqrt{19}+4180\right) \sinh\left( \frac{1}{6}\left( \sqrt{19}+5\right) x\right)\\
&\quad +\left( 4003\sqrt{19}+17480\right) \cosh\left( \frac{1}{6}\left( \sqrt{19}+5\right) x\right) ,\\
&\quad  -95\left( 79\sqrt{19}+344\right) \Bigg\} \left( \tanh\left( \frac{1}{12}\left( \sqrt{19}+5\right) x\right) +1\right) {sech}^{4}\left( \frac{1}{12}\left( \sqrt{19}+5\right) x\right) ,\cdots \end{align}</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(52)
|}

The prior analytical solution leads to the following exact solution [35]

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\omega \left( x,t\right) \,\,=\,\,\frac{-10\beta \mu -10\gamma \mu +\delta }{20\theta \mu }-\frac{\left( 12\delta \mu \right) }{5\theta }\tanh\left( \mu \left( x-Vt\right) \right) -\frac{\left( 6\delta \mu \right) }{5\theta }{\tanh}^{2}\left( \mu \left( x-Vt\right) \right) .</math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" |(53)
|}

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and[[#tab-4|Table 4]] demonstrate that the exact solution of Eq. (32) has a generic type that is equivalent to the aforementioned analytical solutions. In order to comprehend the geometric behavior of our approximation to Eq. (32), shown in [[#img-4|Figure 4]], the exact solution in two and three dimensions is compared to the 2nd iteration of NTM. When <math display="inline">\zeta =1</math>, <math display="inline">\zeta =0.95</math>, <math display="inline">\zeta =0.90</math> and <math display="inline">\zeta =0.80</math>, the NTM solution and the exact solution were also compared.

<div class="center" style="font-size: 75%;">'''Table 4'''. Comparison of case 4 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math></div>

<div id='tab-4'></div>
{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;" 
|-style="text-align:center"
! <math display="inline">x</math> !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math> !!  <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\mathit{\boldsymbol{1}}\right)</math> !! Absolute error !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.95}}\right)</math>  !! <math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}= \mathit{\boldsymbol{0.90}}\right)</math> 
|-
|  style="text-align: center;vertical-align: top;"|2
|  style="text-align: center;vertical-align: top;"|-3.660080833
|  style="text-align: center;vertical-align: top;"|-3.212723611
|  style="text-align: center;vertical-align: top;"|4.473572E-01
|  style="text-align: center;vertical-align: top;"|-3.208190131
|  style="text-align: center;vertical-align: top;"|-3.203946062
|-
|  style="text-align: center;vertical-align: top;"|4
|  style="text-align: center;vertical-align: top;"|-3.739812014
|  style="text-align: center;vertical-align: top;"|-3.701529867
|  style="text-align: center;vertical-align: top;"|3.828214E-02
|  style="text-align: center;vertical-align: top;"|-3.700965987
|  style="text-align: center;vertical-align: top;"|-3.700438105
|-
|  style="text-align: center;vertical-align: top;"|6
|  style="text-align: center;vertical-align: top;"|-3.743393916
|  style="text-align: center;vertical-align: top;"|-3.741658721
|  style="text-align: center;vertical-align: top;"|1.735194E-03
|  style="text-align: center;vertical-align: top;"|-3.741632941
|  style="text-align: center;vertical-align: top;"|-3.741608807
|-
|  style="text-align: center;vertical-align: top;"|8
|  style="text-align: center;vertical-align: top;"|-3.743552259
|  style="text-align: center;vertical-align: top;"|-3.743475523
|  style="text-align: center;vertical-align: top;"|7.673630E-04
|  style="text-align: center;vertical-align: top;"|-3.743474382
|  style="text-align: center;vertical-align: top;"|-3.743473314
|-
|  style="text-align: center;vertical-align: top;"|10
|  style="text-align: center;vertical-align: top;"|-3.743559254
|  style="text-align: center;vertical-align: top;"|-3.743555864
|  style="text-align: center;vertical-align: top;"|3.389871E-06
|  style="text-align: center;vertical-align: top;"|-3.743555813
|  style="text-align: center;vertical-align: top;"|-3.743555766
|-
|  style="text-align: center;vertical-align: top;"|12
|  style="text-align: center;vertical-align: top;"|-3.743559563
|  style="text-align: center;vertical-align: top;"|-3.743559413
|  style="text-align: center;vertical-align: top;"|1.497424E-07
|  style="text-align: center;vertical-align: top;"|-3.743559411
|  style="text-align: center;vertical-align: top;"|-3.743559409
|-
|  style="text-align: center;vertical-align: top;"|14
|  style="text-align: center;vertical-align: top;"|-3.743559576
|  style="text-align: center;vertical-align: top;"|-3.743559570
|  style="text-align: center;vertical-align: top;"|6.614627E-09
|  style="text-align: center;vertical-align: top;"|-3.743559570
|  style="text-align: center;vertical-align: top;"|-3.743559569
|-
|  style="text-align: center;vertical-align: top;"|16
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|2.921902E-10
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|-3.743559577
|-
|  style="text-align: center;vertical-align: top;"|18
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|1.290745E-11
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|-3.743559577
|-
|  style="text-align: center;vertical-align: top;"|20
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|5.702105E-13
|  style="text-align: center;vertical-align: top;"|-3.743559577
|  style="text-align: center;vertical-align: top;"|-3.743559577
|}


<div id='img-4'></div>
{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;" 
|-style="background:white;"
|align="center" | 
{|style="margin: 0em auto 0.1em auto;width:auto;" 
|+
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:4ex.jpg|242px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[File:4app.jpg|236px]] 
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(a) Exact solution of <math display="inline">\omega (x,t)</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(b) NDM solution of <math display="inline">\omega (x,t)</math>
|-
|style="text-align: center;padding:10px;width: 49%;"| [[File:4exapp.jpg|224px]] 
|style="text-align: center;padding:10px;width: 59%;"| [[File:4ex+app.jpg|218px]]
|-
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(c) Absolute error at <math display="inline">t = 0.01</math>
|style="text-align: center;font-size: 75%;padding-bottom:10px;"|(d) Comparison of results at <math display="inline">t = 0.01</math>
|-
| style="text-align: center;padding:10px;width: 49%;"|[[File:4alphaX.jpg|248px]] 
|  style="text-align: center;padding:10px;width: 50%;"|[[File:4alphaT.jpg|242px]] 
|-
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(e) Comparison between exact and various values of <math display="inline">\zeta</math>
|  style="text-align: center;font-size: 75%;padding-bottom:10px;"|(f) Comparison between various values of <math display="inline">\zeta</math> at <math display="inline">x = 1</math>
|}
|-
| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 4'''. Periodic wave analytical (NDM) solutions <math display="inline">\omega (x,t)</math> of Eq. (32) with initial condition (50) and <math display="inline">\alpha = \beta = \gamma = \theta = 1</math>
|}

==6. Conclusion==

Studying and exploring nonlinear physical models using new techniques always help us advance in science and technology. In the current framework, we used NDM to evaluate the BBMPB equation with fractional order. Banach’s fixed-point theory is used to investigate the anticipated issue’s uniqueness theorem and convergence analysis. The anticipated method’s dependability and applicability are demonstrated by presenting four cases. The behaviors for the obtained findings are provided in 2D, 3D graphs, and tables for featured fractional order. These graphs aid to conclude the stimulating behaviors of the analogical models. Furthermore, while solving nonlinear issues, NDM does not require any conversion, perturbation, or consideration of extra polynomials or parameters. The examination of these kinds of occurrences can provide new ideas for investigating more real-world events. It can also generate ideas for employing an accurate method to evaluate nonlinear models related to science and technology. This work elucidates the proposed model, which is notably dependent on time instant and its history and can be convincingly illustrated utilizing fractional notions.

'''Conflict of Interest: '''The author declare that there is no conflict of interest.

'''Data Availability:''' No data were used to support this study.

==Acknowledgments==

This research project was funded by the Deanship of Scientific Research, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding After Publication, grant No (43- PRFA-P-43).

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