\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 147, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2010/147\hfil Fractional diffusion problems]
{Solutions of fractional diffusion problems}
\author[R. W. Ibrahim\hfil EJDE-2010/147\hfilneg]
{Rabha W. Ibrahim}
\address{Rabha W. Ibrahim \newline
School of Mathematical Sciences \\
Faculty of Sciences and Technology\\
UKM, Malaysia}
\email{rabhaibrahim@yahoo.com}
\thanks{Submitted April 20, 2010. Published October 18, 2010.}
\subjclass[2000]{34A12}
\keywords{Fractional calculus; majorant function; diffusion problems;
\hfill\break\indent holomorphic solution; Riemann-Liouville operators}
\begin{abstract}
Using the concept of majorant functions, we prove the existence
and uniqueness of holomorphic solutions to nonlinear
fractional diffusion problems. The analytic continuation
of these solutions is studied and the singularity for two
cases are posed.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\section{Introduction}
Fractional calculus and its applications (that is the theory
of derivatives and integrals of any arbitrary real or complex order)
has importance in several
widely diverse areas of mathematical, physical and engineering
sciences. It generalized the ideas of integer order differentiation
and n-fold integration. Fractional derivatives introduce an
excellent instrument for the description of general properties of
various materials and processes. This is the main advantage of
fractional derivatives in comparison with classical integer-order
models, in which such effects are in fact neglected. Also the
advantages of fractional derivatives become apparent in modelling
mechanical and electrical properties of real materials, as well as
in the description of properties of gases, liquids and rocks, and in
many other fields.
The class of fractional differential equations of
various types plays important roles and tools not only in
mathematics but also in physics, control systems, dynamical systems
and engineering to create the mathematical modelling of many physical
phenomena. Naturally, such equations required to be solved. Many
studies on fractional calculus and fractional differential
equations, involving different operators such as Riemann-Liouville
operators \cite{r4}, Erd\'elyi-Kober operators \cite{r6}, Weyl-Riesz
operators \cite{r18}, Caputo operators \cite{r2} and
Gr\"{u}nwald-Letnikov operators \cite{r20}, have appeared during the
past three decades. The existence of positive solution and
multi-positive solutions for nonlinear fractional differential
equation are established and studied \cite{r23}. Moreover, by using
the concepts of the subordination and superordination of analytic
functions, the existence of analytic solutions for fractional
differential equations in complex domain are suggested and posed in
\cite{r7,r8}.
The mathematical study of fractional diffusion equations began
with the work of Kochubei \cite{r11,r12}. Later this study followed
by the work of Metzler and Klafter \cite{r16} and Zaslavsky
\cite{r22}. Recently, Mainardi et all obtained the time fractional
diffusion equation from the standard diffusion equation \cite{r14}.
Our aim in this paper is to consider the existence and uniqueness of
nonlinear diffusion problems of fractional order in the complex
domain by employing the concept of the majorant functions. The
problems are taken in sense of Riemann-Liouville operators. Also,
the analytic continuation of solutions are studied. Finally, the
singularity for two cases are posed. In the fractional diffusion
problems, we replace the first order time derivative by a fractional
derivative. Fractional diffusion problems are useful in physics
\cite{r5}.
\section{Preliminaries}
One of the most frequently used tools in the
theory of fractional calculus is furnished by the Riemann-Liouville
operators (see \cite{r9,r10,r17,r19,r20,r21}).
\begin{definition} \label{def2.1} \rm
The fractional (arbitrary)
order integral of the function $f$ of order $\alpha >0$ is
defined by
\[
I^{\alpha}_{a}f(t)=\int^{t}_{a}
\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}f(\tau)d\tau.
\]
When $a=0$, we write $I^{\alpha}_{a}f(t)=f(t)*\phi_{\alpha}(t)$,
where $(*)$ denotes the convolution product (see \cite{r19}),
$\phi_{\alpha}(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}, \, t>0$ and
$\phi_{\alpha}(t)=0, \,\,t\leq 0$ and $\phi_{\alpha}\rightarrow
\delta(t)$ as $\alpha \rightarrow 0$ where $\delta(t)$ is the delta
function.
\end{definition}
\begin{definition} \label{def2.2} \rm
The fractional (arbitrary)
order derivative of the function $f$ of order $0 \leq \alpha <1$ is
defined by
\[
D^{\alpha}_{a}f(t)=\frac{d}{dt}\int^{t}_{a}\frac{(t-\tau)^{-\alpha}}
{\Gamma(1-\alpha)}f(\tau)d\tau=\frac{d}{dt}I^{1-\alpha}_{a}f(t).
\]
\end{definition}
\begin{definition} \label{def2.3}\rm
The majorant relations described as: if $a(x) = \sum a_{i}x^{i}$ and
$A(x) = \sum A_{i}x^{i}, A_{i} \geq 0, \, \forall \, i$, then we say
that $a(x) \ll A(x)$ if and only if $|a_{i}| \leq A_{i}$ for each
$i$. Likewise, if $g(t, x) = \sum g_{ik}(t-\varepsilon)^{i}x^{k}$
and $G(t, x) = \sum G_{ik}(t - \varepsilon)^{i}x^{k}$, then we say
that $g(t, x) \ll_{\varepsilon}G(t, x)$ if and only if $|g_{ik}|
\leq G_{ik} $ for all $i$ and $k$.
\end{definition}
Now we
define the following family of majorant functions: for each $i \in
\mathbb{N}$, we set
\begin{equation}\label{e2.1}
\Phi^{(i)}(z)=\sum^{\infty}_{n=0}\frac{z^{n}}{(n+1)^{i+2}},
\quad (|z|<1).
\end{equation}
Note that for each $i \in \mathbb{N}$, the family $\Phi^{(i)}$
converges for all $|z| < 1$. Moreover, this family of functions
enjoys some interesting majorant relations, as is stated in the
following proposition.
\begin{proposition} \label{prop2.4} \rm
The following relations hold.
\begin{itemize}
\item[(i)] $ \Phi^{(0)}(z) \Phi^{(0)}(z) \ll \Phi^{(0)}(z)$;
\item[(ii)] $ \Phi^{(0)}(z)\gg \Phi^{(1)}(z)\gg \Phi^{(2)}(z)\gg
\dots$;
\item[(iii)] $ \frac{1}{2^{i+2}}\Phi^{(i-1)}(z) \ll \frac{d}{dz}
\Phi^{(i)}(z)\ll \Phi^{(i-1)}(z)$,\\
$\frac{1}{2^{i+2}}\Phi^{(i-2)}(z)\ll
\frac{d^{2}}{dz^{2}} \Phi^{(i)}(z)\ll \Phi^{(i-2)}(z), \dots$;
\item[(iv)] $\Phi^{(i)}(z)\Phi^{(i)}(z) \ll \Phi^{(i)}(z)$;
\item[(v)] $\frac{1}{1- \varepsilon z}\Phi^{(i)}(z)\ll
C_{i,\varepsilon} \Phi^{(i)}(z)$, ($0 < \varepsilon <1$,
$C_{i,\varepsilon} >0$);
\item[(vi)] $\frac{\Phi^{(i-2)}(z)}{(2 \mu)^{3(i+2)}} \ll
D^{\alpha}_{a}\Phi^{(i)}(z)$,
\end{itemize}
for sufficient large $\mu \geq 1$.
\end{proposition}
\begin{proof}
The first two relations are verified
using the definition of $\Phi^{(i)}(z)$. Since
\[
\frac{n+1}{(n+2)^{i+2}} < \frac{n+2}{(n+2)^{i+2}}
= \frac{1}{(n+2)^{i+1}}
\leq \frac{1}{(n+1)^{i+1}}
\]
then we obtain (iii). Similarly for (iv).
To prove (v), by arbitrary choice of $\varepsilon$ we consider
\[
\varepsilon^{n} \leq \frac{C_{i,\varepsilon}}{(n+1)^{i+2}}
\]
which implies
\[
\frac{1}{1- \varepsilon z}= \sum^{\infty}_{n=0}\varepsilon^{n}
z^{n} \ll C_{i,\varepsilon} \Phi^{(i)}(z).
\]
But this is equivalent to saying that for all $n$
\[
\frac{1}{1- \varepsilon z}\Phi^{(i)}(z)\ll
C_{i,\varepsilon} \Phi^{(i)}(z).
\]
Finally, by using the approximation
\[
D^{\alpha}_{a}\Phi^{(i)}(t)=
\sum^{\infty}_{n=0}\frac{\Gamma(n+1)}
{\Gamma(n+1-\alpha)(n+1)^{i+2}}t^{n-\alpha}, \quad
( 0 < t < 1)
\]
we obtain the desired relation (vi)
for sufficient large $\mu \geq 1$.
\end{proof}
Similarly we can verify the following property.
\begin{proposition} \label{prop2.5}
If $f(z)$ is holomorphic in a neighborhood of $|z|\leq r_{0}$,
then $f(z)$ is majorized by
\[
f(z) \ll \frac{M}{1-\frac{z}{r_{0}}} \ll \frac{M}{1-\frac{\varepsilon z}{r} }\times
\Phi^{(i)}(\frac{z}{r}) \ll MC_{i,\varepsilon}
\Phi^{(i)}(\frac{z}{r}),
\]
for any $0< r< \varepsilon r_{0}$.
\end{proposition}
\section{Fractional Diffusion Problems}
Let $F(t, z, u, v)$,
$t \in J=[a,A]$ be a function which is
holomorphic in a neighborhood of the point $(a, b, c, d) \in J
\times \mathbb{ C}^{3}$, and let $\varphi(z)$ be a function which is
holomorphic in a neighborhood of $z = b$ and satisfies
$\varphi(b) = c$ and
$ \frac{\partial^{2} \varphi}{\partial z^{2}} (b) = d$.
Consider the initial value problem:
\begin{equation} \label{e3.1}
\begin{gathered}
\frac{\partial ^{\alpha}u(t,z)}{\partial t^{\alpha}}=
F(t,z,u,\frac{\partial^{2} u}{\partial z^{2}}), \\
u(a,z)=\varphi(z), \quad \text{in a neighborhood of } z = b. \\
\end{gathered}
\end{equation}
For $0 < \alpha<1$, problem \eqref{e3.1} is called sub-diffusion
problem \cite{r15} and for $1 < \alpha<2$, it is called intermediate
processes (see \cite{r3,r13}).
Then we have the following unique solvability result.
\begin{theorem}\label{thm3.1}
Initial value problem \eqref{e3.1} has one and only
one solution $u(t, z)$ which is holomorphic in a neighborhood
of $(a, b) \in J \times \mathbb{C}$.
\end{theorem}
\begin{proof}
Translating the setting from the point $(a, b)$ into the origin
$(0,0)$. Also we perform a change of variable by setting
$w(t, z) = u(t,z)- \varphi(z)$, where $w(t, z)$ is the new
unknown function. Then
the initial value problem \eqref{e3.1} is equivalent to the problem
\begin{equation} \label{e3.2}
\begin{gathered}
\frac{\partial ^{\alpha}w(t,z)}{\partial t^{\alpha}}=
G(t,z,w,\frac{\partial^{2} w}{\partial z^{2}}), \\
w(0,0)=0, \quad \text{in a neighborhood of } z = 0.
\end{gathered}
\end{equation}
Here, the function $G(t, z,w, \frac{\partial^{2} w}{\partial z^{2}})$
is holomorphic in a neighborhood of the origin
$ \in I \times \mathbb{C}^{3}, \,\, t \in I=[0,1]$ and $G(0,z,0,0)\equiv 0$ near $z=0$. Thus it is
sufficient to consider the reduced initial value problem
\eqref{e3.2}. Equation 3.2 has a unique solution takes the form
\[
w(t,z)= \sum^{\infty}_{k=0}w_{k}(z)t^{k}, \quad (t \in I).
\]
The uniqueness comes from the idea given in \cite{r1}; Solutions to
fractional Cauchy problems are obtained by subordinating the
solution to the original Cauchy problem.
We proceed to prove that $w(t,z)$ converges.
Let $r_{0} > 0 $ and $\rho> 0$ be small enough and suppose
that the function $G(t, z,w, v)$ is holomorphic in a neighborhood of
the set $\{(t, z,w, v) \in I \times \mathbb{C}^{3}; t <\tau \leq 1,
|z| \leq r_{0}, |w| \leq \rho $ and $|v| \leq \rho \}$. Suppose
further that $G$ is bounded by $M$ in this domain. Since $G$ is
holomorphic, we may expand it into
\[
G(t, z,w, v)= \sum _{p,q,s}a_{p,q,s}(z) t^{p} w^{q} v^{s},
\quad
\Big(t \in I, \,\, (w, v) \in (\mathbb{C}\times \mathbb{C} )\Big).
\]
By Cauchy's inequality and the fact that the coefficient
$a_{p,q,s}(z)$ is holomorphic in a neighborhood of
$\{z \in\mathbb{C};0< |z| \leq r_{0}\}$, we have
\begin{equation}\label{e3.3}
a_{p,q,s}(z)\ll
\frac{M}{\tau^{p}\rho^{q+s}}\frac{1}{1-(\frac{z}{r_{0}})^{2}}.
\end{equation}
Now the problem returns to find a function $g(t,z)$ satisfying the
majorant relations
\begin{equation} \label{e3.4}
\begin{gathered}
\frac{\partial ^{\alpha}g(t,z)}{\partial t^{\alpha}}\gg
\sum _{p,q,s}\frac{M}{\tau^{p}\rho^{q+s}}
\frac{1}{1-(\frac{z}{r_{0}})^{2}}t^{p}g^{q}
(\frac{\partial^{2} g}{\partial z^{2}})^{s},
\quad t \in I \\
g(0,0) \gg 0,
\end{gathered}
\end{equation}
then the function $g(t, z)$ majorizes the formal solution $w(t, z)$.
Assume $0< r< r_{0} $ and define
\begin{equation}\label{e3.5}
g(t,z)= L \Phi^{(2)}\Big( t+ (\frac{z}{r})^{2}\Big), \quad (L>0).
\end{equation}
By taking the fractional derivative for both sides of \eqref{e3.5}
with respect to $t$ we obtain
\begin{equation}\label{e3.6}
\frac{\partial^{\alpha} g(t,z)}{\partial t^{\alpha}}
= L \frac{\partial^{\alpha} \Phi^{(2)}\big( t+
(\frac{z}{r})^{2}\big)}{\partial t}, \quad (L>0 ).
\end{equation}
Then by Proposition \ref{prop2.4} (vi) we have
\begin{equation}\label{e3.7}
\frac{\partial^{\alpha} g(t,z)}{\partial t^{\alpha}} \gg \frac{
L}{C_{\mu}} \Phi^{(0)}\Big( t+ (\frac{z}{r})^{2}\Big),
\end{equation}
where $C_{\mu}:=(2\mu)^{12}$.
For a constant $K_{0}>0$ again in view of Proposition \ref{prop2.4} (ii) and
(iii) we realize that
\begin{equation}\label{e3.8}
\begin{aligned}
&\sum _{p,q,s}\frac{M}{\rho^{q+s}}
\frac{1}{1-(\frac{z}{r_{0}})^{2}}\frac{1}{1-\frac{t}{\tau}}
g^{q}(\frac{\partial^{2} g}{\partial z^{2}})^{s}\\
& \ll \sum _{p,q,s}\frac{M}{\rho^{q+s}}
\frac{1}{1-(\frac{z}{r_{0}})^{2}-\frac{t}{\tau}}\{L \Phi^{(2)}\Big( t+
(\frac{z}{r})^{2}\Big)\}^{q}
\{\frac{2L}{r^{2}} \Phi^{(0)}\Big( t+
(\frac{z}{r})^{2}\Big)\}^{s}\\
& \ll \sum _{p,q,s}\frac{M}{\rho^{q+s}}
\frac{1}{1-(\frac{z}{r_{0}})^{2}-\frac{t}{\tau}}\{L \Phi^{(0)}
\Big( t+ (\frac{z}{r})^{2}\Big)\}^{q}
\{\frac{2L}{r^{2}}\Phi^{(0)}\Big( t+
(\frac{z}{r})^{2}\Big)\}^{s}\\
&\ll \frac{MK_{0}}{1 - L/\rho- 2L/\rho r^{2}}\Phi^{(0)}\Big( t+
(\frac{z}{r})^{2}\Big)
\end{aligned}
\end{equation}
if $\frac{L}{\rho}+\frac{2L}{\rho r^{2}} <1$.
Comparing \eqref{e3.7} and \eqref{e3.8}, if the relation
\begin{equation}\label{e3.9}
\frac{ L}{C_{\mu}} \geq \frac{MK_{0}}{1 - L/\rho- 2L/\rho r^{2}}
\end{equation}
holds then the majorant relations in \eqref{e3.4} will be satisfied
by $g(t,z)$ defined in \eqref{e3.5}. Note that relation
\eqref{e3.9} holds by choosing a
sufficiently small $L$, the condition $\frac{L}{\rho}+\frac{2L}{\rho
r^{2}} <1$ is satisfied. Hence $g(t,z)$ in \eqref{e3.5} majorizes
the formal solution $w(t, z)$. This now implies that $w(t,z)$
converges in a domain containing $\{ (t,z) \in I \times \mathbb{C};
|t+ (\frac{z}{r})^{2}|< 1 \}$.
\end{proof}
\section{Analytic Continuation of solution}
Let $\Omega$ be a neighborhood of the origin
$(0,0)$. Let $F(t, z, u, v)$, $t \in I$, be a holomorphic
function in $\Omega \times \mathbb{C}_{u} \times \mathbb{C}_{v}$ and
consider the nonlinear fractional partial differential equation
\begin{equation}\label{e4.1}
\frac{\partial^{\alpha} u}{\partial t} =F(t,z,u,\frac{\partial^{2}
u}{\partial z^{2}}).
\end{equation}
Then we may expand it into the convergent series
\begin{equation}\label{e4.2}
F(t,z,u,v)=\sum_{j,p} a_{j,p}(t,z) u^{j}v^{p}.
\end{equation}
Let $S_{0}=\{(j, p) \in \mathbb{N}^{2}; a_{j,p} (t,z) \neq
0\}$ and $S=\{(j, p) \in S_{0}; j+p \geq 2\}$. Note that $F$ is
linear if and only if $S=\emptyset; $ it is nonlinear otherwise.
Assume henceforth that $F$ is nonlinear, that is $S $ is nonempty.
In the following, we will write the coefficients as
\begin{equation}\label{e4.3}
a_{j,p}(t,z)= t^{k_{j,p}}b_{j,p}(t,z),
\end{equation}
where $k_{j,p}$ is a nonnegative integer and $b_{j,p}(0, z) = 0$.
Using
\eqref{e4.1} may now be written as
\begin{equation}\label{e4.4}
\frac{\partial^{\alpha} u}{\partial t^{\alpha}} =\sum_{j,p}
t^{k_{j,p}}b_{j,p}(t,z)u^{j}(\frac{\partial^{2} u}{\partial
z^{2}})^{p}.
\end{equation}
For $\kappa \in \mathbb{R}$ we define the quantity
\begin{equation}\label{e4.5}
\delta(\kappa) := \inf_{j,p \in S} \Big(k_{(j,p) }+1+\kappa(j+p-1)
\Big).
\end{equation}
Note that when $\kappa=0$, then $\delta(\kappa) \geq 1$.
Moreover, if
\[
\kappa > \sup _{(j,p)\in S}\frac{-(k_{(j,p) }+1)}{j+p-1},
\]
then $\delta(\kappa)$ is positive.
In this section, our aim is to show that any solution $u(t,
z)= O(t^{\kappa})$ of the problem \eqref{e4.4} which is holomorphic
in $\Omega$, is analytically continued up to some neighborhood of
the origin.
\begin{theorem}\label{thm4.1}
Suppose $u(t, z)$ is a solution of \eqref{e4.4} which is holomorphic
in $\Omega$. If for some $\kappa \in \mathbb{R}$ satisfying
$\delta(\kappa) >0$, we have
\[
\sup_{z \in \mathbb{C}} |u(t,z)|= O(t^{\kappa}), \quad
(t \rightarrow 0),
\]
then the solution $u(t, z)$ can be extended analytically as a
holomorphic solution of \eqref{e4.4} up to a neighborhood of the
origin.
\end{theorem}
\begin{proof} Assume that $u(t,z)$ is a
solution for the problem \eqref{e4.4} which is holomorphic in
$\Omega$. Furthermore, we suppose that expansion \eqref{e4.2} is
valid in the domain $D$ where
\[
D:= \{(t,z,u,v): t \leq 2\tau , \, |z|\leq 2r, \, |u|
\leq \rho, |v| \leq \rho \},
\]
such that $\tau <1$, $2r<1$ and $\rho$ is positive number.
Let $M$ be a bound of $F$ in $D$.
Now we consider the following initial value problem in
$w(t, z):= \sum^{\infty}_{k=0} w_{k}(z) (t-\varepsilon)^{k}$
\begin{equation} \label{e4.6}
\begin{gathered}
\frac{\partial ^{\alpha}w(t,z)}{\partial t^{\alpha}}=
\sum_{j,p}
t^{k_{j,p}}b_{j,p}(t,z)w^{j}(\frac{\partial^{2} w}{\partial
z^{2}})^{p},
\quad t \in I \\
w(\varepsilon,z)=u(\varepsilon,z).
\end{gathered}
\end{equation}
Our aim is to show that the formal solution $w(t, z)$ converges in
some domain containing the origin. This then poses that $u(t,z )$ is
analytically continued by $w(t, z)$ up to some neighborhood of the
origin. First, since $u(t,z ) = O(t^{\kappa}) $ as $t \rightarrow
0$, there exists a constant $N$ such that $|u(\varepsilon,z )| \leq
N \varepsilon^{\kappa}$ uniformly in $z$. Hence in view of
Proposition \ref{prop2.5}, for some constant $C_{1}$, we have
\begin{equation}\label{e4.7}
u(\varepsilon,z) \ll N \varepsilon^{\kappa} C_{1}
\Phi^{(2)}(\frac{t-\varepsilon}{c\tau}+(\frac{z}{r})^{2}).
\end{equation}
Assume that $\kappa <1$ (without lose generality).
To construct an inequality to be satisfied by the majorant function,
we will first majorize the expression $t^{k_{j,p}}b_{j,p}(t,z)$ by
using $\Phi^{(0)}(z)$. Let
\[
Z:= \frac{t-\varepsilon}{c\tau}+(\frac{z}{r})^{2}
\]
then $t$ is majorized by
\begin{equation}\label{e4.8}
t=\varepsilon+(t-\varepsilon)\ll_{\varepsilon} \Big( \varepsilon+
4c\tau \Big)\Big(
1+\frac{t-\varepsilon}{4c\tau} \Big)
\ll_{\varepsilon}\Big( \varepsilon+ 4c\tau \Big)\Phi^{(0)}(Z).
\end{equation}
Now, we may expand the function
$b_{j,p}(t,z)$ as follows
\[
b_{j,p}(t,z)= \sum^{\infty}_{m=0} b^{(m)}_{j,p}(z) t^{m},
\]
where each $b^{(m)}_{j,p}$ is holomorphic in a neighborhood of
$\{|z| \leq 2r\} $ and satisfies
\[
|b^{(m)}_{j,p}(z)| \leq \frac{M}{\rho^{j+p}(2\tau)^{m+k_{j,p}}}.
\]
This estimate implies
\begin{equation} \label{e4.9}
b^{(m)}_{j,p}(z) \ll \frac{MC_{1}
\Phi^{(0)}(Z)}{\rho^{j+p}(2\tau)^{m+k_{j,p}}}
\end{equation}
where $C_{1}$ is the same constant as in \eqref{e4.7}.
Combining relations \eqref{e4.8} and \eqref{e4.9} and by using
Proposition \ref{prop2.4} (i), we obtain
\begin{equation}\label{e4.10}
\begin{split}
t^{k_{j,p}}b_{j,p}(t,z)
& \ll_{\varepsilon}\sum^{\infty}_{m=0}\Big[
(\varepsilon+ 4c\tau )\Phi^{(0)}(Z) \Big]^{m+k_{j,p}}\Big[
\frac{MC_{1}
\Phi^{(0)}(Z)}{\rho^{j+p}(2\tau)^{m+k_{j,p}}}\Big]\\
& \ll _{\varepsilon}
\frac{C_{1}M}{\rho^{j+p}}\Phi^{(0)}(Z)\sum^{\infty}_{m=0} (
4c)^{m+k_{j,p}},
\end{split}
\end{equation}
by choosing $\varepsilon= \frac{c\tau}{2}$ and $0 0$ and $c> 0$, and setting
$\varepsilon =\frac{c\tau}{2}$, the function
\[
W(t,z)= \varepsilon^{\kappa} NC_{1} \Phi^{(2)}(Z)
\]
satisfies the majorant relations given in
\eqref{e4.11}. Hence $W(t, z)$ is holomorphic in a domain containing
the origin; consequently must be true for $w(t, z)$.
\end{proof}
\section{Singularity}
Again, we state our assumptions in studying the equation
\begin{equation}\label{e5.1}
t\frac{\partial ^{\alpha}w(t,z)}{\partial t^{\alpha}}=
G(t,z,w,\frac{\partial ^{2}w}{\partial z^{2}}).
\end{equation}
Let $G$ be a function holomorphic in some neighborhood of the origin
in $I \times \mathbb{C}^{3}$ and suppose $G(0, z, 0, 0)$ is
identically zero near $z = 0$.
In this section, our aim is to seek
a holomorphic solution $w(t, z)$ of \eqref{e5.1} for special case, which satisfies
$w(0, z)\equiv 0$. First we write $G$ as
\begin{equation}\label{e5.2}
G(t,z,w,\frac{\partial^{2} w}{\partial z^{2}})
= a(z)t+b(z)w+ c(z)\frac{\partial^{2} w}{\partial z^{2}}
+R(t,z,w,\frac{\partial ^{2}w}{\partial z^{2}}),
\end{equation}
where $R$ is the remainder of the Taylor expansion of $G$.
Then we have the following result.
\begin{theorem}\label{thm5.1}
Assume that the coefficient $c(z)\equiv 0$ in \eqref{e5.2}.
If $ b(z)$ does not take values in $\mathbb{N}\cup \{0\}$ at
the origin, then \eqref{e5.1}
has a unique holomorphic solution satisfying
$w(0, z)\equiv 0$.
\end{theorem}
\begin{proof}
Consider a formal solution of the
form $w(t,z) = \sum^{\infty}_{k=0} w_{k}(z) t^{k}$. Then we expand
$G(t,z,w,\frac{\partial ^{2}w}{\partial z^{2}})$ as follows
\[
G(t,z,w,\frac{\partial ^{2}w}{\partial z^{2}})=a(z)t+b(z)
w+ \sum_{p+q+j \geq 2} a_{p,q,j}(z)t^{p} w^{q} (\frac{\partial^{2}
w}{\partial z^{2}})^{j}.
\]
Suppose that this expansion is convergent
in a neighborhood of the set
\[
\mathcal{S}:=\{(t,z,w, \frac{\partial ^{2}w}{\partial z^{2}}):
t \leq \tau, |z| \leq r_{0}, |w|\leq \rho,
|\frac{\partial ^{2}w}{\partial z^{2}}|\leq \rho \}
\]
and that $G$ is bounded in $\mathcal{S}$ by $M$. Since $ b(z)$
does not take values in $\mathbb{N}\cup \{0\}$ at the origin
then there exists a constant $B$ such that
\[
|\frac{k}{k-b(z)}| \leq B, \quad for \, all\,\,
k \in \mathbb{N}\cup \{0\} \,\, and\,\, |z|\leq r_{0}.
\]
Then any function $\omega(t,z)$ satisfying the following relations
is a majorant of the formal solution:
\begin{equation} \label{e5.3}
\begin{gathered}
t\frac{\partial ^{\alpha}\omega(t,z)}{\partial t^{\alpha}} \gg
\frac{BM}{1-\frac{z}{r_{0}}}\frac{t}{\tau}+
\sum_{p+q+j\geq 2}\frac{BM}{\tau^{p}\rho^{q+j}}
\frac{t^{p}\omega^{q}}{1-\frac{z}{r_{0}}}(\frac{\partial^{2}
\omega}{\partial z^{2}})^{j}
\\
\omega|_{t=0}=0.
\end{gathered}
\end{equation}
Let $0< r< r_{0}$. Assume that
\begin{equation}\label{e5.4}
\omega(t,z)= Lt \Phi^{(2)}\Big( t+ (\frac{z}{r})^{2}\Big),
\quad (L>0),
\end{equation}
yields
\[ \frac{\partial^{\alpha}
\omega(t,z)}{\partial t^{\alpha}}= L \frac{\partial^{\alpha}t
\Phi^{(2)}\Big( t+ (\frac{z}{r})^{2}\Big)}{\partial t^{\alpha}},
\quad (L>0).
\]
Applying the Leibniz rule for fractional
differentiation \cite[Eq. 2.202]{r20} and using the relation
(vi), the left hand side of \eqref{e5.3} becomes
\begin{equation}\label{e5.5}
t \frac{\partial^{\alpha}
\omega(t,z)}{\partial t^{\alpha}}
\gg \binom{\alpha}{0} L t^{2}\frac{\Phi^{(0)}(t)}{C_{\mu}}+
\binom{\alpha}{1} L t \frac{\Phi^{(0)}(t)}{C_{\mu}}
\gg C_{\alpha,\mu}L \Phi^{(0)}(t)(t^{2}+t) .
\end{equation}
Meanwhile, the right hand side becomes
\begin{equation}\label{e5.6}
\begin{split}
&\frac{BM}{1-(\frac{z}{r_{0}})^{2}}\frac{t}{\tau}+
\sum_{p+q+j\geq 2}\frac{BM}{\tau^{p}\rho^{q+j}}
\frac{t^{p}\omega^{q}}{1-(\frac{z}{r_{0}})^{2}}(\frac{\partial^{2}
\omega}{\partial z^{2}})^{j}\\
&\ll \frac{BM}{1-(\frac{z}{r_{0}})^{2}}\frac{t}{\tau}\\
&\quad + \sum_{p+q+j\geq 2}\frac{BM}{1-(\frac{z}{r_{0}})^{2}}
(\frac{t}{\tau})^{p}\Big[ \frac{Lt \Phi^{(2)}
\big( t+ (\frac{z}{r})^{2}\big)}{\rho}\Big]^{q}
\Big[ \frac{2Lt}{r^{2} \rho} \frac{d^{2} \Phi^{(2)}
\big(t+ (\frac{z}{r})^{2}\big)}{dz^{2}}\Big]^{j}\\
& \ll \sum^{\infty}_{p=1}\frac{BM}{1-(\frac{z}{r_{0}})^{2}}
(\frac{t}{\tau})^{p} + \sum_{(p+q+j\geq 2,q+j\geq
1)}\frac{BM}{1-(\frac{z}{r_{0}})^{2}}(\frac{t}{\tau})^{p}\Big[
\frac{Lt }{\rho}\Big]^{q}
\Big[ \frac{2Lt}{r^{2} \rho} \Big]^{j} \Phi^{(0)}\big( t+
(\frac{z}{r})^{2}\big).
\end{split}
\end{equation}
But
\begin{equation}\label{e5.7}
\sum^{\infty}_{p=1}\frac{BM}{1-(\frac{z}{r_{0}})^{2}}
(\frac{t}{\tau})^{p}
\ll \frac{BM}{1-(\frac{z}{r_{0}})^{2}
-\frac{t}{\tau}}(\frac{t}{\tau})\Phi^{(1)}(t)
\ll \frac{BMC_{1}}{\tau} t\Phi^{(0)}(t)
\end{equation}
and
\begin{equation}\label{e5.8}
\begin{split}
&\sum_{(p+q+j\geq 2,q+j\geq
1)}\frac{BM}{1-(\frac{z}{r_{0}})^{2}}(\frac{t}{\tau})^{p}\Big[
\frac{Lt
}{\rho}\Big]^{q}
\Big[ \frac{2Lt}{r^{2} \rho} \Big]^{j} \Phi^{(0)}
\big( t+ (\frac{z}{r})^{2}\big) \\
&\ll\Big( \frac{t}{\tau} +\frac{Lt}{\rho} +\frac{2Lt}{r^{2} \rho}
\Big)^{2} \frac{BM}{1-(\frac{z}{r_{0}})^{2}-\frac{t}{\tau}-\frac{Lt}
{\rho}-\frac{2Lt}{r^{2}\rho}}\Phi^{(0)}(t)\\
& \ll \Big( \frac{t}{\tau} +\frac{Lt}{\rho} +\frac{2Lt}{r^{2} \rho}
\Big)^{2} C_{2}BM \Phi^{(0)}(t),
\end{split}
\end{equation}
where
\[
\frac{1}{\tau}+\frac{L}
{\rho}+\frac{2L}{r^{2}\rho}\leq \frac{1}{\tau}.
\]
Comparing the majorant relations \eqref{e5.7} and \eqref{e5.8}
to the one in relation \eqref{e5.6} that $\omega(t,z)$ satisfies
\eqref{e5.3} if we could force
\[
L \geq \frac{BMC_{1}}{\tau C_{\alpha,\mu}}
\]
and
\[
L \geq \frac{\big( \frac{1}{\tau} +\frac{L}{\rho} +\frac{2L}{r^{2} \rho}
\big)^{2} C_{2}BM }{C_{\alpha,\mu}}.
\]
The last two conditions in
are satisfied by choosing $L$ large enough, fixing it, and then
choosing a sufficiently small value for $C_{1}, C_{2}$ and
$C_{\alpha,\mu}$. We thus have shown that the function $\omega(t,z)$
defined in \eqref{e5.4} majorizes the formal solution $w(t,z)$. This
implies that the formal solution converges in some neighborhood of
the origin.
\end{proof}
If $c(z)$ is not identically zero in \eqref{e5.2}, then we
write
\begin{equation}\label{e5.9}
c(z)=z^{p}\widetilde{c}(z),
\end{equation}
where $p$ is a nonnegative integer and $\widetilde{c}(z) \neq 0$. We
now state the following result.
\begin{theorem}\label{thm5.2}
Suppose $p=1$ in \eqref{e5.9}. If
a positive constant $\nu$ exists such that
\begin{equation}\label{e5.10}
|k- b(0)-\widetilde{c}(z)\ell| \geq \nu (k+\nu+1), \quad
(k,\ell)\in \mathbb{N}\cup \{0\} \times \mathbb{N},
\end{equation}
then
\eqref{e5.1} has one and only one holomorphic solution satisfying
$w(0, z)\equiv 0$.
\end{theorem}
\begin{proof}
Equation \eqref{e5.1} may be written as
\begin{equation}\label{e5.11}
\begin{split}
&t\frac{\partial ^{\alpha}w}{\partial t^{\alpha}}-b(0)w
-\widetilde{c}(0)(z\frac{\partial ^{2}w}{\partial z^{2}})\\
&=z \beta(z)w+z \gamma(z) (z\frac{\partial^{2} w}{\partial
z^{2}})+a(z)t+ \sum_{p+q+j \geq 2} a_{p,q,j}(z)t^{p} w^{q}
(\frac{\partial ^{2}w}{\partial z^{2}})^{j} ,
\end{split}
\end{equation}
where
\[
b(z) = b(0) + z \beta(z) \quad \text{and}\quad
\widetilde{c}(z) = \widetilde{c}(0) + z \gamma(z).
\]
Suppose that this expansion is convergent in a neighborhood of the
set
\[
\mathcal{S}:=\{(t,z,w, \frac{\partial ^{2}w}{\partial z^{2}}):
t \leq \tau, 0< r \leq|z| \leq r_{0}, |w|\leq \rho,
|\frac{\partial ^{2}w}{\partial z^{2}}| \leq \rho \}
\]
and that $G$ is bounded in $\mathcal{S}$ by $M$. Moreover, assume that
$a(z), \beta(z)$ and $\gamma(z)$ are bounded by $A, B$ and $C$
respectively. Consider a formal solution of the form $w(t,z) =
\sum^{\infty}_{k=0} w_{k}(z) t^{k}$. Now, it can be shown that this
formal solution is majorized by any function $W(t, z)$ satisfying
these relations:
\begin{equation} \label{e5.12}
\begin{gathered}
\begin{aligned}
&\nu \Big( t\frac{\partial^{\alpha}}{\partial t^{\alpha}}+ z
\frac{\partial^{2}}{\partial z^{2}}+1\Big)W(t,z)\\
& \gg \frac{BzW+At}{1-(\frac{z}{r_{0}})^{2}}
+\frac{Cz}{1-(\frac{z}{r_{0}})^{2}}(z\frac{\partial
^{2}W}{\partial z^{2}})
+ \sum_{p+q+j\geq 2}\frac{M}{\tau^{p}\rho^{q+j}}
\frac{t^{p}W^{q}}{1-(\frac{z}{r_{0}})^{2}}
(\frac{\partial^{2} W}{\partial z^{2}})^{j}
\end{aligned}\\
W|_{t=0}=0
\end{gathered}
\end{equation}
such that $W$ can be found in the form
\begin{equation}\label{e5.13}
W(t,z)= Lt \Phi^{(2)}\Big( t+ (\frac{z}{r})^{2}\Big),
\quad (L>0),
\end{equation}
The summation in \eqref{e5.12} is
estimated as in Theorem \ref{thm5.1}. Our aim is to estimate the terms
\[
\frac{BzW+At}{1-(\frac{z}{r_{0}})^{2}}
+\frac{Cz}{1-(\frac{z}{r_{0}})^{2}}(z\frac{\partial^{2}
W}{\partial z^{2}})\quad \text{and} \quad \nu
z(\frac{\partial^{2} W}{\partial z^{2}})
\]
in \eqref{e5.12}. We thus
have, for some constant $K> 0$,
\begin{equation}\label{e5.14}
\begin{split}
&\frac{BzW+At}{1-(\frac{z}{r_{0}})^{2}}
+\frac{Cz}{1-(\frac{z}{r_{0}})^{2}}(z\frac{\partial^{2}
W}{\partial z^{2}})\\
&\ll BLK zt \Phi^{(0)}\Big( t+
(\frac{z}{r})^{2}\Big) + CLK zt \frac{2z}{r^{2}}
\Phi_{\mu}^{(0)}\Big( t+(\frac{z}{r})^{2}\Big)\\
& \ll (BLK + 2CLK) zt \Phi^{(0)}\Big( t+ (\frac{z}{r})^{2}\Big).
\end{split}
\end{equation}
On the other hand, the left-hand side is estimated by using
Proposition \ref{prop2.4}, (iii) as follows:
\begin{equation}\label{e5.15}
\nu z(\frac{\partial^{2} W}{\partial z^{2}})\gg \frac{2\nu Ltz}{r}
\frac{d ^{2}\Phi^{(2)}\Big( t+( \frac{z}{r})^{2}\Big)}{dz^{2}}\gg
\frac{\nu Ltz}{8 r}\Phi^{(0)}\Big( t+ (\frac{z}{r})^{2}\Big).
\end{equation}
Therefore, in order for $W(t, z)$ to satisfy the majorant relations
in \eqref{e5.12} we must impose the condition
\[
K(B+2C) \leq \frac{\nu}{8r}.
\]
This completes the proof.
\end{proof}
\subsection*{Acknowledgments}
The author is thankful to the anonymous referee for his/her helpful
suggestions for the improvement of this article.
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\end{document}