\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 10, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/10\hfil One-phase Stefan problem]
{One-phase Stefan problem with a latent heat depending on
the position of the free boundary and its rate of change}

\author[J. Bollati,  D. A. Tarzia \hfil EJDE-2018/10\hfilneg]
{Julieta Bollati,  Domingo A. Tarzia}

\address{Julieta Bollati \newline
Depto. Matem\'atica - CONICET, FCE,
 Univ. Austral, Paraguay 1950,
S2000FZF Rosario, Argentina}
\email{JBollati@austral.edu.ar}

\address{Domingo A. Tarzia \newline
Depto. Matem\'atica - CONICET, FCE,
 Univ. Austral, Paraguay 1950,
S2000FZF Rosario, Argentina}
\email{DTarzia@austral.edu.ar}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted May 26, 2017. Published January 8, 2018.}
\subjclass[2010]{35R35, 80A22, 35C05, 35C06}
\keywords{Stefan problem; threshold gradient; variable latent heat;
\hfill\break\indent one-dimensional consolidation; explicit solution;
 similarity solution}

\begin{abstract}
 From the one-dimensional consolidation of fine-grained soils with
 threshold gradient, it can be derived a special type of Stefan problems
 where the seepage front, because of the presence of this threshold gradient,
 exhibits the features of a moving boundary. In this type of problems,
 in contrast with the classical Stefan problem, the latent heat is
 considered to depend inversely to the rate of change of the seepage
 front. In this paper, we study a one-phase Stefan problem with a latent
 heat that depends on the rate of change of the free boundary and on
 its position. The aim of this analysis is to extend prior  results,
 finding an analytical solution that recovers, by specifying
 some parameters, the solutions that have already been examined in the
 literature regarding Stefan problems with variable latent heat.
 Computational examples are presented to examine the effect
 of this parameters on the free boundary.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}

This article is a continuation of  the work done by Zhou et al.\ in \cite{ZBL},
where a  one-dimensional consolidation process with a threshold gradient
is studied. This problem is essentially a moving boundary problem where
the seepage front, which moves downward gradually, plays the role of the
free boundary due to the presence of this threshold gradient.
This kind of problems are known in the literature as Stefan problems.
They have been widely studied in the last century due to the fact that
they arise in many significant areas of engineering, geoscience and
industry \cite{AlSo}-\cite{Gu}, \cite{Lu,Me1992,Ru,Ta4}.
A review of the literature on this topic was presentend in \cite{Ta2}.

The classical Stefan problem intends to describe the process of a material
undergoing a phase change like, for example, the melting process on an ice bar.
Finding a solution to this problem consists in solving the heat-conduction
equation in an unknown region  which has also to be determined, imposing an
initial condition, boundary conditions and the Stefan condition at the interface.

In this article, it will be considered the one-phase Stefan problem in a
semi-infinite material with variable latent heat. The reduction to a one-phase
problem is referred to the case in which it is assumed that one of the phases
is at the phase-change temperature. The new mathematical feature of the
problem to be solved is concerned with  the fact that the latent heat is
assumed to depend on the position of the free boundary as well as on its
rate of change which is our novelty, i.e $L=L(s(t),\dot{s}(t))$ (where $s(t)$
is the free front).
It is known that in the classical formulation of the Stefan problem the
latent heat is constant. The idea of a variable latent heat is motivated by
the following previous works:

$\bullet$   In  \cite{Pr} it was considered a Stefan problem with a latent
heat given as a function of the position of the interface $L=\varphi(s(t))$.
Such assumption corresponds to the practical case when the influence of
phenomena such as surface tension, pressure gradients and nonhomogenity of
 materials are taken into account. Sufficient conditions that ensure the
existence and uniqueness of solution were studied in this paper.

$\bullet$  In \cite{VSP} it was provided an analytical solution to the
one-phase Stefan problem with a latent heat defined as a linear function
of the position, i.e $L=\gamma s(t)$ (with $\gamma$  a given constant).
This hypothesis  makes  physical sense in the study of shoreline movement
in a sedimentary basin.
The extension to the two-phase problem was done in \cite{SaTa} .

$\bullet$  In \cite{ZWB} it was considered a one-phase Stefan problem,
with temperature and flux boundary condition at the fixed face,  where the
latent heat was not constant but, rather a power function of the position,
i.e. $L=\gamma s^{n}(t)$ (with $\gamma$ a given constant and $n$ an arbitrary
 non-negative integer). The extension to a non-integer exponent was done
in \cite{ZhXi}. Moreover, in \cite{BoTa2017}  the same problem with a
convective (Robin) condition at the fixed face was considered, obtaining
the results of \cite{ZWB} and \cite{ZhXi} as a limit case.

$\bullet$ In \cite{ZBL} it was studied the one-dimensional consolidation problem
with a threshold gradient. This problem is reduced to a one-phase Stefan
problem with a latent heat expressed as $L=\frac{\gamma}{\dot{s}(t)}$.
That is to say the latent heat depends on the rate of the moving boundary.
It must be noted that the case considered in \cite{ZBL} is not properly a
Stefan problem because the velocity of the moving boundary disappears,
and it has to be treated as a free boundary problem with implicit
conditions \cite{Fa,Sc}.


Based on the bibliography mentioned above it is quite natural from a
mathematical point of view to define a one-phase Stefan problem with a
latent heat given by $L=\gamma s^{\beta}(t)\dot{s}^{\delta}(t)$
(with $\gamma$ a given constant and $\beta$ and $\delta$ arbitrary real constants).

It is worth pointing out that this formulation constitutes a  mathematical
generalization of the one-phase classical Stefan problem and the problems
studied in \cite{VSP,ZBL,ZhXi}.

The aim of this article is to proof in Section 3 the existence and uniqueness
of the explicit solution of the problem given by equations
\eqref{EcCalor}-\eqref{FreeBound} in Section 2. Moreover, in Section 4
 we will consider some special cases and computational examples for
different values of the parameters involved in the problem
\eqref{EcCalor}-\eqref{FreeBound}.  The analytical solution that will
be obtained in this work will recover in one formula, by  choosing
different values for $\beta$ and $\delta$, the solutions obtained in:
the classical Stefan problem ($\beta=\delta=0$), the problem considered
in  \cite{VSP} ($\beta=1$, $\delta=0$), the problems solved in
\cite{ZBL,ZhXi} ($\beta\in \mathbb{R}^{+}_0$, $\delta=0$) and the problem
studied in \cite{ZBL} ($\beta=0$, $\delta=-1$).


\section{Formulation of the problem}

 This article is intended to study the one-dimensional one-phase Stefan
problem for the fusion of a semi-infinite material $x>0$ in which it is
involved a variable latent heat. From a mathematical point of view
the problem can be formulated as follows: Find  the free boundary $s=s(t)$
 (separation between phases) and the temperature $T=T(x,t)$
in the liquid portion of the material that satisfy  the one-dimensional
heat conduction equation:
\begin{equation} \label{EcCalor}
a^2 T_{xx}(x,t)=T_t(x,t), \quad 0<x<s(t),
\end{equation}
 subject to the  boundary condition
\begin{equation}
T(0,t) = t^{\alpha/2}T_0, \label{TempIni} \quad t>0
\end{equation}
the temperature condition at the interface
\begin{equation}
T(s(t),t)= 0,  \quad t>0 \label{TempCambioFase}
\end{equation}
the Stefan condition at the interface
\begin{equation}
-kT_x(s(t),t)=L(s(t),\dot{s}(t)) \dot{s}(t), \quad t>0  \label{CondStefan}
\end{equation}
and the initial condition
\begin{equation}
s(0)=0. \label{FreeBound}
\end{equation}
Here the parameters $a^2$ (diffusion coefficient) and $k>0$
(thermal conductivity) are known constants.
The phase change temperature is 0 and the imposed temperature at the
fixed face $x=0$ is given by $t^{\alpha/2}T_0>0$, where we assume
that $\alpha$  is a non-negative real exponent.

The remarkable feature of this problem is related to the condition at the
 interface given by the Stefan condition \eqref{CondStefan}, where the
latent heat by unit of volume will be defined by
\begin{equation}
L(s(t),\dot{s}(t))=\gamma s(t)^{\beta} \dot{s}(t)^{\delta}, \label{LatentHeat}
\end{equation}
where $\gamma$ is a given constant, and $\beta$ and $\delta$ are
arbitrary real constants.

\section{Explicit solution of the problem}

To solve  problem \eqref{EcCalor}-\eqref{FreeBound}, we use a similarity
transformation to the one given in \cite{ZWB,ZhXi}:
\begin{equation}
T(x,t)=t^{\alpha/2}\varphi\left( \eta\right), \quad \text{with } \quad
\eta=\frac{x}{2a\sqrt{t}}. \label{Transform-1}
\end{equation}
Computing the derivatives of $T$:
\begin{gather}
T_{xx}(x,t)=\frac{t^{(\alpha/2-1)}}{4a^2} \varphi'' (\eta) ,\\
T_t(x,t)=\frac{\alpha}{2}t^{(\alpha/2-1)}\varphi
(\eta)-t^{(\alpha/2-1)} \varphi'(\eta) \frac{\eta}{2},
\end{gather}
we obtain that the temperature given by \eqref{Transform-1} satisfies
the heat equation \eqref{EcCalor} if and only if $\varphi$ is the solution
of the  ordinary differential equation
\begin{equation}
\varphi''(\eta)+2\eta \varphi'(\eta)-2\alpha \varphi(\eta) =0, \label{EcuDif-1}
\end{equation}
whose general solution, in this case, can be written as
(see the proof in the Appendix A)
\begin{equation} \label{genSol2}
\varphi(\eta)=c_1 M \Big(-\frac{\alpha}{2},\frac{1}{2},-\eta^2\Big)
+c_{2}\eta M\Big(-\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\eta^2\Big),
\end{equation}
where $c_{1}$ and $c_{2}$ are arbitrary  constants. The function $M(a,b,z)$,
called Kummer function, is  defined by
\begin{equation}
 M(a,b,z)=\sum_{s=0}^{\infty}\frac{(a)_s}{(b)_s s!}z^s,  \label{M}
\end{equation}
in which $b$ cannot be a non-positive integer, and where $(a)_s$ is the Pochhammer
 symbol  defined by:
\begin{equation}
 (a)_s=a(a+1)(a+2)\dots (a+s-1), \quad (a)_0=1
\end{equation}
All the properties of Kummer's functions to be used in this paper can be found
in \cite{OLBC}.

Therefore  $T(x,t)$ is given by
\begin{equation} \label{Temperatura}
T(x,t)=t^{\alpha/2}\Big[ c_1 M\Big(-\frac{\alpha}{2},\frac{1}{2},-\eta^2 \Big)
+c_2 \eta M\Big( -\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2}, -\eta^2\Big)\Big].
\end{equation}
where $c_1$ and $c_2$ are constants that must be determined in order that
$T(x,t)$ satisfies the conditions \eqref{TempIni}-\eqref{CondStefan}.


From equation \eqref{TempIni}, taking into account that
 $M\left(-\frac{\alpha}{2},\frac{1}{2},0 \right)=1$, it is obtained
\begin{equation} \label{c1}
c_1=T_0.
\end{equation}



From condition \eqref{TempCambioFase}, we have
$\varphi\big(\frac{s(t)}{2a\sqrt{t}} \big)=0$ for all $t>0$. Then
\begin{equation} \label{fronteraLibre}
s(t)=2a\xi \sqrt{t},
\end{equation}
where $\xi$ is a positive constant that has to be determined.
Bearing in mind that the free boundary $s(t)$ is defined by \eqref{fronteraLibre},
it can be deduced from
\eqref{TempCambioFase} and \eqref{c1} that
\begin{equation} \label{c2}
c_2=\frac{-T_0 M\big(-\frac{\alpha}{2},\frac{1}{2},-\xi^2 \big)}
{\xi M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\xi^2\big)}.
\end{equation}
Until know, $s(t)$ and $c_2$ are given in function of $\xi$.
To determine $\xi$,  we apply the Stefan condition \eqref{CondStefan}
which has not been considered yet. For that purpose, $T_x(x,t)$ must be calculated,
\begin{equation} \label{derivTemperatura}
 T_x(x,t)
=\frac{t^{(\alpha-1)/2}}{a}
\Big[c_1 \alpha M\big( -\frac{\alpha}{2}+1,\frac{3}{2},-\eta^2\big)
+\frac{c_2}{2}M\big( -\frac{\alpha}{2}+\frac{1}{2},
\frac{1}{2},-\eta^2\big) \Big]
\end{equation}

From the Stefan condition, taking into account \eqref{fronteraLibre}
and \eqref{derivTemperatura}, we obtain
\begin{equation}
\begin{aligned}
&-\frac{k t^{(\alpha-1)/2}}{a}\Big[c_1 \alpha M
 \big( -\frac{\alpha}{2}+1,\frac{3}{2},-\xi^2\big)
 +\frac{c_2}{2}M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{1}{2},
 -\xi^2\big) \Big]\\
&= \gamma 2^{\beta} \xi^{\beta+\delta+1}a^{\beta+\delta+1}t^{(\beta-\delta-1)/2}.
\end{aligned}\label{condStefan-2}
\end{equation}
As $c_1$, $c_2$ and $\xi$ does not depend on $t$, \eqref{condStefan-2} makes
sense if and only if  $t^{(\alpha-1)/2}=t^{(\beta-\delta-1)/2}$.
This  leads to
\begin{equation} \label{alpha-beta-delta}
\alpha=\beta-\delta.
\end{equation}

Therefore, assuming that \eqref{alpha-beta-delta} holds, condition \eqref{CondStefan}
leads to
\begin{equation}
\begin{aligned}
&\frac{kT_0}{2a\xi M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\xi^2\big)}
 \Big[ -2\alpha \xi^2 M\big(-\frac{\alpha}{2}+1,\frac{3}{2},-\xi^2 \big)
M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\xi^2\big)\\
&\quad +M\big( -\frac{\alpha}{2},\frac{1}{2},-\xi^2\big)
M\big(-\frac{\alpha}{2}+\frac{1}{2},\frac{1}{2},-\xi^2 \big) \Big]\\
&=\gamma 2^{\beta}a^{\beta+\delta+1}\xi^{\beta+\delta+1}.
\end{aligned} \label{condStefan-3}
\end{equation}
Taking note of the  identity proved in \cite{ZhXi},
\begin{align*}
e^{-\xi^2}
&=   -2\alpha \xi^2 M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},
-\xi^2\big)M\big(-\frac{\alpha}{2}+1,\frac{3}{2},-\xi^2 \big)  \\
&\quad +M\big(-\frac{\alpha}{2},\frac{1}{2},-\xi^2 \big)
M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{1}{2},-\xi^2\big),
\end{align*}
and the relationship presented in \cite{OLBC},
\begin{equation}
M\big(-\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\xi^2\big)
=e^{-\xi^2} M\big( \frac{\alpha}{2}+1,\frac{3}{2},\xi^2\big),
\end{equation}
equation \eqref{condStefan-3} becomes
\begin{equation}
\frac{kT_0}{\gamma a^{\beta+\delta+2}2^{\beta+1}}
\frac{1}{\xi M\big(\frac{\alpha}{2}+1,\frac{3}{2},\xi^2 \big)}
= \xi^{\beta+\delta+1}.
\end{equation}
That is to say $\xi$ is a positive solution of the equation
\begin{equation} \label{ecuacionXi}
\frac{kT_0}{\gamma a^{\beta+\delta+2}2^{\beta+1}} f(z)
= z^{\beta+\delta+1}, \quad z>0,
\end{equation}
where
\begin{equation}
f(z)=\frac{1}{z M\big(\frac{\alpha}{2}+1,\frac{3}{2},z^2 \big)}\,.
\end{equation}
Furthermore, from \cite{OLBC} knowing that
\begin{equation}
\frac{d}{dz} \big[ zM\big(\frac{\alpha}{2}+1,\frac{3}{2},z^2 \big)\big]
=M\big( \frac{\alpha}{2}+1,\frac{1}{2},z^2\big)
\end{equation}
it follows that
\begin{equation}
f'(z)=-f^2(z) M\big( \frac{\alpha}{2}+1,\frac{1}{2},z^2\big).
\end{equation}
In this way, it can be said that the left-hand side of equation \eqref{ecuacionXi}
 given by $LE(z)=\frac{kT_0}{\gamma a^{\beta+\delta+2}2^{\beta+1}} f(z)$ satisfies
\begin{gather}
(LE)'(z)=-\frac{kT_0}{\gamma a^{\beta+\delta+2}2^{\beta+1}}f^2(x)
 M\big( \frac{\alpha}{2}+1,\frac{1}{2},z^2\big)<0, \label{LE-1}\\
 LE(0)= +\infty, \label{LE-2}\\
 LE(+\infty)= 0, \label{LE-3}
\end{gather}
meanwhile the right hand side of \eqref{ecuacionXi} given by
 $RI(z)=z^{\beta+\delta+1}$ satisfies
\begin{gather}
(RI)'(z)= (\beta+\delta+1)z^{\beta+\delta}>0,
\quad (\text{ if } \beta+\delta+1\geq 0) \label{RI-1}\\
RI(0)= 0, \label{RI-2}\\
RI(+\infty)= +\infty. \label{RI-3}
\end{gather}
Thus, from \eqref{LE-1}-\eqref{LE-3} and \eqref{RI-1}-\eqref{RI-3},
one can conclude that equation \eqref{ecuacionXi} has a unique positive
solution $\xi$ provided that $\beta+\delta+1\geq 0$.

It should be mentioned that because of \eqref{alpha-beta-delta},
i.e $\alpha=\beta-\delta$, the fact that $\alpha\geq 0$, and the request
that $\beta+\delta+1\geq 0$, the results obtained in this paper are
valid only if $\beta\geq \max\left(\delta, -\delta-1 \right)$.

The above arguments can be summarized in the following theorem.

\begin{theorem} \label{Teo2.1}
Let $\beta$ and $\delta$ be arbitrary  constants satisfying
$\beta\geq \max\left(\delta, -\delta-1 \right)$. Taking $\alpha=\beta-\delta$,
there exists a unique solution  of a similarity type for the one-phase
Stefan problem \eqref{EcCalor}-\eqref{FreeBound}  given by
\begin{gather}
T(x,t)=  t^{\alpha/2}\big[ c_{1} M\big(-\frac{\alpha}{2},
\frac{1}{2},-\eta^2\big)+c_{2} \eta
 M\big(-\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\eta^2\big)\big], \label{Temp-Teo}\\
s(t) =2a \xi \sqrt{t}, \label{Frontera-Teo}
\end{gather}
where  $\eta=\frac{x}{2a\sqrt{t}}$ and the constants $c_{1}$ and
$c_{2}$ are given by
\begin{equation}
c_{1}=T_0, \quad
 c_{2}=\frac{-T_0 M\big(-\frac{\alpha}{2},\frac{1}{2},-\xi^2 \big)}
{\xi M\big( -\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\xi^2\big)},
\end{equation}
and the dimensionless coefficient $\xi$ is obtained as the unique
positive solution of the  equation
\begin{equation}
\frac{kT_0}{\gamma a^{\beta+\delta+2}2^{\beta+1}} f(z)
= z^{\beta+\delta+1}, \quad z>0, \label{ecu-teo}
\end{equation}
in which $f$ is the real function defined by
\begin{equation}
f(z)=\frac{1}{z M\left(\frac{\alpha}{2}+1,\frac{3}{2},z^2 \right)},
\quad z>0. \label{f-teo}
\end{equation}
\end{theorem}


\section{Special cases and computational examples}

 This section is meant to highlight the problems that are generalized
in this work by showing that the solutions already reached in the
literature can be obtained from the one we present by just choosing the
appropriate parameters $\beta$, $\delta$ and thus $\alpha$.
For each case it it going to be done a computational example in order
to see how the parameter $\xi$, that characterizes the free boundary,
varies with respect to $\delta$, for a fixed $\beta$.

Properties found in \cite{OLBC} and \cite{ZhXi} will be helpful
in the subsequent arguments:
\begin{gather}
M(0,b,z)=1 \label{Prop-1} \\
M(a,b,z)=e^z M(a,b,-z) \label{Prop-2} \\
M\big(-\frac{n}{2},\frac{1}{2},-z^2 \big)
=2^{n-1} \Gamma\big(\frac{n}{2}+1 \big)
\left[i^n \operatorname{erfc}(z)+i^nerfc(-z) \right]
 \label{Prop-3} \\
zM\big(-\frac{n}{2}+\frac{1}{2},\frac{3}{2},-z^2 \big)
=2^{n-2}\Gamma\big( \frac{n}{2}+\frac{1}{2}\big)
 \left[i^n \operatorname{erfc}(-z)-i^n \operatorname{erfc}(z) \right] \label{Prop-4}
\end{gather}
with $n\in\mathbb{N}$, and where $i^n \operatorname{erfc}(\cdot)$ is the repeated integral
of the complementary error function defined by
\begin{gather}
 i^0 \operatorname{erfc}(z)=erfc(z)=1- \operatorname{erf}(z), \quad
\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}}
\int_0^z e^{-u^2}du, \\
 i^n \operatorname{erfc}(z)=\int_{z}^{+\infty} i^{n-1}\operatorname{erfc}(t)dt
\end{gather}

Let us analyze the explicit solution achieved in each of the following problems:


\subsection*{Classical one-phase Stefan problem}
The latent heat is given by $L=\gamma$ constant, so the solution in this
case can be recovered from the solution given by \eqref{Temp-Teo}-\eqref{f-teo}
by taking $\beta=0, \delta=0$ and thus $\alpha=0$.
It must be pointed out that in this case, using  that
\begin{equation}
M\big(0,\frac{1}{2},-\eta^2 \big) =1, \quad \text{and} \quad
 M\big( \frac{1}{2},\frac{3}{2},-\eta^2\big)
=\frac{\sqrt{\pi}}{2\eta} \operatorname{erf}(\eta),
\end{equation}
we get a temperature with the form $T(x,t)=c_1+c_2 \operatorname{erf}(\eta)$,
like in the classical literature \cite{AlSo,Ca,Cr,Gu,Lu,Ru,Ta4}.

\subsection*{Problem of the shoreline movement in a sedimentary basin
\cite{VSP, SaTa}}
In this problem it was considered a latent heat that varies linearly with
the position, that is to say $L=\gamma s(t)$. Therefore the solution can be
obtained from Theorem \ref{Teo2.1} by choosing $\beta=1$, $\delta=0$ and
thus $\alpha=1$. Using \eqref{Prop-3}, and the properties:
$i\operatorname{erf}(z)=\frac{e^{-z^2}}{\sqrt{\pi}}+z \operatorname{erfc}(z)$ and
$\Gamma\left( \frac{3}{2}\right)=\frac{\sqrt{\pi}}{2}$, the temperature becomes:
\begin{equation}
T(x,t)	=  c_1\Big[ \sqrt{t}e^{-\eta^2} +\frac{\sqrt{\pi}}{2}x
\operatorname{erfc}(\eta)  \Big]
 + \frac{c_2}{2} x  \label{alpha=1}
\end{equation}
in accordance to the solution shown in \cite{VSP} and \cite{SaTa}.

\subsection*{Problem with a latent heat defined as a power function of the
position solved in \cite{ZWB} and \cite{ZhXi}}

In these papers, the latent heat $L$ is defined as a power function of the
position, i.e, $L=\gamma s(t)^{\beta}$ with $\gamma$ constant and $\beta$
a non-negative real exponent. Choosing $\beta\in \mathbb{R}_0^{+}$,
$\delta=0$, and then $\alpha=\beta$, the solutions given in \cite{ZWB}
and  \cite{ZhXi} are automatically recovered.

\subsection*{One-dimensional consolidation problem with threshold gradient \cite{ZBL}}

In this work, it is considered a one-dimensional consolidation problem with a
 threshold gradient which can be transformed into a one-phase Stefan problem
with a latent heat that depends on the rate of change of the moving boundary.
It is studied the case in which $L=\frac{\gamma}{\dot{s}(t)}$.
 We remark here that this problem is not a Stefan problem because the velocity
of the free boundary does not appear but it is a free boundary problem
for the heat equation with implicit free boundary conditions
\cite{Fa,Sc}. Fixing $\beta=0$, $\delta=-1$ and so $\alpha=1$,
 the solution of this problem can be obtained from Theorem \ref{Teo2.1}
the temperature can be expressed as \eqref{alpha=1}.
In addition, taking into account \eqref{Prop-2}, it is obtained that
\begin{equation}
f(z)= \frac{1}{zM\big( \frac{3}{2},\frac{3}{2},z^2\big)}
=\frac{1}{z e^{z^2} M\big(0,\frac{3}{2},-z^2 \big)}
= \frac{e^{-z^2}}{z}
\end{equation}
in agreement to the solution in \cite{ZBL}.
\smallskip

Once it has been compared the solution obtained in this paper with the solutions
 presented in the literature, we are going to run some computational examples.
To solve the Stefan problem \eqref{EcCalor}-\eqref{FreeBound} it is necessary
to solve  equation \eqref{ecu-teo} which is equivalent to find the unique
zero of the  function
\begin{equation}
H(z)=\frac{kT_0}{\gamma a^{\beta+\delta+2}2^{\beta+1}} f(z)- z^{\beta+\delta+1},
\quad z>0, \label{H-Newton}
\end{equation}
in which $f$ is the function defined by \eqref{f-teo}.

We are going to apply Newton's method  with the iteration formula
\begin{equation}
z_k=z_{k-1}-\frac{H(z_{k-1})}{H'(z_{k-1})}.
\end{equation}
 to solve \eqref{H-Newton}.
For the computational examples we consider  the corresponding thermal
parameters for the water in liquid state i.e., $k=0.58$ $[W/(m ^{\circ}C)]$
and $a^2=1.39\times 10^{-7}$ $[m^2/s]$. Without loss of generality we
assume $\gamma=1$. Newton's Method will be implemented using Matlab software
to find the unique positive solution of equation \eqref{ecu-teo}.
It is worth pointing out that in this programming language, the Kummer
function $M(a,b,z)$ is represented by the `hypergeom' command.
The stopping criterion to be used here is the boundedness of the
absolute error $\vert z_k-z_{k-1}\vert<10^{-15}$.


Figure \ref{fig1} shows the variation of $\xi$ (solution of \eqref{H-Newton})
with respect to $\delta$, choosing different values for the coefficient
that characterizes the temperature at the fixed face ($T_0=1,5$ or $10$
$[^{\circ}C/s^{\alpha/2}]$) and fixing $\beta=0$.

Looking at Figure \ref{fig1}, it is obvious  that
solution of $\xi$ for the Classical Stefan problem is obtained for
$\delta=0$.
For $\delta=1$  the solution is given by \cite{VSP}, and
for any other $\delta \in \mathbb{R}_0^{+}$
the solution is given by \cite{ZWB} and \cite{ZhXi}.

\begin{figure}[htb]
 \centering
 \includegraphics[width=0.7\textwidth]{fig1} % Figure1.eps
\caption{Variation of $\xi$ with $\delta$, for $\beta = 0$ and $T_0 = 1, 5, 10$.}
\label{fig1}
\end{figure}

Figure \ref{fig2}, shows the variation of $\xi$ with respect to $\beta$
($-1\leq \beta\leq 1$), choosing different values for the coefficient
that characterizes the temperature at the fixed face ($T_0=1,5$ or
$10$ $[^{\circ}C/s^{\alpha/2}]$) and fixing $\delta=0$.
The case for $\beta=-1$ corresponds to the solution of the problem
analyzed in \cite{ZBL}.

\begin{figure}[htb]
\centering
 \includegraphics[width=0.7\textwidth]{fig2}
\caption{Variation of $\xi$ with $\beta$, for $\delta = 0$ and $T_0 = 1, 5, 10$.}
\label{fig2}
\end{figure}

The results obtained  indicate that $\xi$ increases with $\beta$ increasing
and $\delta=0$, and the same happens when $\beta=0$ is fixed and
$\delta$ varies between $-1$ and $1$.
Moreover, it can be assured that the greater the value of $T_0$, the
higher is the value obtained for $\xi$ (parameter that characterizes
the free boundary) implying that the fusion process occurs faster.

\section{Appendix}

We prove that the general solution of the
ordinary differential equation
\begin{equation}
\varphi''(\eta)+2\eta \varphi'(\eta)-2\alpha \varphi(\eta) =0. \label{A-Ecu}
\end{equation}
is
\begin{equation}\label{A-Sol}
\varphi(\eta)=c_1 M \big(-\frac{\alpha}{2},\frac{1}{2},-\eta^2\big)
+c_{2}\eta M\big(-\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},-\eta^2\big),
\end{equation}
regardless of $\alpha$ being an integer, or a non-integer non-negative number,
where $c_1$ and $c_2$ are arbitrary constants.


\subsection*{$\alpha$ non-negative, non-integer}

Introducing the new variable $w(\eta)=-\eta^2$ as in \cite{ZhXi}
and defining $g(w)=\varphi(\eta(w))$ we obtain that \eqref{A-Ecu}
is equivalent to the Kummer's differential equation
\begin{equation}
wg''(w)+g'(w)\big( \frac{1}{2}-w\big) +\frac{\alpha}{2}f(w)=0. \label{A-Ecu-2}
\end{equation}
whose general solution, according to \cite{OLBC}, is
\begin{equation}
g(w)=\widehat{c_{1}}M \big(-\frac{\alpha}{2},\frac{1}{2},w\big)
+\widehat{c_{2}}U\big(-\frac{\alpha}{2},\frac{1}{2},w\big),
\end{equation}
where $\widehat{c_1}$ and $\widehat{c_2}$ are arbitrary constants.
Because $U$ can be defined as
\begin{equation}
U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)
+\frac{\Gamma(b-1)}{\Gamma(a)} z^{1-b}M(a-b+1,2-b,z) \label{U}.
\end{equation}
we obtained that the general solution of \eqref{A-Ecu-2} is
\begin{equation}
g(w)=\overline{c_{1}}M \big(-\frac{\alpha}{2},\frac{1}{2},w\big)
+\overline{c_{2}}w^{1/2} M\big(-\frac{\alpha}{2}+\frac{1}{2},\frac{3}{2},w\big),
\end{equation}
where $\overline{c_{1}}$ and $\overline{c_{2}}$ are arbitrary constants,
arriving in this way to a $\varphi$ solution of \eqref{A-Ecu} defined
by \eqref{A-Sol}.

\subsection*{$\alpha=n$ non-negative integer}

According to \cite{ZWB}, the general solution of \eqref{A-Ecu} is
\begin{equation}
\varphi(\eta)=  \widehat{c_1} i^n \operatorname{erfc}(\eta)+\widehat{c_2}i^n \operatorname{erfc}(-\eta).
 \label{A-Sol-3}
\end{equation}
where $i^n \operatorname{erfc}(\cdot)$ is the family of the repeated integrals of the
 complementary error function.

Let $c_1$ and $c_2$ be arbitrary constants. Taking:
\begin{gather}
\widehat{c_1}=c_1 2^{n-1}\Gamma\big(\frac{n}{2}+1 \big)-c_2 2^{n-2}
\Gamma\big(\frac{n}{2}+\frac{1}{2} \big) \\
\widehat{c_2}=c_1 2^{n-1}\Gamma\big(\frac{n}{2}+1 \big)
+c_2 2^{n-2} \Gamma\big(\frac{n}{2}+\frac{1}{2} \big)
\end{gather}
in \eqref{A-Sol-3}  leads to
\begin{equation}
\begin{aligned}
\varphi(\eta)
&=\big[c_1 2^{n-1}\Gamma\big(\frac{n}{2}+1 \big)-c_2 2^{n-2}
 \Gamma\big(\frac{n}{2}+\frac{1}{2} \big) \big] i^n \operatorname{erfc}(\eta) \\
&\quad+  \big[c_1 2^{n-1}\Gamma\big(\frac{n}{2}+1 \big)
 + c_2 2^{n-2} \Gamma\big(\frac{n}{2}+\frac{1}{2} \big) \big] i^n \operatorname{erfc}(-\eta), \\
&= c_1 2^{n-1} \Gamma\big(\frac{n}{2}+1 \big)
 \big[ i^n \operatorname{erfc}(\eta)+i^n \operatorname{erfc}(-\eta) \big] \\
&\quad +c_2 2^{n-2} \Gamma\big(\frac{n}{2}+\frac{1}{2} \big)
\big[ i^n \operatorname{erfc}(-\eta)-i^n \operatorname{erfc}(\eta) \big], \\
&= c_1 M\big( -\frac{n}{2},\frac{1}{2},-\eta^2\big)
 +c_2\eta M\big(-\frac{n}{2}+\frac{1}{2}, \frac{3}{2},-\eta^2\big).
\end{aligned}
\end{equation}
arriving to a solution of \eqref{A-Ecu} given by a $\varphi$ defined
as \eqref{A-Sol}, using the properties stated in \eqref{Prop-3}-\eqref{Prop-4}.

\begin{center}
\begin{tabular}{|ll|}
\hline
\multicolumn{2}{|c|}{Nomenclature}\\
\hline
$a^2$  &  Diffusivity coefficient ($a^2=k/\rho c$),$[m^2/s]$\\
$c$	 &  Specific heat capacity, $[m^2/^{\circ}C s^2]$\\
$k$   & Thermal conductivity, $[W/(m ^{\circ}C)]$ \\
$s$   & Position of the free front, $[m]$\\
$t$ &  Time, $[s]$\\
$T$  & Temperature, $[^{\circ}C]$ \\
$T_0$  & Coefficient that characterizes the temperature at the \\
 &fixed face,  $[^{\circ}C/s^{\alpha/2}]$ \\
$x$ &  Spatial coordinate, $[m]$\\
$\alpha$ &  Power of the time that characterizes the temperature at the fixed  \\
& boundary, dimensionless\\
$\beta$ &  Power of the position that characterizes the latent heat per unit \\
 & of volume, dimensionless\\
$\delta$ &  Power of  the velocity that characterizes the latent heat per
 unit \\
 & of volume, dimensionless\\
$\gamma$ &  Coefficient that characterizes the latent heat per unit of volume, \\
 & $[s^{\delta-2} kg /(m^{\beta+\delta+1})]$\\
$\rho$ &  Density, $[kg/ m^3]$ \\
$\xi$  & Coefficient that characterizes the free interface, dimensionless\\
$\eta$ & Similarity variable in expression \eqref{Transform-1}, dimensionless.\\
\hline
\end{tabular}
\end{center}

\subsection*{Conclusions}
In this work we analyzed a Stefan problem with a latent heat that depends
on the position of the free boundary as well as on its rate of change.
An explicit solution has been found using the similarity technique and
the theory of Kummer functions. This exact solution gathers in one formula
the solutions obtained in the previous papers
\cite{VSP,ZBL,ZWB,ZhXi}, constituting a generalization of them.
The exact solution is worth finding it, since it can be used to
provide a benchmark for verifying the accuracy of numerical methods that
approximate the solution of Stefan problems.

We have also applied Newton's method to the problem
\eqref{EcCalor}-\eqref{FreeBound}  to estimate the parameter $\xi$
that characterizes the free front numerically.
The solutions given in the literature have also been recovered.
In addition it was observed that this parameter increases with respect
to the parameter $\beta$, fixing $\delta=0$ and vice versa.
 Also, it can be noted that if the coefficient that characterizes
the initial temperature $T_0$ becomes greater, $\xi$ also does,
meaning that the phase-change happens quicker, validating mathematically
what it seems obvious from the physical point of view.


\subsection*{Acknowledgements}

This work was  sponsored by projects PIP No 0275
from CONICET-UA, and PICTO Austral 2016 No 0090, Rosario, Argentina.
This work was also supported by grant AFOSR-SOARD FA9550-14-1-0122.


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\end{document}