Analytical solution to fractional differential equation arising in thermodynamics



<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>'''Analytical solution to fractional differential equation arising in thermodynamics'''</big></div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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>

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 whereas 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

<span id='introduction'></span>

==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 analyses and examination of these kinds of equations are based on the survey of 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 of nonlinear analysis in relation to the everyday life 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 computation, 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 natural decomposition method (NDM) in the classical order [14-16]. Then this method was generalized and presented in the 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, the shear in second-order fluids, fluid flow in fissured rock, and propagation of long waves with tiny amplitudes, among other things [21-24]. The generalised Benjamin-Bona-Mahony-Burgers (BBMB) equation is a significant particular instance of pseudo-parabolic-type equations which 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>\begin{matrix}{\omega }_{t}-{\omega }_{xxt}-\alpha {\omega }_{xx}+\gamma {\omega }_{x}+g{\left( \omega \right) }_{x}&=&0,\end{matrix}</math>
|}
|  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(1)
|}


where <math display="inline">\gamma</math>  denotes any value genuine constant, <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 Kortewegde Vries equation

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}{\omega }_{t}+{\omega }_{xxt}+{\omega }_{x}+\omega {\omega }_{x}&=&0.\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}{\omega }_{t}-{\omega }_{xxt}-\alpha {\omega }_{xx}+\gamma {\omega }_{x}+\theta \omega {\omega }_{x}+\beta {\omega }_{xxx}&=&0.\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}{\omega }_{t}-{\omega }_{xxt}+\gamma {\omega }_{x}+\theta \omega {\omega }_{x}&=&0,\end{matrix}</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 researches [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 Section5, 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.

<span id='preliminaries-to-fc'></span>

==2 Preliminaries to FC==

<span id='definition-idefinition-2.1'></span>

===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 as [3]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{J}_{a}^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{1}{\Gamma \left( \zeta +1\right) }\int_{a}^{\theta }\omega \left( \theta \right) {\left( d\theta \right) }^{\zeta }=\frac{1}{\Gamma \left( \zeta \right) }\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)
|}


<span id='definition-iidefinition-2.2'></span>

===2.2 Definition II ===

The Caputo fractional differential operator of order <math display="inline">\zeta >0</math> is defined as [3]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>{D}_{\ast }^{\zeta }\omega \left( \theta \right) =\left\{ \begin{matrix}\frac{{d}^{k}}{d{\theta }^{k}}\omega \left( \theta \right) ,&\zeta =k\in N,\\\frac{1}{\Gamma \left( k-\zeta \right) }\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)
|}


<span id='definition-iii'></span>

===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 as [31]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <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)
|}


<span id='definition-iv'></span>

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


<span id='definition-v'></span>

===2.5 Definition V===

The effect of the natural transform on the Caputo operator is given as [32]

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <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)
|}


<span id='construction-of-fractional-ndm'></span>

==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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}{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,\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\omega \left( x,0\right) &=&v\left( x\right) ,\end{matrix}</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) and applying definition 5, we get

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}{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] .\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\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\} .\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\omega (x,t)=\sum_{n=0}^{\infty }{\omega}_{n}(x,t),  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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\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\} .\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\omega \left( x,t\right) &=&\sum _{n=0}^{\infty }{\omega }_{n}\left( x,t\right) .\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\chi &=\left\{ {\epsilon }^{3}+\alpha {\varphi }^{2}-\gamma \varphi -\theta \left( X+Y\right) -\beta {\varphi }^{3}\right\} \Xi &\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\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 .\end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\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) },\quad \quad \quad \quad \quad \quad \quad \quad \end{matrix}</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;" 
|-
| 
{| style="text-align: center;margin:auto;width: 100%;"
|-
| <math>\begin{matrix}\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| ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \, \, \end{matrix}</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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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) ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \, \, \end{matrix}(23)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &\leq &\frac{{\chi }^{\phi }}{1-\chi }\left\| {\omega }_{1}-{\omega }_{0}\right\| .\quad \quad \quad \quad \quad \quad \, (24)\end{matrix}</math>
|}
</div>

==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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &=&\underbrace{max}_{\phi \in J}\left| {\omega }_{\mu }-{\omega }_{\sigma }\right| \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\&=&\underbrace{max}_{\phi \in J}\vert {N}^{-}\lbrace \left( \frac{{z}^{\zeta }}{{s}^{\zeta }}\right) {N}^{+}[\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) -\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) ]\rbrace \vert \quad \quad \quad \quad \quad \quad \quad \quad (25)\\&\leq &\underbrace{max}_{\phi \in J}\left[ {N}^{-}\left\{ \left( \frac{{z}^{\zeta }}{{s}^{\zeta }}\right) {N}^{+}\left[ \vert \frac{{\partial }^{3}{\omega }_{\mu -1}}{\partial {x}^{2}\partial t}-\frac{{\partial }^{3}{\omega }_{\sigma -1}}{\partial {x}^{2}\partial t}\vert +\alpha \vert \frac{{\partial }^{2}{\omega }_{\mu -1}}{\partial {x}^{2}}-\frac{{\partial }^{2}{\omega }_{\sigma -1}}{\partial {x}^{2}}\vert -\gamma \vert \frac{{\partial }^{2}{\omega }_{\mu -1}}{\partial x}-\frac{{\partial }^{2}{\omega }_{\sigma -1}}{\partial x}\vert \right. \right. \right. \\&&\left. \left. \left. -\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] .\quad \quad \quad \quad \quad \quad \quad \quad \quad (26)\end{matrix}</math>
|}
</div>

Transform by using the convolution principle for NT, we obtain

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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{matrix}\quad \quad \quad \quad \quad (27)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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. -\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{matrix}\quad \quad \, \, (28)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; 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\| .\quad \quad \, \, (29)</math>
|}
</div>

By using triangular inequality, we obtain

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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{matrix}\quad \quad \quad \quad (30)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\Vert {\omega }_{\mu }-{\omega }_{\sigma }\Vert &\leq &\frac{{\chi }^{\sigma }}{1-\chi }\Vert {\omega }_{1}-{\omega }_{0}\Vert .\end{matrix}\quad \quad \quad \quad \, \, (31)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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}-\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}},0<\zeta \leq 1,\end{matrix}\quad \, (32)</math>
|}
</div>

in the operator form, with initial condition

<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) &=&\phi \left( x\right) .\end{matrix}\quad \quad (33)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}{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] .\quad \quad \quad \quad \, \, (34)\end{matrix}</math>
|}
</div>

By using the natural transformation, we find that

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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] .\end{matrix}\, \, (35)</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}{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] .\quad \quad (36)\end{matrix}</math>
|}
</div>

By using the inverse NT to above Eq.

<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) &=&\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\} .\quad \quad (37)\end{matrix}</math>
|}
</div>

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

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>\begin{matrix}\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\} .\quad \, \, (38)\end{matrix}</math>
|}
</div>

<span id='application-5.1.'></span>

===5.1 Application 1===

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 }{\theta }tanh\left( \frac{-\beta -\gamma }{2\delta }x\right) ,\quad \quad (39)\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}^{\zeta }\left( cosh\left( 2x\right) -3\right) {sech}^{4}\left( x\right) }{\Gamma \left( \zeta +1\right) },\end{matrix}\quad \quad (40)</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 }{sech}^{7}\left( x\right) }{4\Gamma \left( 2\zeta +1\right) }\left\{ 3\left[ 546sinh\left( x\right) -93sinh\left( 3x\right) +sinh\left( 5x\right) \right] +64cosh\left( x\right) -32cosh\left( 3x\right) \right\} \\&&+\frac{6\zeta {t}^{2\zeta -1}}{\Gamma \left( 2\zeta +1\right) }\left( -34cosh\left( 2x\right) +cosh\left( 4x\right) +45\right) {sech}^{6}\left( x\right) ,\cdots \end{matrix}\quad \quad (41)</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>\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) .\quad \, \, (42)</math>
|}
</div>

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and 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 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.

<span id='application-5.2.'></span>

Table 1: Comparison of case 1 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=</math><math>1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math>.

{| style="width: 86%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: bottom;"|x
|  style="border-top: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{1}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: bottom;"|''Absolute error''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.95}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.90}}\right)</math> 
|-
|  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;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;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;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;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;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;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;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;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="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|20
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|-4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|-4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|0
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|-4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|-4
|}

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image1.jpeg|258px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image2.jpeg|258px]] 
|-
|  style="vertical-align: top;"|(a) Exact solution of ω (''x'',''t'').
|  style="vertical-align: top;"|(b) NDM solution of ω (''x'',''t'').
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image3.jpeg|240px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image4.jpeg|246px]] 
|-
|  style="vertical-align: top;"|(c) Absolute error at t = 0.01. 
|  style="vertical-align: top;"|(d) Graphical representation between exact and NDM solutions at t = 0.01.
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image5.jpeg|276px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image6.jpeg|270px]] 
|-
|  style="vertical-align: top;"|(e) Graphical representation between exact solution and various values of ζ at t = 0.01.
|  style="vertical-align: top;"|(f) Graphical representation between exact solution and various values of ζ at x = 1.
|-
|  colspan='2'  style="vertical-align: top;"|Figure 1: Periodic wave analytical (NDM) solutions ω (x,τ) of Eq. (32) with initial condition (39) and α = β = γ = θ = 1.
|}


===5.2 Application 2===

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 }{\theta }coth\left( \frac{-\beta -\gamma }{2\delta }x\right) ,\end{matrix}\quad \quad (43)</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}^{\zeta }\left( cosh\left( 2x\right) +3\right) {csch}^{4}\left( x\right) }{\Gamma \left( \zeta +1\right) },\quad \quad (44)\end{matrix}</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 \quad (45)</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 }{\theta }coth\left( \frac{-\beta -\gamma }{2\delta }\left( x+\beta t\right) \right) .\end{matrix}\quad \quad \quad (46)</math>
|}
</div>

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and 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 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.

<span id='application-5.3.'></span>


Table 2: Comparison of case 2 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=</math><math>1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math>.

{| style="width: 86%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|x
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{1}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|''Absolute error''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.95}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.90}}\right)</math> 
|-
|  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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|20
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.019605E-16
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|}
{| style="width: 100%;border-collapse: collapse;" 
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image7.jpeg|222px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image8.jpeg|216px]] 
|-
|  style="vertical-align: top;"|(a) Exact solution of ω (''x'',''t'').
|  style="vertical-align: top;"|(b) NDM solution of ω (''x'',''t'').
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image9.jpeg|240px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image10.jpeg|264px]] 
|-
|  style="vertical-align: top;"|(c) Absolute error at t = 0.01. 
|  style="vertical-align: top;"|(d) Graphical representation between exact and NDM solutions at t = 0.01.
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image11.jpeg|282px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image12.jpeg|252px]] 
|-
|  style="vertical-align: top;"|(e) Graphical representation between exact solution and various values of ζ at t = 0.01.
|  style="vertical-align: top;"|(f) Graphical representation between exact solution and various values of ζ at x = 1.
|-
|  colspan='2'  style="vertical-align: top;"|Figure 2: Periodic wave analytical (NDM) solutions ω (x,τ) of Eq. (32) with initial condition (42) and α = β = γ = θ = 1.
|}


===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 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 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.

<span id='application-5.4.'></span>

Table 3: Comparison of case 3 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=</math><math>1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math>.

{| style="width: 87%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|x
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{1}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|Absolute error
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.95}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.90}}\right)</math> 
|-
|  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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|20
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|7.907517E-17
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-4.000000000
|}


{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image13.jpeg|222px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image14.jpeg|216px]] 
|-
|  style="vertical-align: top;"|(a) Exact solution of ω (''x'',''t'').
|  style="vertical-align: top;"|(b) NDM solution of ω (''x'',''t'').
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image15.jpeg|204px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image16.jpeg|198px]] 
|-
|  style="vertical-align: top;"|(c) Absolute error at t = 0.01. 
|  style="vertical-align: top;"|(d) Graphical representation between exact and NDM                        solutions at t = 0.01.
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image17.jpeg|228px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image18.jpeg|222px]] 
|-
|  style="vertical-align: top;"|(e) Graphical representation between exact solution and various values of ζ at t = 0.01.
|  style="vertical-align: top;"|(f) Graphical representation between exact solution and various values of ζ at x = 1.
|-
|  colspan='2'  style="vertical-align: top;"|Figure 3: Periodic wave analytical (NDM) solutions ω (x,τ) of Eq. (32) with initial condition (46) and α = β = γ = θ = 1.
|}

===5.4 Application 4===

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) &=&\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 },\end{matrix}\quad \quad \quad \, \, (51)</math>
|}
</div>

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

<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{{t}^{\zeta }}{10800\Gamma \left( \zeta +1\right) }\left\{ -3\left( 959\sqrt{19}+4180\right) sinh\left( \frac{1}{6}\left( \sqrt{19}+5\right) x\right) +\left( 4003\sqrt{19}+17480\right) cosh\left( \frac{1}{6}\left( \sqrt{19}+5\right) x\right) \right. ,\\&&\left. -95\left( 79\sqrt{19}+344\right) \right\} \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{matrix}\, \, (52)</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) &=&\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) .\quad (53)\end{matrix}</math>
|}
</div>

For the specific instance when <math display="inline">\zeta =1</math>, the approximative findings and 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 Figure 4, 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.


<span id='_GoBack'></span>Table 4: Comparison of case 4 for the exact solution with NDM solution and absolute errors using numerical calculations at <math display="inline">t=</math><math>1</math> and <math display="inline">\alpha =\beta =\gamma =\theta =1</math>''.''

{| style="width: 86%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|x
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{Ex}}}</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{1}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|''Absolute error''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \mathit{\boldsymbol{0.95}}\right)</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\omega }}}_{\mathit{\boldsymbol{NDM}}}\left( \mathit{\boldsymbol{\zeta }}=\right. </math><math>\left. \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="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|20
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-3.743559577
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-3.743559577
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|5.702105E-13
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-3.743559577
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-3.743559577
|}


{| style="width: 100%;border-collapse: collapse;" 

|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image19.jpeg|222px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image20.jpeg|216px]] 
|-
|  style="vertical-align: top;"|(a) Exact solution of ω (''x'',''t'').
|  style="vertical-align: top;"|(b) NDM solution of ω (''x'',''t'').
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image21.jpeg|204px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image22.jpeg|198px]] 
|-
|  style="vertical-align: top;"|(c) Absolute error at t = 0.01. 
|  style="vertical-align: top;"|(d) Graphical representation between exact and NDM                        solutions at t = 0.01.
|-
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Hagag_831060637-image23.jpeg|228px]] 
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Hagag_831060637-image24.jpeg|222px]] 
|-
|  style="vertical-align: top;"|(e) Graphical representation between exact solution and various values of ζ at t = 0.01.
|  style="vertical-align: top;"|(f) Graphical representation between exact solution and various values of ζ at x = 1.
|-
|  colspan='2'  style="vertical-align: top;"|Figure 4: Periodic wave analytical (NDM) solutions ω (x,τ) of Eq. (32) with initial condition (50) and α = β = γ = θ = 1.
|}



<span id='conclusion'></span>

==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).

==References==

[1] R. Hilfer, Applications of Fractional Calculus in Physics, Academic press, Orlando, 1999.

[2] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.

[3] I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999).

[4] D. Baleanu, Z. B. Guvenc and J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications (Springer, Dordrecht, 2010).

[5] M. Caputo, Elasticita e Dissipazione (Bologna, Zanichelli, 1969).

[6] K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993).

[7] J. Liouville, Memoire surquelques questions de geometrieet de mecanique, etsurun nouveau genre de calcul pour resoudreces questions, J. Ecole. Polytech. 13 (1832) 1.

[8] G. F. B. Riemann, Versuch einer allgemeinen Auffassung der Integration und Differentiation, in Gesammelte Mathematische Werke (Vieweg + Teubner Verlag, Leipzig, 1896).

[9] C. Cattani, Haar wavelet-based technique for sharp jumps classification, Math. Comput. Model. 39 (2004) 255.

[10] J. Singh, D. Kumar, Z. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput. 316 (2018) 504 515.

[11] D. Baleanu, B. Ghanbari, J. H. Asad, A. Jajarmi and H. M. Pirouz, Planar system-masses in an equilateral triangle: Numerical study within fractional calculus, CMES Comput. Model. Eng. Sci. 124(3) (2020) 953.

[12] S. A. El-Wakil, A. Elhanbaly, M. A. Abdou. Adomian decomposition method for solving fractional nonlinear differential equations, Applied Mathematics and Computation 182 (1) (2006) 313-324.

[13] V. Daftardar-Gejji, H. Jafari. Adomian decomposition: a tool for solving a system of fractional differential equations, Journal of Mathematical Analysis and Applications 301 (2) (2005) 508-518.

[14] M. Rawashdeh, and S. Maitama, Solving coupled system of nonlinear PDE’s using the natural decomposition method, International Journal of Pure and Applied Mathematics 92 (2014) 757-776.

[15] M. Rawashdeh, and S. Maitama, Finding exact solutions of nonlinear PDEs using the natural decomposition method, Mathematical Methods in the Applied Sciences 40 (2017) 223-236.

[16] R. Khandelwal, P. Kumawat, and Y. Khandelwal, A study of natural transform based on decomposition method for solving nonlinear ordinary differential equation, Int. J. Stat. Appl. Math. 3 (2018) 664-669.

[17] M. S. Rawashdeh, The fractional natural decomposition method: Theories and applications, Math. Methods Appl. Sci. 40 (2017) 2362.

[18] R. Shah, et al., Natural transform decomposition method for solving fractional-order partial differential equations with proportional delay, Mathematics 7 (2019) 532.

[19] M. Rawashdeh, The fractional natural decomposition method: theories and applications, Mathematical Methods in the Applied Sciences 40 (2017) 2362-2376.

[20] M. Elbadri, et al., A new solution of time-fractional coupled KdV equation by using natural decomposition method, In Abstract and Applied Analysis, 2020 (2020).

[21] M. O. Korpusov, and A. G. Sveshnikov, Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type, Journal of Mathematical Sciences 148 (2008) 1-142.

[22] S. A. Dubey, Numerical solution for nonlocal Sobolev-type differential equations, In Electron. J. Differ. Equ. Conf 19 (2010) 75-83.

[23] E. Kaikina, P. Naumkin, and I. Shishmarev, The Cauchy problem for an equation of Sobolev type with power non-linearity, Izvestiya: Mathematics 69 (2005) 59.

[24] G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Mathematical Methods in the Applied Sciences 20 (1997) 271-289.

[25] D. H. Peregrine, Calculations of the development of an undular bore, Journal of Fluid Mechanics 25 (1966) 321-330.

[26] T. Benjamin, J. Bona, and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 272 (1972) 47-78.

[27] R. Camassa, and D. Holm, An integrable shallow water equation with peaked solitons, Physical review letters 71 (1993) 1661.

[28] D. Rollins, Painlevé analysis and Lie group symmetries of the regularized long-wave equation, Journal of mathematical physics 32 (1991) 3331-3332.

[29] Y. Kodama, On solitary-wave interaction, Physics Letters A 123 (1987) 276-282.

[30] W. Hereman et al., Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, Journal of Physics A: Mathematical and General 19 (1986) 607.

[31] F. Mainardi, On the initial value problem for the fractional diffusion equation, in: Wave and Stability in Continuous Media, S. Rionero, T. Ruggeeri (Eds.), Word Scientific, Singapore, 246, 1994.

[32] D. Loonker and P. K. Banerji, Solution of fractional ordinary differential equations by natural transform, International Journal of Mathematical Engineering and Science 12(2) (2013) 1-7.

[33] P. Singh and D. Sharma, Convergence and error analysis of series solution of nonlinear partial differential equation, Nonlinear Engineering 7 (2018) 303-308.

[34] E. Kreyszig, Further Applications: Banach Fixed Point Theorems, Erwin Kreyszig, Introductory Functional Analysis with Applications. New York, NY: Wiley Classic Libraries; (1989) 299-321.

[35] Ö. F. Gözükızıl, and Ş. Akçağıl, The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions, Advances in Difference Equations 2013(1) (2013) 1-18.35