\documentclass[10pt]{article}

\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}





\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
            linkcolor = blue,
            urlcolor  = blue,
            citecolor = blue,
            anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
  \errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother


\usepackage{natbib}




\renewenvironment{abstract}
  {{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
  {\bigskip}

\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}


\usepackage{authblk}


\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}


\usepackage[english]{babel}



\usepackage{float}






%%% Your header encountered errors when converting with LaTeXML
%%% Please uncomment it incrementally and check for problems
%%%   If you need assistance, contact us at help@authorea.com
%%% 
%
%\iflatexml
%\documentclass[11pt,a4paper]{article}
%\fi
%
%\usepackage{amsfonts}
%
%\usepackage[latin5]{inputenc}
%
%\usepackage{amsmath}
%
%\usepackage{amssymb}
%
%\usepackage{graphicx}
%
%\usepackage{subfigure}
%
%\usepackage{caption}
%
%\usepackage{float}
%
%% \usepackage{mathtools} % Not (yet) supported by LaTeXML.
%
%\usepackage{color}
%
%\usepackage{cite}
%
%% \usepackage{mathrsfs} % Not (yet) supported by LaTeXML.
%
%\usepackage{geometry}
%\setcounter{MaxMatrixCols}{10}
%\setlength\parindent{0pt}
%\DeclareMathOperator{\sech}{sech}
%\newtheorem{theorem}{Theorem}


\begin{document}

\title{Hyperbolic Tangent Ansatz Method to Partial Differential Equations by
Using Symbolic Computation}



\author[1]{Ozlem Ersoy Hepson}%
\affil[1]{Eskisehir Osmangazi Universitesi Fen Edebiyat Fakultesi}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
Space--time fractional forms of the modified Korteweg-de Vries equation, the modified Equal Width equation and Benney-Luke equations are solved by using simple hyperbolic tangent ansatz approach. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer--ordered ODEs. Then, the hyperbolic tangent ansatz is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.%
\end{abstract}%



\sloppy


\textit{Keywords:} Ansatz method; fractional modified Korteweg--de Vries
equation; fractional modified Equal Width equation; fractional Benney-Luke
equation; Conformable fractional derivative.\newline
\textit{MSC2010:} 35C07;35R11;35Q53. \newline
\textit{PACS:} 02.30.Jr; 42.65.--k; 04.20.Jb

\section*{Introduction}

\bigskip Fractional calculus has been used in almost all fields
of engineering, mathematics and other natural sciences especially in fluid mechanics, plasma waves, plasma physics etc.. Recently, several mathematical methods have been applied to PDEs. Two-dimensional differential transform method was implemented to obtain approximate analytical solutions of fractional modified Korteweg-de Vries(fmKdV) equation \hyperref[csl:1]{(Kuralay \& Bayram, 2010)}. Homotopy analysis method and its modification were used to solve fmKdV equation \hyperref[csl:2]{(Abdulaziz et al., 2009)}. Similarly the homotopy-perturbation method was also capable of getting approximate analytical solution of fmKdV equation \hyperref[csl:3]{(Abdulaziz et al., 2009)}. In \hyperref[csl:4]{(Wang et al., 2008)}, the travelling wave solutions were expressed in terms of hyperbolic, trigonometric and rational functions. Then ($G^{\prime }/G$)-expansion method was applied for the analytical solutions of the fmKdV equations. 
Exact solutions of the fmKdV equation were investigated by the generalized Kudryashov
method by Bulut et al. \hyperref[csl:5]{(Bulut et al., 2014)}. In that paper, soliton solutions and
hyperbolic function solutions were constructed by using the properties of exponential functions.

The conformable fractional mKdV equation

\begin{equation}
D_{t}^{\alpha }u(x,t)+pu^{2}(x,t)D_{x}^{\beta }u(x,t)+qD_{x}^{3\beta}u(x,t)=0,
\label{kdv} 
\end{equation}
where $u$ is defined for $x>0,t>0$ and the coefficients $p,q\in \mathbb{R}-\{0\}$, is considered. In this equation, the
conformal derivative operators $D_{x}^{\beta }$ and $D_{t}^{\alpha }$ represent the $\beta $.th and $\alpha $.th
order derivatives ($\beta, \alpha \in (0,1]$).

The fractional modified Equal Width (fmEW) equation in conformable
fractional form as

\begin{equation}
D_{t}^{\alpha }u(x,t)+ru^{2}(x,t)D_{x}^{\beta }u(x,t)+sD_{xxt}^{\beta\beta\alpha }u(x,t)=0
\label{ew} 
\end{equation}
is also regarded as the second equation to be solved exactly for $x>0,t>0$ and $r,s$ are parameters. The bright
soliton solutions and singular solutions were obtained using ansatz method
when the derivatives are in sense of modified Riemann-Lioville derivative\hyperref[csl:6]{(Hosseini \& Ayati, 2016)}. Hosseini and Ayati
derived exact solutions of fmEW equation via the Kudryashov
method. For this purpose, they introduced fractional complex transformation
to reduce it into the integer ordered ordinary differential equation. Korkmaz introduced some more exact solutions in forms of various hyperbolic functions \hyperref[csl:7]{(Korkmaz, 2017)}.

The space-time fractional Benney-Luke (fBL) equation in conformable
fractional form can be interpreted as

\begin{align}
D_{t}^{2\alpha }u(x,t)-D_{x}^{2\beta }u(x,t)+\mu D_{x}^{4\beta
}u(x,t)
\notag\\
-\eta D_{xt}^{2\beta2\alpha}u(x,t)+D_{t}^{\alpha
}u(x,t)D_{x}^{2\beta }u(x,t)
\notag\\
+2D_{x}^{\beta }u(x,t)D_{xt}^{\beta \alpha
}u(x,t)=0 \label{bl} 
\end{align}
is considered as the last equation to be solved exactly, where $x>0,t>0$ and $\mu $ and $\eta $ are positive parameters, linked to the inverse bond number, whose
characteristics is to capture the effects of surface tension and the gravity
forces \hyperref[csl:8]{(U. et al., 2017)}. This equation describes two-way water wave propogation in
the presence of surface tension \hyperref[csl:9]{(Quintero \& Grajales, 2008)}. The existence and analyticicity
of lump solutions for generalized Benney-Luke equations was proved by \hyperref[csl:10]{(Quintero, 2002)}. 
In \hyperref[csl:11]{(\textit{{Travelling Wave}}, 2012)}, some travelling wave solutions of two well-known BL
equations were derived via tanh-coth method analytically. An expansion method 
was applied to the BL equation for travelling wave solutions expressed by the trigonometric, hyperbolic and rational functions in \hyperref[csl:12]{(Islam, 2015)}. Also, Kudryashov method and the modified extended tanh expansion were also applied for Benney-Luke equation \hyperref[csl:13]{(Ali \& Nuruddeen, 2017)}.

This paper is organized as follows: In Section 2, preliminaries and essential tools are given. 
In section 3, the ansatz method is mentioned briefly. Then, in Sections 4,5,6 the ansatz method is given for space--time fmKdV, fmEW and Benney--Luke equations, respectively. In the last sections,the findings are summerized.

\section*{Conformable Fractional Derivative and its Properties}

Conformable fractional derivative of order $\alpha$th (similarly $\beta $th) $u$ is described as

\begin{equation}
D_{t}^{\alpha }(u(t))=\lim\limits_{\tau \rightarrow 0}{\dfrac{u(t+\tau
t^{1-\alpha })-u(t)}{\tau }},\,\alpha \in (0,1]
\end{equation}
in the positive half space for a function $u:[0,\infty )\rightarrow \mathbb{R}$\hyperref[csl:14]{(Khalil et al., n.d.)}. The $\alpha $th conformable derivative supports several properties
given below.

\begin{theorem}
Let $\alpha \in (0,1]$, and assume that $u$ and $v$ are $\alpha$-differentiable in the positive half plane (interval) $t>0$. Then,

\begin{itemize}
\item $D_{t}^{\alpha }(au+bv)=aD_{t}^{\alpha }(u)+bD_{t}^{\alpha }(v)$

\item $D_{t}^{\alpha }(t^{p})=pt^{p-\alpha },\forall p\in \mathbb{R}$

\item $D_{t}^{\alpha }(\lambda )=0$, for all constant function $u(t)=\lambda $

\item $D_{t}^{\alpha }(uv)=uD_{t}^{\alpha }(v)+vD_{t}^{\alpha }(u)$

\item $D_{t}^{\alpha }(\frac{u}{v})=\dfrac{vD_{t}^{\alpha }(u)-uD_{t}^{\alpha
}(v)}{v^{2}}$

\item $D_{t}^{\alpha }(u)(t)=t^{1-\alpha }\frac{du}{dt}$

\end{itemize}

for $\forall a,b\in \mathbb{R}$\hyperref[csl:15]{(Atangana et al., 2015}; \hyperref[csl:16]{Çenesiz et al., 2016}; \hyperref[csl:17]{Abdeljawad, 2015)}.

\end{theorem}

The conformable fractional derivative supports many significant properties like Laplace
transform, Gronwall's inequality, chain rule, various integration rules, exponential function, and Taylor series expansion\hyperref[csl:17]{(Abdeljawad, 2015)}.

\begin{theorem}
Let $u$ be an $\alpha $-differentiable function in conformable sense. Also
suppose that $v$ is differentiable in classical sense and is defined in the
range of $u$. Then,

\begin{equation}
D_{t}^{\alpha }(u\circ v)(t)=t^{1-\alpha }v^{\prime }(t)u^{\prime }(v(t))

\end{equation}

\end{theorem}

\section*{Method of Solution}

The implementation of the ansatz method can be achieved after reducing the
fractional PDE

\begin{equation}
P_{1}(u,D_{t}^{\alpha }u,D_{x}^{\beta }u,D_{tt}^{\alpha\alpha }u,D_{xx}^{\beta\beta}u,D_{tx}^{\alpha\beta}u,...)=0
\end{equation}
to an ODE of the form

\begin{equation}
P_{2}(U,U^{\prime },U^{\prime \prime },\ldots )=0  \label{ode} 
\end{equation}
by using a compatible wave transformation. In the present study, we choose
the wave transformation

\begin{equation}
u(x,t)=U(\xi ),\quad \xi =a(\frac{x^{\beta }}{\beta }-\nu \frac{t^{\alpha }}{\alpha })  \label{wt} 
\end{equation}
to satisfy the compatibility of the conformable derivative. The next step is
determination of the power $B$ of the ansatz in the predicted solution

\begin{equation}
U(\xi )=A\tanh ^{B}{\xi },\,A\neq 0,\,B\neq 0  \label{ansatz} 
\end{equation}
by substituting it into (\ref{ode}). Substituting the ansatz with determined the balance between
the non linear term and the derivative term with highest order, B, into (7) leads an
algebraic system of equations to specify the relation among the parameters.
The particular goal is to determine the parameters $A$, $a$ and $\nu $ in terms of the others. 
Once the relations between the parameters are determined, the solution to (\ref{ode}) can be expressed
explicitly. The particular goal is to determine the parameters in PDEs in terms of the original variables.

\section*{The solutions of the conformable fmKdV Equation}

The wave transformation (\ref{wt}) reduces the fmKdV equation (\ref{kdv})

\begin{equation}
-{a\nu \frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi \right) +p{a}U\left(
\xi \right) ^{2}{\frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi \right)
+qa^{3}{\frac{\mathrm{d}^{3}}{\mathrm{d}{\xi }^{3}}}U\left( \xi \right) =0
\label{odejm} 
\end{equation}
Integrating this equation once gives

\begin{equation}
-{a\nu }U\left( \xi \right) +\frac{1}{3}\,p{a}U\left( \xi \right) ^{3}+qa^{3}{\frac{\mathrm{d}^{2}}{\mathrm{d}{\xi }^{2}}}U\left( \xi \right) =K
\label{ewint} 

\end{equation}

where $K$ is the constant of integration. Substituting the predicted
solution (\ref{ansatz}) into (\ref{ewint}) results in

\begin{equation*}

\begin{aligned}
-avA \left( \tanh \left( \xi \right)  \right) ^{B}+1/3\,pa{A}^{3}
 \left(  \left( \tanh \left( \xi \right)  \right) ^{B} \right) ^{3}+{
\frac {q{a}^{3}A \left( \tanh \left( \xi \right)  \right) ^{B}{B}^{2}
 \left( 1- \left( \tanh \left( \xi \right)  \right) ^{2} \right) ^{2}}
{ \left( \tanh \left( \xi \right)  \right) ^{2}}} \\
-2\,q{a}^{3}A \left( 
\tanh \left( \xi \right)  \right) ^{B}B \left( 1- \left( \tanh \left( 
\xi \right)  \right) ^{2} \right) -{\frac {q{a}^{3}A \left( \tanh
 \left( \xi \right)  \right) ^{B}B \left( 1- \left( \tanh \left( \xi
 \right)  \right) ^{2} \right) ^{2}}{ \left( \tanh \left( \xi \right) 
 \right) ^{2}}}=K
   
\end{aligned}

\end{equation*}

B is determined as $1$ by balancing $U^{3}$ and $U^{''}$. Thus, the predicted solution is set as

\begin{equation*}
\left( \frac{1}{3}pa{A}^{3}+2\,q{a}^{3}A \right)  \left( \tanh  \xi
 \right) ^{3}+ \left( -avA-2\,q{a}^{3}A \right) \tanh \left( 
\xi \right) =K
\end{equation*}

Since the predicted solution is different from zero, the last equation is
satisfied if only the coefficients of powers of the hyperbolic tangent
function are zero. Thus,

\begin{eqnarray}
\frac{1}{3}pa{A}^{3}+2\,q{a}^{3}A&=&0 \label{e3} \\
-avA-2\,q{a}^{3}A&=&0\label{e4} \\
K&=&0  \label{e5} 

\end{eqnarray}
Solution of this algebraic system for $A$, $a$ and $\nu $ gives

\begin{equation*}
A=\pm \sqrt {-6\,{\frac {q}{p}}}
\end{equation*}

\begin{equation*}
v=-2qa^2
\end{equation*}

for nonzero $p$, $q$ and arbitrarily chosen $a$. Using these data,
one can write the solution of (\ref{ode}) as

\begin{equation*}
u_{1,2}(\xi )=\pm \sqrt {-6\,{\frac {q}{p}}}\tanh \left( \xi \right) 
\end{equation*}
Returning the original variables gives the solution to the fmKdV(\ref{kdv}) as

\begin{equation}
u_{1,2}(x,t)=\pm \sqrt {-6\,{\frac {q}{p}}}\tanh \left( a \left( {\frac {{x}^{\beta}}{
\beta}}+2\,{\frac {q{a}^{2}{t}^{\alpha}}{\alpha}} \right)  \right) 
\label{sol} 
\end{equation}
for nonzero $p,q$. This solution is the general form of hyperbolic tangent
function type solution family. Choosing the parameters properly, many
solutions in different characteristics covering complex ones can be
generated.

Some real solutions are illustrated for particular choices of the parameters. The effect of the derivative order $\alpha$ and $\beta$ to the solution is indicated in the following figures. The solution (\ref{sol}) is chosen to demonstrate a particular solution by choosing the parameters as $p=2$, $q=-1$ and $a=1$. The solutions are plotted on $xtu-$space for various values of $\alpha$ and $\beta$. Thus, the solution is reduced to

\begin{equation}
u(x,t)= \sqrt {6}\tanh  \left( {\frac {{x}^{\beta}}{\beta}}-2\,{\frac {{t}^{
\alpha}}{\alpha}}  \right) \label{sss} 

\end{equation}

When the derivative order $\alpha$ and $\beta$ are both chosen as $0.5$, the wave profile determined in the solution (\ref{sss}) moves towards to the left with a variable velocity depending on time variable $t$, Fig {\ref{fig1}(a)}. The increase the $\beta$ value to $1$, makes the shape of the wave sharper slightly, Fig \ref{fig1}(b). Moreover, the velocity of the wave also increases.
In the plot Fig \ref{fig2}(a) and Fig \ref{fig2}(b), the order parameters are chosen as $\alpha =1$ while $\beta =0.5$ and $\beta =0.1$ respectively. The wave profile travels to the left along the $x$-axis with a constant velocity as time proceeds. When $\beta$ is chosen as $1$, the wave becomes sharper besides its velocity increases but still constant.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig1/Fig1}
\caption{{\label{fig1} (a) $\alpha=0.5$, $\beta=0.5$ (b) $\alpha=0.5$, $\beta=1$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig3/Fig3}
\caption{{\label{fig2} (a) $\alpha=1$, $\beta=0.5$ (b) $\alpha=1$, $\beta=1$%
}}
\end{center}
\end{figure}

\section*{The solutions of the conformable fmEW Equation}

The solution algorithm starts by transforming the fmEW to an ODE by using the compatible wave transformation (\ref{wt}) . Thus, $u(x,t) \rightarrow U(\xi)$. This transformation reduces the fmEW (\ref{ew}) to

\begin{equation}
-{a\nu \frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi \right) +r{a}U\left(
\xi \right) ^{2}{\frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi \right)
-sa^{3}v{\frac{\mathrm{d}^{3}}{\mathrm{d}{\xi }^{3}}}U\left( \xi \right) =0
\label{OD} 

\end{equation}

Integrating both sides of the equation gives

\begin{equation}
-avU \left( \xi \right) +\frac{1}{3}ra \left( U \left( \xi \right)  \right) 
^{3}-s{a}^{3}v{\frac {d^{2}}{d{\xi}^{2}}}U \left( \xi \right) =K
\label{blint} 

\end{equation}

where $K$ is the constant of integration. Assume that (\ref{blint}) has a solution of the form  (\ref{ansatz})
where $A$ is a nonzero real and $B$ is positive integer.
Substituting the predicted solution  and its derivatives into the equation (\ref{blint}) gives

\begin{equation}

\begin{aligned}
-avA \left( \tanh \left( \xi \right)  \right) ^{B}+\frac{1}{3}ra{A}^{3}
 \left(  \left( \tanh \left( \xi \right)  \right) ^{B} \right) ^{3}-{
\frac {s{a}^{3}vA \left( \tanh \left( \xi \right)  \right) ^{B}{B}^{2}
 \left( 1- \left( \tanh \left( \xi \right)  \right) ^{2} \right) ^{2}}
{ \left( \tanh \left( \xi \right)  \right) ^{2}}} \\
+2\,s{a}^{3}vA \left( \tanh \left( \xi \right)  \right) ^{B}B \left( 1- \left( \tanh
 \left( \xi \right)  \right) ^{2} \right) +{\frac {s{a}^{3}vA \left( 
\tanh \left( \xi \right)  \right) ^{B}B \left( 1- \left( \tanh \left( 
\xi \right)  \right) ^{2} \right) ^{2}}{ \left( \tanh \left( \xi
 \right)  \right) ^{2}}}=K
\label{ewi} 

\end{aligned}

\end{equation}

B is obtained as $1$ by balancing $U^{3}$ and $U^{''}$. Thus, the following solution is determined as

\begin{eqnarray}
\left( \frac{1}{3}ra{A}^{3}-2\,s{a}^{3}vA \right)  \left( \tanh \left( \xi
 \right)  \right) ^{3}+ \left( -avA+2\,s{a}^{3}vA \right) \tanh
 \left( \xi \right) =K
\label{eq2} 

\end{eqnarray}
Collecting the coefficients of powers of the $\tanh$ function yields to a system of algebraic
equations in $r,s,A,v,a,K$ of the form

\begin{eqnarray}
\frac{1}{3}ra{A}^{3}-2\,s{a}^{3}vA &=&0 \label{eq3} \\
-avA+2\,s{a}^{3}vA&=&0 \label{eq4} \\
K&=&0 \label{eq5} 

\end{eqnarray}
is obtained. Solving the resultant system (\ref{eq3}-\ref{eq5}) for $\{A,a,v,K\}$, the relation among these parameters are stated as

\begin{eqnarray}
a&=& \pm \frac {1}{\sqrt{2s}} \\
v&=& \frac{1}{3}r{A}^{2}

\end{eqnarray}
for arbitrarily chosen $r,s$. Thus, the solution to the equation (\ref{ode}) is determined as

\begin{equation}
u_{3,4}(\xi)=A\tanh \left( \xi \right)

\end{equation}
Returning the original variables $(x,t)$ from $\xi$ gives the solution to the EW(\ref{ew}) as

\begin{equation}
u_{3,4}(x,t)=\pm A\tanh \left( \frac {1}{\sqrt{2s}} \left( {\frac {{x}^{\beta
}}{\beta}}-{\frac {r{A}^{2}{t}^{\alpha}}{3\alpha}} \right) 
 \right) \label{ss1} 

\end{equation}

The plots of the solutions for some fixed values of the parameters ($A=5,r=1,s=1$)

\begin{equation}
U(x,t)=\tanh \left( \frac{1}{2}\sqrt {2}\sqrt {\frac{1}{s}} \left( x-\frac{25}{3}rt \right) 
 \right)

\end{equation}
The particular solutions determined by choosing some the independent parameters as $A=1$ and $r=s=1$ are depicted for some selections of derivative orders. This solution represents a classical front wave profile propagating to the left, Fig \ref{fig3} (a),(b) - Fig \ref{fig4}(a),(b) for all choices of order parameters $\alpha$ and $\beta$. When both order parameters are $0.5$, the wave has a slightly bending shape, Fig \ref{fig3}(a). Fixing $\alpha =0.5$ but increasing $\beta$ to $1$ causes the shape of the wave to become sharper and the velocity becomes larger, but still non linear Fig \ref{fig3} (b). To fix the velocity order parameter $\alpha =1$ causes a linear propagation of the wave. Increasing $\beta$ from $0.5$ to $1$, its shape becomes sharper and its velocity also increases Fig \ref{fig4}(a),(b).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig5/Fig5}
\caption{{\label{fig3} (a) $\alpha=0.5$, $\beta=0.5$ (b) $\alpha=0.5$, $\beta=1$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig7/Fig7}
\caption{{\label{fig4} (a) $\alpha=1$, $\beta=0.5$ (b) $\alpha=1$, $\beta=1$%
}}
\end{center}
\end{figure}

\section*{The solutions of the conformable fBL Equation}

The wave transformation (\ref{wt}) reduces the fBL equation (\ref{bl})

\begin{equation}
{a}^{2}({\nu }^{2}-1){\frac{\mathrm{d}^{2}}{\mathrm{d}{\xi }^{2}}}U\left(
\xi \right) +{a}^{4}(\mu -\eta v^{2}){\frac{\mathrm{d}^{4}}{\mathrm{d}{\xi }^{4}}}U\left( \xi \right) -3a^{3}v{\frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi \right) {\frac{\mathrm{d}^{2}}{\mathrm{d}{\xi }^{2}}}U\left( \xi
\right) =0
\end{equation}
Integrating this equation once gives

\begin{equation*}
{a}^{2}({\nu }^{2}-1){\frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi
\right) +{a}^{4}(\mu -\eta v^{2}){\frac{\mathrm{d}^{3}}{\mathrm{d}{\xi }^{3}}}U\left( \xi \right) -\frac{3}{2}a^{3}v({\frac{\mathrm{d}}{\mathrm{d}{\xi }}}U\left( \xi \right) )^{2}=K
\end{equation*}
where $K$ is the constant of integration. Substituting the predicted
solution (\ref{ansatz}) into (\ref{bl}) results in

\begin{eqnarray*}
&&\frac{a^{2}A\tanh (\xi )^{B}Bv^{2}}{\tanh (\xi )}-a^{2}v\tanh (\xi
)^{B}B\tanh (\xi )v^{2}-\frac{a^{2}A\tanh (\xi )^{B}B}{\tanh (\xi )} \\
&&+a^{2}A\tanh (\xi )^{B}B\tanh (\xi )+\frac{a^{4}\mu A\tanh (\xi
)^{B}B^{3}(1-\tanh (\xi )^{2})^{3}}{\tanh (\xi )^{3}} \\
&&-\frac{6a^{4}\mu A\tanh (\xi )^{B}B^{2}(1-\tanh (\xi )^{2})^{2}}{\tanh
(\xi )}-\frac{3a^{4}\mu A\tanh (\xi )^{B}B^{2}(1-\tanh (\xi )^{2})^{3}}{\tanh (\xi )^{3}} \\
&&+4a^{4}\mu A\tanh (\xi )^{B}B\tanh (\xi )(1-\tanh (\xi )^{2})+\frac{4a^{4}\mu A\tanh (\xi )^{B}B(1-\tanh (\xi )^{2})^{2}}{\tanh (\xi )} \\
&&+\frac{2a^{4}\mu A\tanh (\xi )^{B}B(1-\tanh (\xi )^{2})^{3}}{\tanh (\xi
)^{3}}-\frac{a^{4}\eta v^{2}A\tanh (\xi )^{B}B^{3}(1-\tanh (\xi )^{2})^{3}}{\tanh (\xi )} \\
&&+\frac{6a^{4}\eta v^{2}A\tanh (\xi )^{B}B^{2}(1-\tanh (\xi )^{2})^{2}}{\tanh (\xi )}+\frac{3a^{4}\eta v^{2}A\tanh (\xi )^{B}B^{2}(1-\tanh (\xi
)^{2})^{3}}{\tanh (\xi )^{3}} \\
&&-4a^{4}\eta v^{2}A\tanh (\xi )^{B}B\tanh (\xi )(1-\tanh (\xi )^{2}) \\
&&-\frac{4a^{4}\eta v^{2}A\tanh (\xi )^{B}B(1-\tanh (\xi )^{2})}{\tanh (\xi )}-\frac{2a^{4}\eta v^{2}A\tanh (\xi )^{B}B(1-\tanh (\xi
)^{2})^{3}}{\tanh (\xi )^{3}} \\
&&-\frac{3}{2}\frac{a^{3}vA^{2}(\tanh (\xi )^{B})^{2}B^{2}(1-\tanh (\xi
)^{2})^{2}}{\tanh (\xi )^{2}} \\
&=&K

\end{eqnarray*}

B is obtained as $1$ by balancing $U^{3}$ and $U^{''}$. Thus, the following solution is determined as

\begin{eqnarray*}
&&\left( -6a^{4}\mu A+6a^{4}\eta v^{2}A-\frac{3}{2}a^{3}vA^{2}\right) \tanh
(\xi )^{4} \\
&&+\left( a^{2}A-8a^{4}\eta v^{2}A+8a^{4}\mu
A-a^{2}Av^{2}+3a^{3}vA^{2}\right) \tanh (\xi )^{2} \\
&&+a^{2}Av^{2}-\frac{3}{2}a^{3}vA^{2}-a^{2}A-2a^{4}\mu A+2a^{4}\eta v^{2}A \\
&=&K

\end{eqnarray*}

Since the predicted solution is different from zero, the last equation is
satisfied if only the coefficients of powers of the hyperbolic tangent
function are zero. Thus,

\begin{eqnarray}
\left( -6a^{4}\mu A+6a^{4}\eta v^{2}A-\frac{3}{2}a^{3}vA^{2}\right)&=&0 \label{bl3} \\
\left( a^{2}A-8a^{4}\eta v^{2}A+8a^{4}\mu A-a^{2}Av^{2}+3a^{3}vA^{2}\right)&=&0 \label{bl4} \\
a^{2}Av^{2}-\frac{3}{2}a^{3}vA^{2}-a^{2}A-2a^{4}\mu A+2a^{4}\eta v^{2}A-K&=&0 \label{bl5} 

\end{eqnarray}

Solution of this algebraic system for $A$, $a$ and $\nu $ gives

\begin{eqnarray}
A&=& \pm \frac{4(-\mu +\eta )a}{(4a^{2}\eta -1)\sqrt{-\frac{1-4a^{2}\mu }{4a^{2}\eta -1}}}\\
v&=& \pm \sqrt{-\frac{1-4a^{2}\mu }{4a^{2}\eta -1}}

\end{eqnarray}
for nonzero $\mu,\eta$ and arbitrarily chosen $a$. Using these data,
one can write the solution of (\ref{ode}) as

\begin{equation*}
u_{5,6,7,8}(\xi )=\pm \frac{4(-\mu +\eta )a}{(4a^{2}\eta -1)\sqrt{-\frac{1-4a^{2}\mu }{4a^{2}\eta
-1}}}\tanh (\xi )
\end{equation*}
Returning the original variables gives the solution to the space-time fractional BL equation (\ref{bl}) ) as

\begin{equation}
u_{5,6,7,8}(x,t)=\pm \frac{4(-\mu +\eta )a\tanh (a\left( \frac{x^{\beta }}{\beta }\pm \sqrt{-\frac{1-4a^{2}\mu }{4a^{2}\eta -1}}\frac{t^{\alpha }}{\alpha }\right) )}{(4a^{2}\eta -1)\sqrt{-\frac{1-4a^{2}\mu }{4a^{2}\eta -1}}}
\end{equation}
This solution is the general form of hyperbolic tangent
function type solution family. Choosing the parameters properly, many
solutions in different characteristics covering complex ones can be
generated.

\begin{equation}
u(x,t)=-\frac{4(-\mu +\eta )\tanh \left( x-\sqrt{-\frac{1-4\mu }{4\eta -1}}t\right) 
}{(4\eta -1)\sqrt{-\frac{1-4\mu }{4\eta -1}}}

\end{equation}

The plots of the solutions to $xt$ plane for some fixed values of the parameters ($\mu=5,\eta=\frac{1}{2}$,$a=1$) .
These particular solutions models a one dimensional wave propagation along the space axis. When $\alpha$ and $\beta$ both are $0.5$, the shape of the wave is slightly soft, Fig \ref{fig5} (a). When $\alpha=0.5$ and $\beta=1$, the wave becomes sharper and faster, Fig \ref{fig5} (b). When $\alpha =1$, the wave is soft and slow for $\beta =0.5$, Fig \ref{fig6} (a) but it becomes sharper and faster when  $\beta =1$, Fig \ref{fig6} (b).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig9/Fig9}
\caption{{\label{fig5} (a) $\alpha=0.5$, $\beta=0.5$  (b) $\alpha=0.5$, $\beta=1$%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Fig11/Fig11}
\caption{{\label{fig6} (a) $\alpha=1$, $\beta=0.5$ (b) $\alpha=1$, $\beta=1$%
}}
\end{center}
\end{figure}

\section*{Conclusion}
Some conformable space--time fractional PDEs are solved by taking hyperbolic tangent ansatz.
FmKdV, fmEW and fBL equations are reduced to some nonlinear ODEs of integer order by using compatible wave
transformations. Substituting the solution into the resultant ODEs and finding relations between the parameters of the equations give the exact solutions of PDEs. The exact solutions are successfully found for
particular choices of the $\alpha$ and $\beta$ and the other parameters.As a conclusion, the method is
applicable to PDEs in the theory of fractional calculus.

\selectlanguage{english}
\FloatBarrier
\section*{References}\sloppy
\phantomsection
\label{csl:1}\textit{{Approximate Analytical Solution}}. (2010).

\phantomsection
\label{csl:2}(2009).

\phantomsection
\label{csl:3}\textit{{Approximate}}. (2009).

\phantomsection
\label{csl:4}\textit{{The (g/g )-Expansion}}. (2008).

\phantomsection
\label{csl:5}\textit{{Exact Solutions}}. (2014).

\phantomsection
\label{csl:6}\textit{{Exact Solutions of Space-Time}}. (2016).

\phantomsection
\label{csl:7}\textit{{Exact Solutions of Space-Time Fractional}}. (2017).

\phantomsection
\label{csl:8}(2017).

\phantomsection
\label{csl:9}\textit{{Instability of}}. (2008).

\phantomsection
\label{csl:10}\textit{{Existence and Analyticity of Lump}}. (2002).

\phantomsection
\label{csl:11}(2012).

\phantomsection
\label{csl:12}\textit{{Applications of the Exp($\phi$)-Expansion Method to Find Exact Traveling Wave Solutions of the}}. (2015).

\phantomsection
\label{csl:13}\textit{{Analytical Treatment for the Conformable Space-Time Fractional}}. (2017).

\phantomsection
\label{csl:14}

\phantomsection
\label{csl:15}\textit{{New}}. (2015).

\phantomsection
\label{csl:16}(2016).

\phantomsection
\label{csl:17}(2015).

\end{document}

