\documentclass[a4paper,12pt]{book} \usepackage{amsmath, amsfonts, amssymb}

\documentclass[a4paper,12pt]{book}

\usepackage{amsmath, amsfonts, amssymb}
\usepackage[english, russian]{babel}
\usepackage{graphicx}
%\usepackage{pifont,setspace,hyperref,amsthm,amsmath,amssymb,color,multirow,graphicx,subfigure }
\usepackage{graphics}
\usepackage{graphicx}
\textheight = 230mm \textwidth = 160mm \topmargin = -2mm
\oddsidemargin=6mm \evensidemargin=6mm \makeindex
\begin{document}
%\textit{ÓÄÊ XXX.XX}
\begin{center}\textbf{ A non-local problem for a time fractional differential equation with the Hilfer operator on metric graph}
\medskip
% Àâòîðû
% Last names and initials (first and middle names) of the authors

\textbf{Ruzhansky M.$^{1}$, Sobirov Z. A.$^{2}$, Khujakulov J. R.$^{3}$}\\

%\medskip
% Affiliation and email
$^{1}$\textit{Ghent Universty,Ghent, Belgium;} \\
$^2$\textit{University of Geological Sciences, Tashkent, Uzbekistan;} \\
$^{3}$\textit{ V.I.Romanovskiy Institute of Mathematics, Tashkent, Uzbekistan.} \\
$^1$\texttt{ [e-mail]},\,\,\ $^2$\texttt{[e-mail]},\,\,\,\, $^3$\texttt{[e-mail]}
\smallskip
\end{center}
\begin{center}
\textbf{Abstract}
\end{center}
\begin{quote}
\emph{The main task of the present research is a non-local boundary-value problem for a time-fractional differential equation involving Hilfer fractional derivative on a metric star graph. The main goal of this work is to study the uniqueness and the existence of solution of the formulated problem. Using by the method of separation of variables we find a solution of the investigated problem in the form of the Fourier series. Sufficient classes of given functions, which provides the existence and uniqueness of solution of the considered problem is defined. Uniqueness of solution is proved, by using an a-priori estimates for the solution.}
\end{quote}
\textbf{Keywords.} Hilfer operator; metric graph; non-local problem; method of separation of variables; Mittag-Leffler function; a-priori estimation; fractional derivatives and integrals \\
MSC 2010: 35R11, 35B45
\begin{center}
\textbf{1 Introduction}
\end{center}

Fractional differential equations have become an important target for investigations because of their properties that are very

useful to describe memory phenomena in control theory viscoelasticity [1],
anomalous transport and anomalous diffusion [2], modeling physical and biological processes [3].\\
Detailed information of fractional differential equations(FDE) and their application in other fields can be found in I. Podlubniy [4] A. Kilbas, H. M Srivastava, J. J Trujillo [5], Samko Kilbas Marichov [6] and others.
The study of differential equations on metric graphs is one of the new directions of modern mathematics. It should be noted, that the growth at interest to the study of various problems on the metric graphs, is due to the wide range of practical problems of modern physics, biology and other sciences in branch structures (see [7]-[9]).
Initial boundary value problems (IBVP) for Partial differential equations(PDE) fractional and integer order on the metric graphs, were examined by number of authors (e.g., [10]-[14]).\\
A. V. Svetkova, A. I. Shafarevich examined for the wave equation in a homogeneous tree [15] and Yu. V. Pokorny, O. M. Penkin, and others studied differential equations in geometric graphs[16]. Using numerical methods, V. Mehandiratta, M. Mehra [17] studied the fractional differential equations of Caputo-type derivatives on a star metric graph. Direct and inverse problems for fractional differential equations were studied on the star metric graph ( see [12], [13]).\\

Initial value problems related to fractional order equations with the Hilfer derivative have been studied by several authors, e.g. [18-20], and see the references in them. In [21] the study, the time-fractional differential equations with the Hilfer's operator were first considered on metric graphs. It should be noted, that a generalization of the Riemann-Liouville and Caputo products is given by R. Hilfer in [22], which interpolates between the Riemann-Liouville and Caputo fractional derivatives. Some properties and applications of the Hilfer's operator are given in (see, [23],[24],[39]) and the references given in it.\\

This research work deals on the problem with non-local initial conditions for a time FDE with the Hilfer's operator on metric graphs. Notice that boundary- value problems with non-local initial conditions were considered in [25] for reaction-diffusion equations, in [26] for the heat equation, in [27], [28] for degenerate parabolic equations, and in [29] for a mixed parabolic equation, R. Ashurov and Y. Fayziev investigated Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations in [42]. \\
Moreover, non-local initial problem for second order time-fractional and space-singular equation, and non-local boundary value problem for a mixed-type equation involving the bi-ordinal Hilfer fractional differential operators. Were investigated by E. Karimov, M. Mamchuev and M. Ruzhansky[30], and E. Karimov, B Toshtemirov [31].

\begin{center}

\textbf{2. Preliminaries}
\end{center}
\textbf{ Fractional derivatives and integrals}\\
\textbf{Definition 1.} \,\ The Riemann-Liouville (R-L) fractional integrals with order $\alpha$ of a function $f(t)$ is defined by[5]
$$
\left(I_{a+}^\alpha f\right)(t) =\left(D_{a+}^{-\alpha} f\right)(t)= \frac{1}{\Gamma (\alpha )}\int\limits_a^t {\frac{f(\tau)d\tau}{(t- \tau)^{1-\alpha}}} , (x>a,\,\,\ \alpha >0).
$$
\textbf{Definition 2.} \,\ The Riemann-Liouville (R-L) fractional derivatives with order $\alpha$ of a function $f(t)$ is defined by [5]
$$
\left(D_{a+}^{\alpha}f\right)(t):=\left(\frac{d}{dt}\right)^{n}\left(I_{a+}^{n-\alpha}f\right)(t)=
$$
$$
=\frac{1}{\Gamma(n-\alpha)}\left(\frac{d}{dt}\right)^{n}\int\limits_{a}^{t}\frac{f(\tau)d\tau}{(t-\tau)^{\alpha-n+1}} \,\,\ (n=[\alpha]+1, n\in \mathbb{N} \,\ x>a)
$$
\textbf{Definition 3.} \,\ The Caputo fractional derivatives $\left(_CD_{ax}^\alpha f \right)(t)$ of order $(\alpha >0$), $\alpha \notin \mathbb{N} $ are defined by [5]
$$
\left( {_CD_{at}^\alpha f} \right)(t) := \frac{1}{{\Gamma (n - \alpha )}}\int\limits_a^t {\frac{{{f^{(n)}}(\tau)}}{{{{\left( {t - \tau} \right)}^{\alpha - n + 1}}}}d\tau}=:\left(I_{at}^{n-\alpha} \left(\frac{d}{dt}\right)^{n}f\right)(t), \,\,\, x>a,
$$
where $ n\in \mathbb{N}$ and $n = [\alpha ] + 1$.\\

\textbf{Definition 4.} [5]

We consider the weighted spaces of continuous functions
$$
C_{\gamma}[a,b]=\{f:(a,b]\rightarrow \mathbb{R}:\,(t-a)^{\gamma}f(t)\in C[a,b]\}, \,\,\ 0\leq \gamma<1,
$$
and
$$
C_{\gamma}^n[a,b]=\{f\in C^{n-1}[a,b]: f^{(n)}\in C_{\gamma}, n\in \mathbb{N},
$$
$$
C_{\gamma}^0[a,b]=C_{\gamma}[a,b],
$$
with the norms
$$
\|f\|_{C_{\gamma}}=\left\|(t-a)^{\gamma}f(t))\right\|_C
$$
and
$$
\|f\|_{C_{\gamma}^n}=\sum_{k=0}^{n-1}\|f^{(k)}\|_C+\|f^{(n)}\|_{C_{\gamma}}.
$$
These spaces satisfy the following properties.\\
a) $C_{0}^0[a,b]=C[a,b].$\\
b) $C_{\gamma}^n(a,b)\subset A C^n[a,b].$\\
c) $C_{\gamma_1}[a,b]\subset C_{\gamma_2}[a,b], \,\,\ 0\leq \gamma_1 <\gamma_2<1$.\\

\textbf{Lemma 1.(see[5])} If $\alpha>0$, $n=[\alpha]+1$, and $f(x)\in C_{\gamma}^{n}[a,b]$,$0\leq \gamma<1$, then the fractional derivative $D_{a+}^{\alpha}f$ in Definition 2 and Definition 3 exist on (a,b] and

$$
\left(D_{a+}^{\alpha}f\right)(t)=\sum\limits_{k = 0}^{n-1}\frac{f^{(k)}(a)}{\Gamma(1+k-\alpha)}(t-a)^{k-\alpha}+\left(_{C} D_{a+}^{\alpha}f\right)(t).\\
$$

\textbf{Definition 5.} Hilfer fractional derivative $D_{0+}^{\alpha,\mu}$ of order $\alpha$ and type $\mu$ with respect to $t$ is defined by[32]

$$
\left(D_{0+}^{\alpha,\mu}u \right)(t)=I_{0+}^{\mu(n-\alpha)} D^{(n)}\left(I_{0+}^{(1-\mu)(n-\alpha)}u\right)(t), n-1<\alpha$$
whenever the right-hand side exists.\\
The $D_{0+}^{\alpha,\mu}$ derivative is considered as an interpolation between the Riemann-Liouville and Caputo derivative:
$$
D_{0+}^{\alpha,\mu}=\left\{\begin{array}{c} {D_{0+}^{\alpha},\,\,\, \mu=0. } \\ {_CD_{0+}^{\alpha}, \,\,\,\,\ \mu=1.} \end{array}\right.
$$
\textbf{Lemma 2.(see[5] )} Let $\alpha>0$,\,\ $\beta>0$,\,\ $0\leq \gamma<1$. If $f(t)\in C_{\gamma}[a,b]$, then $\left(I_{a+}^{\alpha}I_{a+}^{\beta}f\right)(t)=\left(I_{a+}^{\alpha+\beta}f\right)(t)$ for $t\in(a,b]$. When $f(t)\in C[a,b]$, the equality holds at any point $t\in(a,b]$.

\textbf{Lemma 3.(see[5])} Let $\alpha>0$, $0\leq \gamma<1$. If $f(t)\in C_{\gamma}[a,b]$, then

$$
\left(D_{a+}^{\alpha}I_{a+}^{\alpha}f\right)(t)=f(t)
$$
for $t\in(a,b]$. When $f(t)\in C[a,b]$, the equality holds at any point $t\in(a,b]$.\\

\textbf{Lemma 4.(see[5] )} Let $\alpha>0$, $0\leq \gamma<1$, $n=[\alpha]+1$ and $f_{n-\alpha}(t)=\left(I_{a+}^{n-\alpha}f\right)(t).$ If $f(t)\in C_{\gamma}[a,b]$ and $f_{n-\alpha}(x)\in C_{\gamma}^{n}[a,b]$, then $$

\left(I_{a+}^{\alpha}D_{a+}^{\alpha}f\right)(t)=f(t)-\sum_{j=1}^{n}\frac{f_{n-\alpha}^{(n-j)}(a+)}{\Gamma(\alpha-j+1)}(t-a)^{\alpha-j}, t\in(a,b],
$$
where $f_{n-\alpha}^{(n-j)}(a+)=\lim\limits_{t \to a+}f_{n-\alpha}^{(n-j)}(t)$. If $f(t)\in C[a,b]$ and $f_{n-\alpha}(x)\in C^{n}[a,b]$ then the equality holds at any point $t\in(a,b]$.\\

\textbf{Definition 6.}

We introduce the inner scalar product and the norm on graph as follows:
$$
(f(x),g(x))_\mathbf{{\Gamma}} =\sum_{k} \int\limits_0^{L_{k}} f^{(k)}(x_{k})g^{(k)}(x_{k})dx_{k},
$$
$$
\|f(x)\|_\mathbf{\Gamma}=\sqrt{(f(x),f(x))\mathbf{_{\Gamma}}}.
$$
Here we understand the functions $f(x)$ and $g(x)$ in the following views:
$$
f(x)=\left(\begin{array}{c} {f^{(1)}(x_{1})} \\ {f^{(2)}(x_{2})} \\ {...} \\ {f^{(k)}(x_{k})}\end{array}\right), \,\, g(x)=\left(\begin{array}{c} {g^{(1)}(x_{1})} \\ {g^{(2)}(x_{2})} \\ {...} \\ {g^{(k)}(x_{k})}\end{array}\right)
$$

\textbf{ Mittag-Leffler function}\\

A two-parameter of the Mittag-Leffler function represented in the form[4],
$$
E_{\alpha ,\beta }^{}(z) = \sum\limits_{n = 0}^\infty {\frac{{{z^n}}}{{\Gamma (\alpha n + \beta )}}}, \alpha > 0, \,\ \beta \in \mathbb{C}, \,\ z \in \mathbb{C}.
$$
\textbf{Lemma 5.(see[4])} if $\alpha<2$, $\beta$ is an arbitrary real number, $\mu$ is such that $\pi\alpha/2<\mu$$
\left| {E_{\alpha ,\beta }^{}(z)} \right| \leqslant \frac{C_{1}}{{1 + \left| z \right|}},(\mu\leq\left| arg(z)\right|\leq\pi), \left|z\right|\geq0.
$$
For the Mittag-Leffler type functions takes place the following properties, at $\alpha, \beta, \gamma=const>0$:
$$
E_{\alpha ,\beta }^{}(z) = \frac{1}{{\Gamma (\beta )}} + zE_{\alpha ,\beta + \alpha }^{}(z),
$$
$$
\frac{1}{{\Gamma (\gamma )}}\int\limits_0^z {{{(z - t)}^{\gamma - 1}}E_{\alpha ,\beta }^{}(\lambda {t^\alpha }){t^{\beta - 1}}dt} = {z^{\beta + \gamma - 1}}E_{\alpha ,\beta + \gamma }^{}(\lambda {z^\alpha }),
$$
$$
\frac{d}{dz} {E_{\alpha
,1 }(\lambda z^{\alpha})}= \lambda z^{\alpha-1} E_{\alpha,\alpha}(\lambda z^{\alpha}),\,\,\,
$$
\textbf{3. Problem formulation}\\
Let us consider a graph $\mathbf{\Gamma}=V \bigcup E$ be a connected metric graph, where $E = \{B_k\}_{k=1}^m$ is the set of its edges, and
$V = \{\nu_k\}_{j=1}^n$ is the set of vertices ([36], [41]. Let us determine the coordinates $x_k$ on the edges of the graph using
isometric mapping of these edges onto the intervals $(0, L_k )$, $k = 1, 2, . . . , m.$

We will say that a vertex $\nu$ is in contact with an edge $B_k$ if it is the end of this edge, and denote this as $B_k \sim \nu$. The number of elements of the set $\{b : b \sim \nu, b \in E\}$ is called vertex valency $\nu$. If the valency of a vertex is equal to one, then it is called boundary. Let $ \{\gamma_1, \gamma_2,...,\gamma_{m_1}\}=\partial \Gamma\subset V $ be the boundary vertices of the graph. Further, without loss of generality, we will use $x$ instead of $x_k$ .

$$ Figure.1$$
On the each edges of the over defined graph, we consider fractional differential equations\\
\begin{equation}\label{eq1}
D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)-u_{xx}^{(k)} (x,t)=f^{(k)} (x,t), \ \ \ (x,t) \in \left(B_{k} \times (0,T) \right)
\end{equation}
on the metric graphs, where $D_{0+}^{(\alpha, \mu) }$ is Hilfer's operator, $l-1<\alpha We will study the following problem for equation (\ref{eq1}) on $\mathbf{\Gamma}$.\\

\textbf{Problem.} To find functions $ u^{(k)}(x,t)$ in the domain $B_{k} \times \left(0,T \right)$, satisfy an equation (\ref{eq1}) at $l-1<\alpha
$$ u^{(k)}(x,t)\in C([0,L_{k} ]\times (0,T]),u_x^{(k)}(x,t)\in C([0,L_{k})\times (0,T]), $$
$$[\alpha]\frac{d}{dt}I_{0+}^{(1-\mu)(l-\alpha)} u^{(k)} (x,t)\in C_{\gamma}\left((0,L_k)\times(0,T]\right) , k=\overline{1,m},$$
$$u_{xx}^{(k)}(x,t), D_{0+}^{\alpha, \mu } u^{\left(k\right)} \left(x{\rm ,}t\right)\in C\left((0,L_{k} )\times (0,T)\right), \gamma =\alpha+l\mu-\alpha \mu.$$\\
non-local and initial conditions:\\
$$
I_{0+}^{(l-\gamma)} u^{(k)} (x,t)\Big|_{t=0}=MI_{0+}^{(l-\gamma)} u^{(k)} (x,t)\Big|_{t=T}, \,\, \gamma =\alpha+l\mu-\alpha \mu\,\ x\in \overline{B_{k}},
$$
\begin{equation}\label{eq2}
\end{equation}
$$
[\alpha]\frac{d}{dt}I_{0+}^{(l-\gamma)} u^{(k)} (x,t)\Big|_{t=0}=\varphi^{(k)}(x), \,\,\ k=\overline{1,m} \,\,\ x\in B_{k};
$$

At branch points (i.e., at internal vertices) of the graph, the solution must satisfy

the following conditions\\
$\textbf{(A)}$ Continuity conditions: the values at the inner vertex \quad $\nu$ \quad of all functions \quad$ u^{(k)}(x,t)$ \quad for which \quad $B_k \sim \nu$ \quad are the same and\\
$\textbf{(B)}$ Local flux conservation conditions at the branching points:
the sum of one-sided derivatives at each vertex $\nu$ of all functions $u^{(k)}(x,t)$, for which
$B_k \sim \nu$, is equal to zero:

\begin{equation}\label{eq3}

\sum_{B_k\sim \nu} u^{(k)}_x(x,t)|_{\nu}=0, \quad \nu \in V \setminus \partial \Gamma, t\in(0,T],\, k=\overline{1,n}
\end{equation}

and boundary conditions

\begin{equation}\label{eq4}
u^{(k)} (x,t)|_{\gamma_k}=0, \,\ \quad \gamma_k \in \partial \Gamma, t\in(0,T],\ k=\overline{1,m_1}.
\end{equation}
where $\varphi ^{(k)} \left(x\right)$ are sufficiently smooth given functions.
\begin{center}
\textbf{ Uniqueness of the problem}
\end{center}
\textbf{Theorem 1.}
{\it If, $ M^2 < \frac{1}{l-1+T^{\beta-1}E_{\alpha+l-1,\beta}(T^{\alpha})}$, for $l-1<\alpha$$
\beta=\left\{\begin{array}{c} {1, \,\,\, if \,\, l=1, } \\ {\alpha-1, \,\,\,\,\ if \,\,\, l=2.} \end{array}\right.
$$
}\
\textbf{Remark 1.}The operator $D_{a+}^{\alpha, \mu}u$ which is defined in Definition 5 can be written as
$$
I_{0+}^{\mu (l-\alpha)}D^{(l)}I_{0+}^{(1-\mu)(l-\alpha)}u^{(k)}(x,t)-u_{xx}^{(k)}(x,t)=f^{(k)}(x,t),
$$
$$
I_{0+}^{\gamma-\alpha}D^{(l)}I_{0+}^{l-\gamma}u^{(k)}(x,t)=u_{xx}^{(k)}(x,t)+f^{(k)}(x,t), \gamma=\alpha+l\mu-\alpha \mu.
$$
Introducing notation
\begin{equation}\label{eq5}
I_{0+}^{l-\gamma}u^{(k)}(x,t)=v^{(k)}(x,t), k=\overline{1,m}
\end{equation}
equation (\ref{eq1}) we will rewrite as follows:
$$
I_{0+}^{\gamma-\alpha}D^{(l)}v^{(k)}(x,t)=u_{xx}^{(k)}(x,t)+f^{(k)}(x,t).
$$
Further applying $I_{0+}^{l-\gamma} $ and considering Lemma 2, we deduce
$$
I_{0+}^{l-\alpha}D^l v^{(k)}(x,t)=v_{xx}^{(k)}(x,t)+f_{l}^{(k)}(x,t).
$$
Using Definition 3 we obtain
\begin{equation}\label{eq6}
_CD_{0+}^{\alpha}v^{(k)}(x,t)-v_{xx}^{(k)}(x,t)=f_{l}^{(k)}(x,t),
\end{equation}
where $f_{l}^{(k)}(x,t)=I_{0+}^{l-\gamma}f^{(k)}(x,t)$. Considering
(\ref{eq2})-(\ref{eq4}), and from (\ref{eq5}) we deduce
$$
v^{(k)} (x,t)|_{t=0}=M v^{(k)} (x,t)|_{t=T} ,
$$
\begin{equation}\label{eq7}
\end{equation}
$$
[\alpha] v_{t}^{(k)} (x,t)|_{t=0}=\varphi^{(k)}(x), k=\overline{1,m}, \,\, x \in B_{k}
$$
and vertex conditions\\
$\textbf{(A)}$ \emph{Continuity conditions:
the values at the inner vertex \quad $\nu$ \quad of all functions \quad$ v^{(k)}(x,t)$ \quad for which \quad $B_k \sim \nu$ \quad are the same } and\\
$\textbf{(B)}$ \emph{Local flux conservation conditions at the branching points:
the sum of one-sided derivatives at each vertex $\nu$ of all functions $v^{(k)}(x,t)$, for which
$B_k \sim \nu$, is equal to zero}:

\begin{equation}\label{eq8}

\sum_{B_k\sim \nu} v^{(k)}_x(x,t)|_{\nu}=0, \quad \nu \in V \setminus \partial \Gamma, t\in[0,T],\, k=\overline{1,n}
\end{equation}

and boundary conditions

\begin{equation}\label{eq9}
v^{(k)} (x,t)|_{\gamma_k}=0, \,\ \quad \gamma_k \in \partial \Gamma, t\in[0,T],\ k=\overline{1,m_1}.
\end{equation}
First we consider the case $0<\alpha<1$:
Let us obtain an a priori estimate for the solution of problem (\ref{eq6})-(\ref{eq9}). To this end, we take the
inner products of both sides of Eq. (\ref{eq6}) by the function $ v^{(k)} (x,t)$, and obtain
\begin{equation}\label{eq10}
\left(_CD_{0+}^{\alpha}v(x,t),v(x,t)\right)_{\mathbf{\Gamma}}-\left(v_{xx}(x,t),v(x,t)\right)_{\mathbf{\Gamma}}=\left(f_{1}(x,t),
v(x,t)\right)_{\mathbf{\Gamma}},
\end{equation}
where $f_{1}^{(k)}(x,t)=I_{0+}^{1-\gamma}f^{(k)}(x,t) $. Based on the inequality [38] $v{}_CD_{ot}^{\alpha}v\geq\frac{1}{2}{} _CD_{ot}^{\alpha}v^2$, we have
\begin{equation}\label{eq11}
\int\limits_{0}^{L_k}v^{(k)}(x,t){}_CD_{0+}^{\alpha}v^{(k)}(x,t)dx\geq \frac{1}{2}\int\limits_{0}^{L_k}{}_CD_{0+}^{\alpha}\left(v^{(k)}(x,t)\right)^2dx.
\end{equation}
Therefore
\begin{equation}\label{eq12}
\left(_CD_{0+}^{\alpha}v(x,t),v(x,t)\right)_{\mathbf{\Gamma}}\geq \frac{1}{2}{} _CD_{ot}^{\alpha} \left\|v(x,t)\right \|^2_{\mathbf{\Gamma}}.
\end{equation}
After integrating by parts and using boundary conditions (\ref{eq9}) and from the Definition 6, \\
we get
$$
\left(v_{xx}(x,t),v(x,t)\right)_{\mathbf{\Gamma}}=\sum_{k=1}^{m}\int\limits_{0}^{L_k}v^{(k)}(x,t)v^{(k)}_{xx}(x,t)dx=
$$
$$
-\sum_{B_k\sim \nu}v^{(k)}(x,t) \frac{d}{dx}v^{(k)}(x,t)\Big|_{\nu}-\sum_{k=1}^{m}\int\limits_{0}^{L_k}\left(v^{(k)}_{x}(x,t)\right)^2dx.
$$
Considering gluing conditions (\ref{eq8}), we get
\begin{equation}\label{eq13}
\left(v_{xx}(x,t),v(x,t)\right)_{\mathbf{\Gamma}}=-\left \|v_{x}(x,t)\right \|_{\mathbf{\Gamma}}^2.
\end{equation}
Now, we estimate:
$$
\int\limits_{0}^{L_k}f_kv_{k}dx\leq\varepsilon\int\limits_{0}^{L_k}v_{k}^2dx+\frac{1}{4\varepsilon}\int\limits_{0}^{L_k}f_{k}^2dx,\,\ \varepsilon>0.
$$
and therefore ($\varepsilon=\frac{1}{2}$)
\begin{equation}\label{eq14}
\left(f_{1}(x,t),v(x,t)\right)_{\mathbf{\Gamma}}\leq \frac{1}{2}\left(\left\|v(x,t)\right \|_{\mathbf{\Gamma}}^2+\left \|f_1(x,t) \right\|_{\mathbf{\Gamma}}^2 \right).
\end{equation}

Taking (\ref{eq10}) into account, from (\ref{eq12})-(\ref{eq14}), we get

\begin{equation}\label{eq15}
\frac{1}{2}{}_CD_{0t}^{\alpha}\left\|v(x,t)\right\|_{\mathbf{\Gamma}}^2+\left\|v_x(x,t)\right\|_{\mathbf{\Gamma}}^2\leq \frac{1}{2}\left(\left\|v(x,t)\right\|_{\mathbf{\Gamma}}^2 + \left\|f_1(x,t)\right\|_{\mathbf{\Gamma}}^2\right),
\end{equation}
\begin{equation}\label{eq16}
{}_CD_{0t}^{\alpha}\left\|v(x,t)\right\|_{\mathbf{\Gamma}}^2\leq \left\|v(x,t)\right\|_{\mathbf{\Gamma}}^2 + \left\|f_1(x,t)\right\|_{\mathbf{\Gamma}}^2.
\end{equation}
Based on the statement [38], and from (\ref{eq16}) it follows that
$$
\left\|v(x,t)\right\|_{\mathbf{\Gamma}}^2\leq \left\|v(x,0)\right\|_{\mathbf{\Gamma}}^2 E_{\alpha,1}\left( t^{\alpha}\right) + \Gamma(\alpha)E_{\alpha, \alpha}\left( t^{\alpha}\right)D_{0t}^{-\alpha}\left\|f_1(x,t)\right\|_{\mathbf{\Gamma}}^2
$$
\begin{equation}\label{eq17}
\left\|v(x,T)\right\|_{\mathbf{\Gamma}}^2\leq \left\|v(x,0)\right\|_{\mathbf{\Gamma}}^2 E_{\alpha,1}\left( T^{\alpha}\right) + N_1,
\end{equation}
where
$$
N_1=\Gamma(\alpha)E_{\alpha, \alpha}\left( T^{\alpha}\right)D_{0t}^{-\alpha}\left\|f_1(x,t)\right\|_{\mathbf{\Gamma}}^2 |_{t=T}
$$

considering non-local condition (\ref{eq7}), we have

\begin{equation}\label{eq18}
\frac{1-\mathbb{M}^2 E_{\alpha,1}\left( T^{\alpha}\right)}{M^2}\left\|v(x,0)\right\|_{\mathbf{\Gamma}}^2 \leq N_1.
\end{equation}
Based on the condition of the theorem $M^2 E_{\alpha,1}\left( T^{\alpha}\right)<1 $. Hence

\begin{equation}\label{eq19}

\left\|v(x,0)\right\|_{\Gamma}^2 \leq \frac{M^2 N_1}{1-M^2 E_{\alpha,1}\left( T^{\alpha}\right)}.
\end{equation}
Considering (\ref{eq17})-(\ref{eq19}) we get
\begin{equation}\label{eq20}
\left\|v(x,t)\right\|_{\mathbf{\Gamma}}^2\leq \frac{\Gamma(\alpha)E_{\alpha, \alpha}\left( T^{\alpha}\right)}{1- M^2 E_{\alpha,1}\left( T^{\alpha}\right)} D_{0+}^{-\alpha}\left\|f_1(x,t)\right\|_{\mathbf{\Gamma}}^2 |_{t=T}.
\end{equation}

If $f^{(k)}(x,t)=0$ , then $v^{(k)}(x,t)\equiv0$. Due to (\ref{eq5}) we get $u^{(k)}(x,t)\equiv0$.

Therefore, we can say that the solution to problem (\ref{eq1})-(\ref{eq4}) is unique, for the case $l=1$.\\

Now we consider the case $1<\alpha<2$.

Further, we get a priory estimation for solution of the investigated problem.
\,\, To this end, we take the inner products of both sides of Eq. (\ref{eq6}) by the function $ v_t^{(k)} (x,t)$, and obtain
\begin{equation}\label{eq21}
\left(_CD_{0+}^{\alpha}v(x,t),v_t(x,t)\right)_{\mathbf{\Gamma}}-\left(v_{xx}(x,t),v_t(x,t)\right)_{\mathbf{\Gamma}}=
\left(f_{2}(x,t),v_t(x,t)\right)_{\mathbf{\Gamma}}.
\end{equation}
where $f_{2}^{(k)}(x,t)=I_{0+}^{2-\gamma}f^{(k)}(x,t) $. Based on the inequality \eqref{eq11} and from the Definition 5, we have
$$
\left(_CD_{0+}^{\alpha}v(x,t),v_t(x,t)\right)_{\mathbf{\Gamma}}=\sum_{k=1}^{m}\int_{0}^{L_{k}} v_{t}^{(k)}(x,t){_{C}}D_{0t}^{\alpha-1}v_{t}^{(k)}(x,t)dx\geq
$$
\begin{equation}\label{eq22}
\geq \sum_{k=1}^{m}\int_{0}^{L_{k}}\frac{1}{2} {_{C}}D_{0t}^{\alpha-1}\left(v_{t}^{(k)}(x,t)\right)^{2}dx=\frac{1}{2}{_{C}}D_{0t}^{\alpha-1}\left\|v_{t}(x,t)\right\|_{\mathbf{\Gamma}}^{2}
\end{equation}
and
$$
-\left(v_{xx}(x,t),v_t(x,t)\right)_{\mathbf{\Gamma}}=-\sum_{k=1}^{m}\int_{0}^{L_{k}}v_{t}^{(k)}(x,t)v_{xx}^{(k)}(x,t)dx=
$$
$$
=\frac{1}{2}\sum_{k=1}^{m}\frac{\partial}{\partial t}\int_{0}^{L_{k}}\left(v_{x}^{(k)}(x,t)\right)^{2}dx=\frac{1}{2}\frac{\partial}{\partial t}\left\|v_{x}(x,t)\right\|_{\mathbf{\Gamma}}^{2}.
$$
Similar to \eqref{eq14}, one can obtain
$$
\left(f_{2}(x,t),v_t(x,t)\right)_{\mathbf{\Gamma}}
\leq \frac{1}{2} \left\|v_{t}(x,t)\right\|_{\mathbf{\Gamma}}^{2}+\frac{1}{2} \left\|f_2(x,t)\right\|_{\mathbf{\Gamma}}^{2}.
$$
Taking into account the performed transformations, from the identity \eqref{eq21}, we deduce the inequality
\begin{equation}\label{eq23}
{_{C}}D_{0t}^{\alpha-1}\left\|v_{t}(x,t)\right\|_{\mathbf{\Gamma}}^{2}+\frac{\partial}{\partial t}\left\|v_{x}(x,t)\right\|_{\mathbf{\Gamma}}^{2}\le \left\|v_{t}(x,t)\right\|_{\mathbf{\Gamma}}^{2}+ \left\|f_2(x,t)\right\|_{\mathbf{\Gamma}}^{2}.
\end{equation}
By integrating this relation with respect to $\tau$ from $0$ to $t$, and considering
$$
\int_{0}^{t}{_{C}}D_{0\tau}^{\alpha-1}g^{(k)}(x,\tau)d\tau=-\frac{t^{2-\alpha}}{\Gamma(3-\alpha)}g^{(k)}(x,0)+D_{0t}^{\alpha-2}g^{(k)}(x,t),
$$
we obtain the inequality
$$
D_{0+}^{\alpha-2}\left\|v_{t}(x,t)\right\|_{\mathbf{\Gamma}}^{2}+\left\|v_{x}(x,t)\right\|_{0}^{2}\le
$$
\begin{equation}\label{eq24}
\le\int_{0}^{t}\left(\left\|v_{\tau}(x,\tau)\right\|_{\mathbf{\Gamma}}^{2}+ \left\|f_2(x,\tau)\right\|_{\mathbf{\Gamma}}^{2}\right)d\tau+c_{1}\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2}+
\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2},
\end{equation}
here $c_{1}=T^{2-\alpha}/ \Gamma(3-\alpha)$. Based on \eqref{eq24}, we deduce
$$
\left\|v_{x}(x,t)\right\|_{\mathbf{\Gamma}}^{2}\le
$$
\begin{equation}\label{eq25}
\le \int_{0}^{t}\left(\left\|v_{\tau}(x,\tau)\right\|_{\mathbf{\Gamma}}^{2}+ \left\|f(x,t)\right\|_{\mathbf{\Gamma}}^{2}\right)d\tau+c_{1}\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2}+
\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2},
\end{equation}
$$
D_{0+}^{\alpha-2}\left\|v_{t}(x,t)\right\|_{\mathbf{\Gamma}}^{2}\le
$$
\begin{equation}\label{eq26}
\le\int_{0}^{t}\left\|v_{\tau}(x,t)\right\|_{\mathbf{\Gamma}}^{2}d\tau+\int_{0}^{t}\left\|f_2(x,t)\right\|_{\mathbf{\Gamma}}^{2}d\tau +c_{1}\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2}+\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2}.
\end{equation}
Consequently following [38] we arrive at
$$
\int_{0}^{t}\left\|v_{\tau}(x,\tau)\right\|_{\mathbf{\Gamma}}^{2} d\tau\le
$$
$$
\le\Gamma(\alpha)E_{\alpha-1,\alpha-1}\left(t^{\alpha}\right) D_{0+}^{1-\alpha}\left[\int_{0}^{t}\left\|f(x,\tau)\right\|_{\mathbf{\Gamma}}^{2}d\tau +c_{1}\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2} +\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2}\right]=
$$
$$
=\Gamma(\alpha)E_{\alpha-1,\alpha-1}\left(t^{\alpha}\right)D_{0+}^{-\alpha}\left\|f(x,t)\right\|_{\mathbf{\Gamma}}^{2} +t^{\alpha-1}E_{\alpha-1,\alpha-1}\left(t^{\alpha}\right)\left(c_{1}\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2} +\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2}\right),
$$
therefore
$$
\int_{0}^{t}\left\|v_{\tau}(x,\tau)\right\|_{\mathbf{\Gamma}}^{2} d\tau\le\Gamma(\alpha)E_{\alpha-1,\alpha-1}\left(T^{\alpha}\right)D_{0+}^{-\alpha}\left\|f_2(x,t)\right\|_{\mathbf{\Gamma}}^{2}\Big|_{t=T}+
$$
\begin{equation}\label{eq27}
+c_2\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2} +c_3\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2},
\end{equation}
where $c_2=c_1 T^{\alpha-1}E_{\alpha-1,\alpha-1}\left(T^{\alpha}\right) $, $c_{3}=c_2/c_1$.

If, in \eqref{eq25} we replace $t$ with $T$, we have

\begin{equation}\label{eq28}
\left\|v_{x}(x,T)\right\|_{\mathbf{\Gamma}}^{2}\le N_2+(c_3+1)\left\|v_{x}(x,0)\right\|_{\mathbf{\Gamma}}^{2},
\end{equation}
where
$$
N_2=\Gamma(\alpha)E_{\alpha-1,\alpha-1}\left(T^{\alpha}\right) D_{0+}^{-\alpha}\left\|f_2(x,t)\right\|_{\mathbf{\Gamma}}^{2}|_{t=T}
+\int_{0}^{T}\left\|f_2(x,\tau)\right\|_{\mathbf{\Gamma}}^{2}d\tau+(c_1+c_2\left\|\varphi(x)\right\|_{\mathbf{\Gamma}}^{2}.
$$

From the non-local condition \eqref{eq7} and based on the condition of the Theorem 1,\\

$M^2 \left(1+c_3\right)<1 $, hence
\begin{equation}\label{eq29}
\left\|v_x(x,0)\right\|_{\mathbf{\Gamma}}^2 \leq \frac{M^2 N_2}{1-M^2 \left(1+c_3\right)}.
\end{equation}
We can write \eqref{eq27} in the following form:
$$
\int_{0}^{t}\left\|v_{\tau}(x,\tau)\right\|_{\mathbf{\Gamma}}^{2} d\tau\le
$$
\begin{equation}\label{eq30}
\le A\left[c_2 \|\varphi(x)\|_{\mathbf{\Gamma}}^2 + c_3 M^2 \int_{0}^{T}\left\|f_2(x,\tau)\right\|_{\mathbf{\Gamma}}^{2}d\tau+(1-M^2) D_{0+}^{-\alpha}\left\|f_2(x,t)\right\|_{\mathbf{\Gamma}}^2 \Big|_{t=T}\right]
\end{equation}
where
$$
A=\frac{1}{1- M^2(1+c_3)}.
$$

Using a standart scheme, one can prove that a solution the problem (\ref{eq6})-(\ref{eq9}) is unique. If $f^{(k)}(x,t)=0$ and $\varphi^{(k)}(x)\equiv0$ , then $v^{(k)}(x,t)\equiv0$. Based on (\ref{eq5}) we get $u^{(k)}(x,t)\equiv0$.

Therefore, we can say that the solution to problem (\ref{eq1})-(\ref{eq4}) is unique for the case $l=2$.\\

\begin{center}

\textbf{The Existence}
\end{center}
Using by the method separations of variables for the homogeneous equation we will get ODE of fractional order
$$
D_{0+}^{\alpha,\mu} T(t)+\lambda ^{2} T(t)=0, \,\, l-1<\alpha$$
and ODE of integer order
\begin{equation}\label{eq31}
\frac{d^{2} }{dx^{2} } X^{(k)} (x)+\lambda ^{2} X^{(k)} (x)=0, \,\, \, k=\overline{1,m}
\end{equation}

Moreover, from the conditions (\ref{eq5}), we obtain

At branch points (i.e., at internal vertices) of the graph, the solution must satisfy
the following conditions\\
$\textbf{(A)}$ Continuity conditions:
the values at the inner vertex \quad $\nu$ \quad of all functions \quad$ X^{(k)}(x)$ \quad for which \quad $B_k \sim \nu$ \quad are the same and\\
$\textbf{(B)}$ Local flux conservation conditions at the branching points:
the sum of one-sided derivatives at each vertex $\nu$ of all functions $X^{(k)}(x)$, for which
$B_k \sim \nu$, is equal to zero:

\begin{equation}\label{eq32}

\sum_{B_k\sim \nu} \frac{d}{dx} X^{(k)}(x)|_{\nu}=0, \quad \nu \in V \setminus \partial \Gamma,\, k=\overline{1,n}
\end{equation}

and boundary conditions

\begin{equation}\label{eq33}
X^{(k)} (x)|_{\gamma_k}=0, \,\ \quad \gamma_k \in \partial \Gamma,\ k=\overline{1,m_1}.
\end{equation}

The general solution of equation (\ref{eq31}) has a form:

\begin{equation}\label{eq34}
X^{(k)} (x)=a_{k} \cos \lambda x+b_{k} \sin \lambda x; \,\, x\in B_{k}
\end{equation}

The spectral problem (\ref{eq31})-(\ref{eq33}) in the case of general metric graphs is investigated in ([33],[34], [35],[40]).

In this case the graph is called "quantum" graph and the operator $\frac{d^2}{dx^2}$, defined in each edge of the graph together with conditions (\ref{eq32})-(\ref{eq33}), called to be "edge-based" Laplacian (see [40]).

Next we need to constitute some results from [33] and [40].

Let us define the eigenvalue counting function $\texttt{N}_{\Gamma}(k)$ as a number of eigenvalues of the quantum graph \textbf{$\Gamma$} which are smaller than $k$,
$$
\texttt{N}_{\mathbf{\Gamma}}(k)=\sharp \left\{\lambda \in \sigma(\Gamma): \lambda \leq k \right\}.
$$

This number is guaranteed to be finite since the spectrum of a quantum graph is discrete and bounded from below

([33],[34]). We count the eigenvalues in terms of $k=\sqrt{\lambda}$ as this is more convenient and can be easily related
back to $\lambda$.
The counting function $\texttt{N}_{\Gamma}(k)$ grows linearly in $k$, with the slope proportional to the total lengths
of the graph. This type of result is known as the Weyl's law.

\textbf{Lemma 6.}[36]

Let $\mathbf{\Gamma} $ be a graph with Neumann or Dirichlet conditions at every boundary vertex. Then
$$
\texttt{N}(k)=\frac{\hat L}{\pi} k+ O(1),
$$
where $\hat L =L_{1}+L_{2}+...+L_{m}$ is the total length the graph's edges and the remainder term is bounded above and below by constants independent of $k$.\\

In our case we put $X_{N}(x)=\left( X_{N}^{(1)}(x), X_{N}^{(2)}(x), \quad...,\quad X_{N}^{(m)}(x)\right)^T$ be vector eigenfunctions of the problems (\ref{eq31})-(\ref{eq33}) corresponding to $\lambda_{N}$, $N=1,2,...$\,. \quad

$\lambda_{1}^{2}$,$\lambda_{2}^{2}$, ..., $\lambda_{n}^{2}$ ... is non-decreasing sequence of eigenvalues which take to account the multiplicity(i.e. eigenvalues with multiplicity $k$ is taken $k$-times).
From the above lemma it follows that $\lambda_{N}\sim const\cdot n$ at $n\rightarrow +\infty$.\\
By $ C^\infty \left( \Gamma \right)$, the set of {\it infinitely differentiable functions on \textbf{$\Gamma$},} we mean the set of
continuous functions on \textbf{$\Gamma$} whose restriction to each edge interior is $k$-times uniformly continuously differentiable (as a
function on that real interval) for any $k$=1,2, ...\ .

$$C_{Dir}^{\infty}(\Gamma)=\{f\in C^{\infty}(\Gamma): \,\ f |_{\partial \Gamma}=0 \}, $$

where $\partial \Gamma$ is the set of boundary vertices.
$L_{Dir}^2(\Gamma)$ be the closure of $C_{Dir}^{\infty}$ under the norm
$$
\|u\|^2_{\Gamma}=\sum_{k} \int_0^{L_k} \left|u^{(k)} \right|^2 dx
$$

Now we formulate theorem on completeness of eigenfunctions of the "edge-based" graph Laplacian (or quantum graph) from [40].

\textbf{Theorem 2.}(See Proposition 3.2. in [40] ).\\
{\it Let \textbf{$\Gamma$} is finite graph. There exists eigenpairs $(X_N, \lambda_N)$, $N=1,2, . . .$ for the edge based Laplacian, such that:\\
(1) $ 0\leq \lambda_1 \leq \lambda_2 \leq \cdot \, \cdot\, \cdot ,$\\
(2) the $ X_N$ satisfy the Dirichlet condition,\\
(3) the $X_N$ form a complete orthonormal basis for $L^2_{Dir}(\Gamma)$, and\\
(4) $\lambda_N \rightarrow \infty$.}\ \\

\textbf{Theorem 3.}

{\it If $\varphi ^{(k)} (x), \,\ \in C^{1} [0 ,L_{k}]$, \,\, $\frac{\partial}{\partial x} f^{(k)}(x,t)\in C_{\gamma}\left([0,L_k]\times(0,T)\right)$ besides
$\frac{d^{2}}{dx^{2}}\varphi^{(k)}(x)$ and $\frac{\partial^2}{\partial x^2}f^{(k)}(x,t)$ are absolute integrable
functions in $ (0, L_{k})$ and $(B_{k}\times(0,T))$ respectively. \\Furthermore, continuity and local flux conservation conditions are also valid for functions
$ f^{(k)}(x,t),\,\ \varphi ^{(k)}(x), \,\, $
and
$ M\neq \frac{1}{E_{\alpha,1}(-\lambda_{n}^{2}T^{\alpha})}$ hold, then the solution of the investigated problem exists.}\

\textbf{Proof:}

\,\, Noting that $f^{(k)}(x,t)\in L_2[0;L_k]$ and expand into the Fourier series in terms of eigenfunctions, i.e.
\begin{equation}\label{eq35}
f (x,t)=\sum _{N=1}^{\infty } f_{N}(t) X_{N} (x),\,\, .
\end{equation}
where $f_{N}(t) $ is the coefficients of the Fourier series (\ref{eq35}). Further, introducing a solution of equation (\ref{eq1}) in the form
\begin{equation}\label{eq36}
u^{(k)}(x,t)=\sum _{N=1}^{\infty }X_N (x)W_N(t) \,\, k=\overline{1,m}
\end{equation}
Substituting (\ref{eq36}) into equation (\ref{eq1}), we obtain
$$
\sum _{N=1}^{\infty }\left( D_{0+}^{(\alpha, \mu) } W_{N} (t)+\lambda_N^{2} W_{N} (t)-f_{N}(t) \right)X_{N} \left(x\right) =0.
$$
Since the function $X_{N} (x)$ is an eigenfunction of the investigating problem from we obtain an inhomogeneous differential equation of fractional order
\begin{equation}\label{eq37}
D_{0+}^{(\alpha, \mu) } W_{N} (t)+\lambda _{N}^{2} W_{N} (t)=f_N(t).
\end{equation}
If, we consider the case $0<\alpha<1$, general soluation of the Eq.(\ref{eq37}) has a form (see[37] Lemma 2):
$$
W_{N}(t)=\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
\begin{equation}\label{eq38}
+A_{N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }), \,\ \,
\end{equation}
where $\gamma= \alpha +\mu-\alpha \mu$. Considering (\ref{eq36}) and (\ref{eq38}) we can write the general solution of equation (\ref{eq1}) has the following form:
$$
u^{(k)}(x,t)=\sum _{N=1}^{\infty }\Big[\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
\begin{equation}\label{eq39}
+A_{N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }) \Big]X_{N}^{(k)}(x). \,\,\ k=\overline{1,m}.
\end{equation}
We know, that if $l=1$, condition (\ref{eq2}) is equal to the following one condition:
\begin{equation}\label{eq40}
I_{0+}^{(1-\gamma)} u^{(k)} (x,t)\Big|_{t=0}=M I_{0+}^{(1-\gamma)} u^{(k)} (x,t)\Big|_{t=T}, \,\,\ \gamma = \alpha +\mu-\alpha \mu.
\end{equation}
Applying the operator $I_{0+}^{1-\gamma}$ to the (\ref{eq39}) and considering the Definition 1, we have
$$
I_{0+}^{1-\gamma}u^{(k)}(x,t)=\frac{1}{\Gamma(1-\gamma)}\sum _{N=1}^{\infty }A_{N}X_{N}^{(k)}(x)\int_{0}^t (t-\tau)^{-\gamma}\tau^{\gamma-1}E_{\alpha,\gamma}(-\lambda_{N}^{2}\tau^{\alpha})d\tau+
$$
$$
+\sum _{N=1}^{\infty }A_{N}X_{N}^{(k)}(x)\frac{1}{\Gamma(1-\gamma)}\int_{0}^t (t-\tau)^{-\gamma}\int_{0}^{\tau}(\tau-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}(\tau-s)^{\alpha})f_{N}(s)d\tau ds=
$$
\begin{equation}\label{eq41}
=\sum _{N=1}^{\infty }A_{N}F_{1,N}(t)X_{N}^{(k)}(x)+\sum _{N=1}^{\infty }F_{2,N}(t)X_{N}^{(k)}(x).
\end{equation}
$$
F_{1,N}(t)=\frac{1}{\Gamma(1-\gamma)}\int_{0}^t z^{-\gamma}(t-z)^{\gamma-1}E_{\alpha,\gamma}(-\lambda_{N}^{2}(t-z)^{\alpha})dz=E_{\alpha,1}(-\lambda_{N}^{2}t^{\alpha}),
$$
$$
F_{2,N}(t)=\frac{1}{\Gamma(1-\gamma)}\int_{0}^t f_{N}(s) ds \int_{s}^{t}(t-\tau)^{-\gamma}(\tau-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}(\tau-s)^{\alpha})d\tau
$$
$$
\int_{s}^{t}(t-\tau)^{-\gamma}(\tau-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}(\tau-s)^{\alpha})d\tau=
\left[z=\frac{\tau-s}{t-s}t\right]=
$$
$$
=\left(\frac{t-s}{t}\right)^{\alpha-\gamma}\int_{0}^{t}(t-z)^{-\gamma}z^{\alpha-1}E_{\alpha,\alpha}
\left[-\lambda_{N}^{2}\left(\frac{t-s}{t}\right)^{\alpha}z^{\alpha}\right]dz=
$$
$$
=\Gamma(1-\gamma)(\tau-s)^{(\alpha-\gamma)}E_{\alpha,1+\alpha-\gamma}(-\lambda_{N}^{2}(\tau-s)^{\alpha}),
$$
$$
F_{2,N}(t)=\int_{0}^t (t-\tau)^{(\alpha-\gamma)}E_{\alpha,1+\alpha-\gamma}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}(\tau) d\tau.
$$
Based on the condition (\ref{eq40}), we have
$$
\sum_{N=1}^{\infty}\left(A_{N}F_{1,N}(0)+F_{2,N}(0)\right)X_{N}^{(k)}(x)=M\sum_{N=1}^{\infty}\left(A_{N}F_{1,N}(T)+F_{2,N}(T)\right)X_{N}^{(k)}(x),
$$
hence
$$
A_{N}=MA_{N}F_{1,N}(T)+F_{2,N}(T) \quad or \quad A_{N}=\frac{M}{1-MF_{1}(T)}F_{2}(T),
$$

$$
M\neq \frac{1}{F_{1}(T)} \,\, or \,\, M\neq \frac{1}{E_{\alpha,1}(-\lambda_{N}^{2}T^{\alpha})}.

$$
Finally, the soluation equation (\ref{eq1}) satisfying the conditions (\ref{eq2})-(\ref{eq4}) when $l=1$, has the following form:
$$
u^{(k)}(x,t)=\sum_{N=1}^{\infty}\Big[\frac{M}{1-MF_{1}(T)}F_{2}(T)t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha })+
$$
\begin{equation}\label{eq42}
+\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left(-\lambda_{N}^{2}(t-\tau)^{\alpha}\right)f_{N}(\tau)d\tau \Big]X_{N}^{(k)}(x).
\end{equation}

It is required to prove the convergence of infinite series, corresponding to the functions $u^{(k)}(x,t)$, $u_{xx}^{(k)} (x,t) $, $D_{0+}^{\alpha,\mu } u^{(k)}(x,t)$

in the domain $B_{k}\times(0,T)$.\\
For futher investigations we need the following lemma:\\

\textbf{Lemma 7.} The following estimate holds true:

\begin{equation}\label{eq43}
\left|X_{N}^{(k)}(x)\right|=\left|a_{N,k} \cos \lambda _{N} x+b_{N,k} \sin \lambda _{N} x\right|\le \sqrt{\frac{2}{L_k} }.
\end{equation}

\textbf{Proof:} We have

$$
\|X_{N}(x)\|_\Gamma^2=\sum_{k} \int\limits_0^{L_{k}} \left(X_{N}^{(k)}(x)\right)^2dx_{k}=1.
$$
Therefore
$$
\int\limits_0^{L_{k}} \left(a_{N,k} \cos \lambda _{N} x+b_{N,k} \sin \lambda _{N} x \right)^2dx=
$$
$$
=\frac{a_{N,k}^2+b_{N,k}^2}{2}L_k+\frac{a_{N,k}^2+b_{N,k}^2}{4\lambda _{N}} \sin 2\lambda _{N} L_k -\frac{a_{N,k}b_{N,k}}{4\lambda _{N}}\left(\cos 2 \lambda _{N} L_k-1 \right) \leq 1.
$$
From the last relation, considering Lemma 3, at $N\rightarrow \infty$ we get $ a_{N,k}^2+b_{N,k}^2 \leq \frac{2}{L_k}$.

Hence, we have

$$\left|X_{N}^{(k)}(x)\right|=\left|a_{N,k} \cos \lambda _{N} x+b_{N,k} \sin \lambda_{N} x\right|\le \sqrt{ a_{N,k}^2+b_{N,k}^2} \leq \sqrt{ \frac{2}{L_k} }.$$
This proves Lemma 7.\\

From the Lemma 5, one can get

$$
\left|A_{N}\right|=\left|\frac{M}{1-MF_{1}(T)} F_{2,N}(T)\right|\le
$$
\begin{equation}\label{eq44}
\le\left|\frac{M}{1-MF_{1,N}(T)}\right|\left|F_{2,N}(T)\right|\le \frac{M_{1}}{\lambda_{N}^{2}},
\end{equation}
where $M_{2}=const>0$. From (\ref{eq35}) and based on the conditions of the Theorem 2, we will deduce, that\\
$$
\left|f _{N}(t) \right|= \left|\sum_k\int _{0}^{L_{k} }f^{(k)} (x,t) X_{N}^{(k)} (x)dx \right|=
$$
\begin{equation}\label{eq45}
=\left|\sum_k\frac{1}{\lambda_{N}^{2}} \int _{0}^{L_{k} }\frac{d^2}{dx^2}f^{(k)} (x,t) X_{N}^{(k)} (x)dx \right| \le \frac{const}{\lambda_{N}^{2}}.
\end{equation}

From (\ref{eq39}) and (\ref{eq43})-(\ref{eq45}) we obtain

$$
\left|u^{(k)}(x,t)\right|=\Big|\sum _{N=1}^{\infty }\Big[\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
$$
+A_{N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha })\Big]X_{N}^{(k)}(x)\Big|\le
$$
$$
\le \sum _{N=1}^{\infty }\Big|\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
$$
+A_{N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha })\Big|\left|X_{N}^{(k)}(x)\right|\le
$$
$$
\le const \sum _{N=1}^{\infty }\Big(\int _{0}^{t}\left|t-\tau \right|^{\alpha-1}\left|E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]\right|\left|f_{N}(\tau)\right|d\tau+
$$
$$
+\left|A_{N}\right|t^{\gamma-1} \left|E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha })\right|\Big) \le
$$
$$
\le \sum _{N=1}^{\infty }\frac{m_{3}}{\lambda_{N}^{2}(1+\lambda_{N}^{2})}+\sum _{N=1}^{\infty }\frac{m_{4}}{\lambda_{N}^{2}(1+\lambda_{N}^{2})}\le \sum _{N=1}^{\infty }\frac{m_{5}}{\lambda_{N}^{4}}
$$
where $m_{i}=const>0, i=\overline{3,5}$ and $m_{5}=m_{3}+m_{4}$. Hence, $u^{(k)}(x,t)$ are uniform convergent.
Similarly, we get
$$
\left|u_{xx}^{(k)}(x,t)\right|=\Big|-\sum _{N=1}^{\infty }\lambda_{N}^{2}\Big[\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
$$
+A_{N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha })\Big]X_{N}^{(k)}(x)\Big|\le\sum _{N=1}^{\infty }\frac{m_{6}}{\lambda_{N}^{2}}
$$
We infer that the series, corresponding to function $u_{xx}^{(k)} (x,t)$ will be uniformly convergent due to the Definition 5 owing to
$$
D_{0+}^{\alpha, \mu } u^{(k)} (x,t)=I_{0+}^{\gamma-\alpha}\frac{d}{dt}I_{0+}^{1-\gamma}u^{(k)} (x,t).
$$
If we consider (\ref{eq41}), we get
$$
\frac{d}{dt}I_{0+}^{1-\gamma}u^{(k)}(x,t)=\sum_{N=1}^{\infty}A_{N} X_{N}^{(k)}(x)\frac{d}{dt}E_{\alpha,1}\left(-\lambda_{N}^{2}t^{\alpha}\right)+
$$
$$
+\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\frac{d}{dt}\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,1+\alpha-\gamma}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}(\tau) d\tau=
$$
$$
-\sum_{N=1}^{\infty}A_{N}\lambda_{N}^{2} X_{N}^{(k)}(x)t^{\alpha-1}E_{\alpha,\alpha}\left(-\lambda_{N}^{2}t^{\alpha}\right)
$$
$$
+\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\frac{d}{dt}\int_{0}^t \frac{d}{d\tau}\left[(t-\tau)^{\alpha-\gamma+1}E_{\alpha,2+\alpha-\gamma}(-\lambda_{N}^{2}(t-\tau)^{\alpha})\right]f_{N}(\tau) d\tau=
$$
$$
=-\sum_{N=1}^{\infty}A_{N}\lambda_{N}^{2} X_{N}^{(k)}(x)t^{\alpha-1}E_{\alpha,\alpha}\left(-\lambda_{N}^{2}t^{\alpha}\right)
-\sum_{N=1}^{\infty}f_{N}(0) X_{N}^{(k)}(x)t^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-
$$
$$
-\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau.
$$
$$
I_{0+}^{\gamma-\alpha}\frac{d}{dt}I_{0+}^{1-\gamma}u^{(k)}(x,t)=-\sum_{N=1}^{\infty}A_{N}\lambda_{N}^{2} X_{N}^{(k)}(x)I_{0+}^{\gamma-\alpha}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})-
$$
$$
-\sum_{N=1}^{\infty}f_{N}(0)X_{N}^{(k)}(x)I_{0+}^{\gamma-\alpha}t^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}
(-\lambda_{N}^{2}t^{\alpha})-
$$
$$
-\sum_{N=1}^{\infty}X_{N}^{(k)}(x)I_{0+}^{\gamma-\alpha}\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau=
$$
$$
=-\sum_{N=1}^{\infty}A_{N}\lambda_{N}^{2} X_{N}^{(k)}(x)H_{1,N}(t)-\sum_{N=1}^{\infty}f_{N}(0)X_{N}^{(k)}(x)H_{2,N}(t)-\sum_{N=1}^{\infty}X_{N}^{(k)}(x)H_{3,N}(t).
$$
Here
$$
H_{1,N}(t)=I_{0+}^{\gamma-\alpha}t^{\alpha-1}E_{\alpha,\alpha}\left(-\lambda_{N}^{2}t^{\alpha}\right)=
$$
$$
=\frac{1}{\Gamma{(\gamma-\alpha)}}\int_{0}^t (t-\tau)^{\gamma-\alpha-1)}\tau^{\alpha-1}E_{\alpha,\alpha}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)d\tau
= t^{\gamma-1)}E_{\alpha,\gamma}\left(-\lambda_{N}^{2}\tau^{\alpha}\right),
$$
$$
H_{2,N}(t)=I_{0+}^{\gamma-\alpha}t^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}\left(-\lambda_{N}^{2}t^{\alpha}\right)=
$$
$$
=\frac{1}{\Gamma{(\gamma-\alpha)}}\int_{0}^t (t-\tau)^{\gamma-\alpha-1)}\tau^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)d\tau
= E_{\alpha,1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right),
$$
$$
H_{3,N}(t)=I_{0+}^{\gamma-\alpha}\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau=
$$
$$
= \frac{1}{\Gamma(\gamma-\alpha)}\int_{0}^t (t-\tau)^{\gamma-\alpha-1}\int_{0}^\tau (\tau-s)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(\tau-s)^{\alpha})f_{N}^{'}(s)ds d\tau=
$$
$$
= \frac{1}{\Gamma(\gamma-\alpha)}\int_{0}^t f_{N}^{'}(s)ds\int_{s}^{t} (t-\tau)^{\gamma-\alpha-1} (\tau-s)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(\tau-s)^{\alpha}) d\tau=
$$
$$
=\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau)d\tau.
$$
Finally,
$$
D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)=\sum_{N=1}^{\infty}A_{N}\lambda_{N}^{2}t^{\gamma-1}E_{\alpha,\gamma}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)
X_{N}^{(k)}(x)-
$$
$$
-\sum_{N=1}^{\infty}f_{N}(0)E_{\alpha,1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)X_{N}^{(k)}(x)-
\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau)d\tau.
$$
Now consider $D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)$ for the convergence issue.
$$
\left|D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)\right|=
\Big|\sum_{N=1}^{\infty}A_{N}\lambda_{N}^{2}t^{\gamma-1}E_{\alpha,\gamma}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)
X_{N}^{(k)}(x)-
$$
$$
-\sum_{N=1}^{\infty}f_{N}(0)E_{\alpha,1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)X_{N}^{(k)}(x)-
\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau)d\tau \Big|\leq
$$
$$
\leq \sum_{N=1}^{\infty}\left|A_{N}\lambda_{N}^{2}t^{\gamma-1}E_{\alpha,\gamma}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)
X_{N}^{(k)}(x) \right|+
$$
$$
+\sum_{N=1}^{\infty}\left|f_{N}(0)E_{\alpha,1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)X_{N}^{(k)}(x)\right|+
\sum_{N=1}^{\infty}\left|X_{N}^{(k)}(x)\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau)d\tau \right|\leq
$$
$$
\leq \sum_{N=1}^{\infty}\left|A_{N}\right|\lambda_{N}^{2}\left|t^{\gamma-1}E_{\alpha,\gamma}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)\right|
\left| X_{N}^{(k)}(x) \right|+
$$
$$
+\sum_{N=1}^{\infty}f_{N}(0)\left|E_{\alpha,1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)\right|\left|X_{N}^{(k)}(x)\right|+
\sum_{N=1}^{\infty}\left|X_{N}^{(k)}(x)\right|\int_{0}^t \left|E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})\right|f_{N}^{'}(\tau)d\tau \leq
$$
$$
\leq \sum_{N=1}^{\infty}\left|A_{N}\right|\lambda_{N}^{2}\left|t^{\gamma-1}E_{\alpha,\gamma}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)\right|
\left| X_{N}^{(k)}(x) \right|+
$$
$$
+\sum_{N=1}^{\infty}f_{N}(0)\left|E_{\alpha,1}\left(-\lambda_{N}^{2}\tau^{\alpha}\right)\right|\left|X_{N}^{(k)}(x)\right|+
\sum_{N=1}^{\infty}\left|X_{N}^{(k)}(x)\right|\left|\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau)d\tau\right| \leq
$$
$$
\leq \sum_{N=1}^{\infty}\frac{c_{1}}{1+\lambda_n^{2}}+\sum_{N=1}^{\infty}\frac{c_{2}}{1+\lambda_n^{2}}+
\sum_{N=1}^{\infty}\frac{c_{3}}{\lambda_n^{2}(1+\lambda_n^{2})}\leq
$$
$$
\leq \sum_{N=1}^{\infty}\frac{c_{4}}{\lambda_n^{2}}.
$$
Here $c_{i}=const>0$, $(i=\overline{1,4})$ and $ c_{1}+c_{2}+c_{3}=c_{4}$. from the Lemma 6, according to the asymptotes of $\lambda_n \sim c\cdot n$ ($c$ is const), we can conclude that series of $D_{0+}^{\alpha, \mu}u^{(k)}(x,t)$ is uniformly convergent.

Now we consider the case $l = 2$. In this case, general solution of the Eq.(\ref{eq37}) has a form (see,[37], lemma 2):

$$
W_{N}(t)=\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-\tau)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
\begin{equation}\label{eq46}
+B_{1,N}t^{\gamma-2} E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })+B_{2,N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }), \,\ \,\ 1<\gamma<2
\end{equation}
where $\gamma= \alpha +2\mu-\alpha \mu$. From (\ref{eq36}) and (\ref{eq46}) we can write the general solution of equation (\ref{eq1}) in the following form:
$$
u^{(k)}(x,t)=\sum _{N=1}^{\infty }\Big[\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
\begin{equation}\label{eq47}
+B_{1,N}t^{\gamma-2} E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })+B_{2,N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }) \Big]X_{N}^{(k)}(x), 1<\gamma<2. \,\,\ k=\overline{1,m}.
\end{equation}
Based on conditions (\ref{eq2}), applying the operator $I_{0+}^{2-\gamma}$ to the (\ref{eq47}) and considering the Definition 1 we have
$$
I_{0+}^{2-\gamma}u^{(k)}(x,t)=B_{1,N}\frac{1}{\Gamma(2-\gamma)}\int_{0}^t (t-\tau)^{1-\gamma}\tau^{\gamma-2}E_{\alpha,\gamma-1}(-\lambda_{N}^{2}\tau^{\alpha})d\tau+
$$
$$
+B_{2,N}\frac{1}{\Gamma(2-\gamma)}\int_{0}^t (t-\tau)^{1-\gamma}\tau^{\gamma-1}E_{\alpha,\gamma}(-\lambda_{N}^{2}\tau^{\alpha})d\tau+
$$
$$
+\frac{1}{\Gamma(2-\gamma)}\int_{0}^t (t-\tau)^{1-\gamma}\int_{0}^{\tau}(\tau-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}(\tau-s)^{\alpha})f_{N}(s)d\tau ds=
$$
$$
=B_{1,N}F_{3,N}(t)+B_{2,N}F_{4,N}(t)+F_{5,N}(t).
$$
Here
$$
F_{3,N}(t)=\frac{1}{\Gamma(2-\gamma)}\int_{0}^t (t-\tau)^{1-\gamma}\tau^{\gamma-2}E_{\alpha,\gamma-1}(-\lambda_{N}^{2}\tau^{\alpha})d\tau=E_{\alpha,1}(-\lambda_{N}^{2}t^{\alpha}),
$$
$$
F_{4,N}(t)=\frac{1}{\Gamma(2-\gamma)}\int_{0}^t (t-\tau)^{1-\gamma}\tau^{\gamma-1}E_{\alpha,\gamma}(-\lambda_{N}^{2}\tau^{\alpha})d\tau=tE_{\alpha,2}(-\lambda_{N}^{2}t^{\alpha}),
$$
$$
F_{5,N}(t)=\frac{1}{\Gamma(2-\gamma)}\int_{0}^t f_{N}(s) ds \int_{s}^{t}(t-\tau)^{1-\gamma}(\tau-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}(\tau-s)^{\alpha})d\tau,
$$
$$
\int_{s}^{t}(t-\tau)^{1-\gamma}(\tau-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}(\tau-s)^{\alpha})d\tau=\left[z=\frac{\tau-s}{t-s}t\right]=
$$
$$
=\left(\frac{t-s}{t}\right)^{1+\alpha-\gamma}\int_{0}^{t}(t-z)^{1-\gamma}z^{\alpha-1}E_{\alpha,\alpha}\left[\left(\frac{t-s}{t}\right)^{\alpha}
z^{\alpha}\right]dz=
$$
$$
=\Gamma(2-\gamma)(\tau-s)^{1+\alpha-\gamma}E_{\alpha,2+\alpha-\gamma}(-\lambda_{N}^{2}(t-s)^{\alpha})
$$
or
$$
F_{5,N}(t)=\int_{0}^t (t-\tau)^{1+\alpha-\gamma)}E_{\alpha,2+\alpha-\gamma}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}(\tau) d\tau.
$$
$$
\frac{d}{dt}I_{0+}^{(2-\gamma)}u^{(k)}(x,t)=\sum_{N=1}^{\infty}\Big[-B_{1,N}\lambda_{N}^{2}t^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})+B_{2,N}E_{\alpha,1}(-\lambda_{N}^{2}t^{\alpha})+
$$
\begin{equation}\label{eq48}
+\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,1+\alpha-\gamma}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}(\tau) d\tau \Big]X_{N}^{(k)}(x).
\end{equation}
We assume, that
\begin{equation}\label{eq49}
\varphi(x)=\sum_{N=1}^{\infty} \varphi_{N} X_{N}(x).
\end{equation}

By virtue of (\ref{eq2}) from (\ref{eq48}) we find that

\begin{equation}\label{eq50}
B_{2,N}=\varphi_{N}
\end{equation}
where
$$
\varphi_{N}=\left(\begin{array}{c} {\varphi_{N}^{(1)}} \\ {\varphi_{N}^{(2)}} \\ \varphi_{N}^{(3)} \end{array}\right), \,\,B_{N}=\left(\begin{array}{c} {B_{N}^{(1)}} \\ {B_{N}^{(2)}} \\ {B_{N}^{(3)}} \end{array}\right).
$$
Further integrating by parts two times the functions $\varphi^{(k)}(x)$ and considering conditions of the Theorem 2, we get:
\begin{equation}\label{eq51}
\varphi _{N} =\sum_k\int _{0}^{L_{k} }\varphi^{(k)} (x) X_{N}^{(k)} (x)dx=-\sum_k\frac{1}{\lambda_{N}^{2}} \int _{0}^{L_{k}} \frac{d^{2}}{dx^{2}}\varphi^{(k)} (x) X_{N}^{(k)} (x)dx,
\end{equation}

Then, in (\ref{eq2}), based on on the first condition, we get

$$
\sum_{N=1}^{\infty}\left(B_{1,N}F_{1,N}(0)+\varphi_{N}F_{4,N}(0)+F_{5,N}(0)\right)X_{N}^{(k)}(x)=
$$
$$
=M\sum_{N=1}^{\infty}\left(B_{1,N}F_{1,N}(T)+
\varphi_{N}F_{4,N}(T)+F_{5,N}(T)\right)X_{N}^{(k)}(x),
$$
hence
$$
B_{1,N}=MB_{1,N}F_{1,N}(T)+\varphi_{N} F_{4,N}(T)+F_{5,N}(T)
$$
or
$$
B_{1,N}=\frac{M}{1-MF_{1}(T)}\left(\varphi_{N} F_{4,N}(T)+F_{5,N}(T)\right).
$$

Finally, we get a formal solution of the Eq.(\ref{eq1}) satisfiying conditions (\ref{eq2})-(\ref{eq5}) in the following form:

$$
u^{(k)}(x,t)=\sum_{N=1}^{\infty}\Big[\frac{M}{1-MF_{1,N}(T)}\left(\varphi_{N} F_{4,N}(T)+F_{5,N}(T)\right)t^{\gamma-2} E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })+
$$
\begin{equation}\label{eq52}
+\varphi_{N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }) +\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left(-\lambda_{N}^{2}(t-\tau)^{\alpha}\right)f_{N}(\tau)d\tau \Big]X_{N}^{(k)}(x).
\end{equation}

It is required to prove the convergence of infinite series, corresponding to functions $u^{(k)}(x,t)$, $u_{xx}^{(k)} (x,t) $,

$D_{0+}^{\alpha,\mu } u^{(k)}(x,t)$
in the domain $B_{k}\times(0,T)$ similar to $l=1$.\\
Using the Lemma 5, Lemma 7 and relation (\ref{eq51}), we find
\begin{equation}\label{eq53}
\left|B_{2,N}\right|=\left|\varphi _{N}\right| \le\frac{m_{1}}{\lambda _{N}^{2} } \,\
\end{equation}
$$
\left|B_{1,N}\right|=\left|\frac{M}{1-MF_{1}(T)}\left(\varphi_{N} F_{4,N}(T)+F_{5,N}(T)\right)\right|\le
$$
\begin{equation}\label{eq54}
\le\left|\frac{M}{1-MF_{1}(T)}\right|\left|\varphi_{N} F_{4,N}(T)+F_{5,N}(T)
\right|\le \frac{m_{2}}{\lambda_{N}^{2}}.
\end{equation}
where $m_{1},m_{2}=const>0$.
From (\ref{eq47}), (\ref{eq53}), (\ref{eq54}) and (\ref{eq45}), we obtain
$$
\left|u^{(k)}(x,t)\right|=\Big|\sum _{N=1}^{\infty }\Big[\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
$$
+B_{1,N}t^{\gamma-2} E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })+B_{2,N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }) \Big]X_{N}^{(k)}(x)\Big|\le
$$
$$
\le \sum _{N=1}^{\infty }\Big|\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
$$
+B_{1,N}t^{\gamma-2} E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })+B_{2,N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }) \Big|\left|X_{N}^{(k)}(x)\right|\le
$$
$$
\le const \sum _{N=1}^{\infty }\Big(\int _{0}^{t}\left|t-\tau \right|^{\alpha-1}\left|E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]\right|\left|f_{N}(\tau)\right|d\tau+
$$
$$
+\left|B_{1,N}\right|t^{\gamma-2} \left|E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })\right|+\left|B_{2,N}\right|t^{\gamma-1} \left|E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha })\right|Big) \le
$$
$$
\le \sum _{N=1}^{\infty }\frac{m_{3}}{\lambda_{N}^{3}} \left|f_{N}(\tau)\right|+\sum _{N=1}^{\infty }\frac{m_{4}}{\lambda_{N}^{2}(1+\lambda_{N}^{2})}+\sum _{N=1}^{\infty }\frac{m_{5}}{\lambda_{N}^{2}(1+\lambda_{N}^{2})} \le
$$
$$
\le \sum _{N=1}^{\infty }\frac{m_{6}}{\lambda_{N}^{5}}+\sum _{N=1}^{\infty }\frac{m_{4}}{\lambda_{N}^{4}}+\sum _{N=1}^{\infty }\frac{m_{5}}{\lambda_{N}^{4}} \le\sum _{N=1}^{\infty }\frac{m_{7}}{\lambda_{N}^{4}},
$$
where $m_{i}=const>0, i=\overline{3,7}$ and $m_{7}=m_{4}+m_{5}+m_{6}$. So $u^{(k)}(x,t)$ are convergent.
Similarly, we obtain
$$
\left|u_{xx}^{(k)}(x,t)\right|=\Big|\sum _{N=1}^{\infty }-\lambda_n^2\Big[\int _{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha, \alpha}\left[-\lambda_{N}^{2}(t-z)^{\alpha}\right]f_{N}(\tau)d\tau+
$$
$$
+B_{1,N}t^{\gamma-2} E_{\alpha,\gamma-1} (-\lambda _{N}^{2} t^{\alpha })+B_{2,N}t^{\gamma-1} E_{\alpha,\gamma} (-\lambda _{N}^{2} t^{\alpha }) \Big]X_{N}^{(k)}(x)\Big|\le
$$

$$
\le\sum _{N=1}^{\infty }\frac{m_{7}}{\lambda_{N}^{2}}.

$$
We infer that the function $u_{xx}^{(k)} (x,t)$ be uniformly convergent, as well, owing to
$$
D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)=I_{0+}^{\gamma-\alpha}\frac{d^{2}}{dt^{2}}I_{0+}^{2-\gamma}u^{(k)} (x,t).
$$
Hence, applying the operator $I_{0+}^{\gamma-\alpha}\frac{d}{dt}(\cdot)$ to the (\ref{eq48}), we get
$$
\frac{d}{dt}I_{0+}^{2-\gamma}u^{(k)}(x,t)=\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\frac{d}{dt}\int_{0}^t \frac{d}{d\tau}\left[(t-\tau)^{\alpha-\gamma+1}E_{\alpha,\alpha-\gamma+2}(-\lambda_{N}^{2}(t-\tau)^{\alpha})\right]f_{N}(\tau) d\tau-
$$
$$
-\frac{d}{dt}\sum_{N=1}^{\infty}B_{1,N} X_{N}^{(k)}(x)t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})
+\frac{d}{dt}\sum_{N=1}^{\infty}B_{2,N}X_{N}^{(k)}(x)E_{\alpha,1}(-\lambda_{N}^{2}t^{\alpha})=
$$
$$
=-\sum_{N=1}^{\infty}f_{N}(0) X_{N}^{(k)}(x)t^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-
$$
$$
-\sum_{N=1}^{\infty}X_{N}^{(k)}(x)
\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau-
$$
$$
-\sum_{N=1}^{\infty}B_{1,N}\lambda_{N}^{2} X_{N}^{(k)}(x)t^{\alpha-2}E_{\alpha,\alpha-1}(-\lambda_{N}^{2}t^{\alpha})
+\sum_{N=1}^{\infty}B_{2,N}\lambda_{N}^{2}X_{N}^{(k)}(x)t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha}),
$$

$$
I_{0+}^{\gamma-\alpha}\frac{d^{2}}{dt^{2}}I_{0+}^{2-\gamma}u^{(k)}(x,t)=-\sum_{N=1}^{\infty}f_{N}(0) X_{N}^{(k)}(x)I_{0+}^{\gamma-\alpha}t^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-

$$
$$
-\sum_{N=1}^{\infty}X_{N}^{(k)}(x)
I_{0+}^{\gamma-\alpha}\int_{0}^t (t-\tau)^{\alpha-\gamma}E_{\alpha,\alpha-\gamma+1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau-
$$
$$
-\sum_{N=1}^{\infty}B_{1,N}\lambda_{N}^{2} X_{N}^{(k)}(x)I_{0+}^{\gamma-\alpha}t^{\alpha-2}E_{\alpha,\alpha-1}(-\lambda_{N}^{2}t^{\alpha})
+\sum_{N=1}^{\infty}B_{2,N}\lambda_{N}^{2}X_{N}^{(k)}(x)I_{0+}^{\gamma-\alpha}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})=
$$
$$
=-\sum_{N=1}^{\infty}f_{N}(0) X_{N}^{(k)}(x)E_{\alpha,1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-
$$
$$
-\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau-
$$
$$
-t^{\gamma-2}\sum_{N=1}^{\infty}B_{1,N}\lambda_{N}^{2}E_{\alpha,\gamma-1}(-\lambda_{N}^{2}t^{\alpha})X_{N}^{(k)}(x)-
t^{\gamma-1}\sum_{N=1}^{\infty}B_{2,N}\lambda_{N}^{2}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})X_{N}^{(k)}(x).
$$

Finally,
$$

D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)=-\sum_{N=1}^{\infty} X_{N}^{(k)}(x)\Big[ f_{N}(0) E_{\alpha,1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-\sum_{N=1}^{\infty}\int_{0}^t E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau-
$$
$$
-t^{\gamma-2}\sum_{N=1}^{\infty}B_{1,N}\lambda_{N}^{2}E_{\alpha,\gamma-1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-
t^{\gamma-1}\sum_{N=1}^{\infty}B_{2,N}\lambda_{N}^{2}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})\Big].
$$

Now consider $D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)$ for the convergence issue. Using the inequalities (\ref{eq43}),(\ref{eq45}), (\ref{eq53}),(\ref{eq54}), we heve

$$
\left|D_{0+}^{(\alpha, \mu) } u^{(k)} (x,t)\right|=\Big|\sum_{N=1}^{\infty}X_{N}^{(k)}(x)\Big[f_{N}(0) E_{\alpha,1}\left(-\lambda_{N}^{2}t^{\alpha}\right)-\int_{0}^t
E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau-
$$
$$
-B_{1,N}t^{\gamma-2}\lambda_{N}^{2}E_{\alpha,\gamma-1}(-\lambda_{N}^{2}t^{\alpha})-
B_{2,N}t^{\gamma-1}\lambda_{N}^{2}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})\Big]\Big|\leq
$$
$$
\leq const\Big[\left|\sum_{N=1}^{\infty}f_{N}(0) E_{\alpha,1}\left(-\lambda_{N}^{2}t^{\alpha}\right)\right|+\left|\sum_{N=1}^{\infty}\int_{0}^t
E_{\alpha,1}(-\lambda_{N}^{2}(t-\tau)^{\alpha})f_{N}^{'}(\tau) d\tau\right|+
$$
$$
+\left|t^{\gamma-2}\sum_{N=1}^{\infty}B_{1,N}\lambda_{N}^{2}E_{\alpha,\gamma-1}(-\lambda_{N}^{2}t^{\alpha})\right|+
\left|t^{\gamma-1}\sum_{N=1}^{\infty}B_{2,N}\lambda_{N}^{2}E_{\alpha,\alpha}(-\lambda_{N}^{2}t^{\alpha})\right|\Big]\leq
$$
$$
\leq \sum_{N=1}^{\infty}\frac{const}{\lambda_{N}^{2}}.
$$
From from the Lemma 6, according to the asymptotes of $\lambda_N \sim c\cdot n$ ($c$ is const), we can conclude that series of $D_{0+}^{\alpha,\mu}u^{(k)}(x,t)$ is uniformly convergent.

Since, we proved the uniform convergence of all infinite series, corresponding to the

solution and its appropriate derivatives, we can state that the equation (\ref{eq52}) defines the regular solution of the problem.\\

\begin{center}

\textbf{References}
\end{center}

\noindent 1. \textbf{M. M. Dzhrbashyan, A. B. Nersesyan} \emph{Fractional Derivatives and the Cauchy Problem for Fractional Differential Equations,} Izv. Akad. Nauk Armyan. SSR. 3, No 1 (1968), 3-29.

\noindent 2. \textbf{V. M. Bulavatsky.} \emph{Closed form of the solutions of some boundary problems for anomalous diffusion equation
with Hilfer's generalized derivative.} Cybernetics and Systems Analysis. 30(4), 2014, pp. 570-577.

\noindent 3. \textbf{E.T.Karimov, B.H.Toshtemirov.} \emph{Tricomi type problem with integral conjugation condition for a mixed type

equation with the hyper-Bessel fractional differential operator.} Bulletin of the Institute of Mathematics. 4(1)
9-14, (2019)

\noindent 4. \textbf{ I. Podlubny}, \emph{ Fractional Differential Equations}, Academic Press, San Diego. 1999.

\noindent 5. \textbf{ A. A. Kilbas, H. M. Srivstava, J. J. Trujillo},{\it Theory and applications of fractional differential equations}, in North-Holland Matematics Studies, Vol. 204, Elsevier Science. B.V., Amsterdam. 2006.

\noindent 6. \textbf{ S. G. Samko, A. A. Kilbas, O. I. Marichev}, {\it Fractional Integrals and Derivatives: theory and applications},

Amsterdam: Gordon and Breach science Publishers; 1993.

\noindent 7. \textbf{ Y. Kain, L. Oksanen, E. Soccorsi, M. Yamamoto},

{\it Global uniqueness in an inverse problem for time fractional diffusion equations},
Journal of Differential Equations, Elsevier,2018, 264 (2), pp.1146-1170. doi.org/10.1016/j.jde.2017.09.032.
.

\noindent 8. \textbf{ J. T. Kemppainen}, {\it Existence and uniqueness of the solution for a time-fractional diffusion equation},

Fractional Calculus and Applied Analysis, Vol.14, No.3, 2011, pp. 411-417.

\noindent 9. \textbf{ Y. S. Kivshar, G. P. Agarwal},{\it Optical Solitons: From Fibers to Photonic Crystals}, Academic, San Diego, 2003.

\noindent 10. \textbf{ T. Kottos, U. Smilansky},
{\it Periodic orbit theory and spectral statistics for quantum graphs},
Annals of Physics. Volume \textbf{274}, Issue 1, 1999, pp.76-124.

\noindent 11. \textbf{ S. Gnutzmann, U. Smilansky},{\it Quantum graphs: Applications to quantum chaos and universal spectral statistics},

Advances in Physics, \textbf{55} 2006, No 5-6, pp.527-625.

\noindent 12. \textbf{ O. Kh. Abdullaev, J. R. Khujakulov},

\emph{ On a problem for the time-fractional diffusion equation on a metric graphs},
Uzbek Mathematical Journal. No.4, pp. 3-12, 2017.

\noindent 13. \textbf{ J. R. Khujakulov},

{\it On inverse source problem for time fractional diffusion equation on simple metric graphs},
Uzbek Mathematical Journal. No.2, pp. 99-108, 2020.

\noindent 14. \textbf{ Ch. Seifert},{\it The linearized Korteweg-de-Vries equation on general metric graphs}, The Diversity and

Beauty of Applied Operator Theory, (2018), 449-458.

\noindent 15. \textbf{ A. V. Svetkova, A. I. Shafarevich},{\it The Cauchy Problem for the Wave Equation on Homogeneous Trees },

Mathematical Notes, 2016, Vol. 100, No. 6, pp. 862-869. DOI: 10.1134/S0001434616110262.

\noindent 16. \textbf{ Yu. V. Pokorniy, O. M. Penkin, V. L. Pryadiyev, A. V. Borovskix, K. P. Lazarev, S. A. Shabrov},

{ Differensialniye uravneniya na geometricheskix grafax (Differensial equations on geometric graphs) },
M.: FIZMATLIT, 2005. -272p (in Russian).

\noindent 17. \textbf{ V. Mehandiratta, M. Mehra},

{\it A difference scheme for the time-fractional diffusion equation on a metric star graph}, Applied Numerical Mathematics, 2020.

\noindent 18. \textbf{ K. M. Furati, M. D. Kassim, N. E. Tatar},

{\it Existence and uniqueness for a problem involving Hilfer fractional derivative}, Comput. Math. Appl., 64 (2012), 1616-1626.

\noindent 19. \textbf{Gu H., Trujillo J. J. } {Existence of mild solution for evolution equation with Hilfer fractional derivative.}

Appl. Math. Comput. 2015, 257, pp.344-354.

\noindent 20. \textbf{J. Wang, Y. Zhang } {Nonlocal initial value problems for differential equations with Hilfer fractional

derivative.} Appl. Math. Comput., 266 (2015), 850-859.

\noindent 21. \textbf{Karimov E. T., Sobirov Z. A., Khujakulov J. R. } {Solvability of a problem for a time fractional differential equation with the Hilfer operator on metric graphs} Bulletin of the Institute of Mathematics. 2021, Vol.4, No.4, pp. 9-18.

\noindent 22. \textbf{ Hilfer R., } {Applications of Fractional Calculus in Physics.} World Scientific, Singapore 2000.

\noindent 23. \textbf{ Hilfer R.,}{Experimental evidence for fractional time evolution in glass forming materials} Chem. Phys.,

284 (2002), 399-408.

\noindent 24. \textbf{Hilfer R., Luchko Y.,Tomovski Z.} {Operational method for solution of the fractional differential

equations with the generalized Riemann-Liouville fractional derivatives.} Fractional Calculus and
Applied Analysis, 2009, 12, pp.299-318.

\noindent 25. \textbf{Pao C. V.} {Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions.}

J. Math. Anal. Appl. 195 (1995), 702-718.

\noindent 26. \textbf{Shopolov N. N.} {Mixed problem with non-local initial condition for a heat conduction equation.}

Reports of Bulgarian Academy of Sciences 7 (1981), 935-936.

\noindent 27. \textbf{Rassias, J. M. and Karimov E. T.} {Boundary-value problems with non-local initial condition for degenerate

parabolic equations.} Contemp. Anal. Appl. Math. 1 (2013), 42-48.

\noindent 28. \textbf{Rassias J. M. and Karimov E. T.} {Boundary-value problems with non-local

initial condition for parabolic equations with parameter.} European J. Pure Appl. Math. 6 (2010), 948-957.

\noindent 29. \textbf{Kerefov A. A.} {The Gevrey problem for a certain mixed-parabolic equation.}

Differensialnye Uravneniya 13 (1977), 76-83. (in Russian)

\noindent 30. \textbf{E. Karimov, M. Mamchuev and M. Ruzhansky} {Non-local initial problem for second order time-fractional

and space-singular equation} Hokkaido Mathematical Journal Vol. 49 (2020) p. 349-361.

\noindent 31. \textbf{Erkinjon Karimov, Bakhodirjon Toshtemirov} {Non-local boundary value problem for a mixed-type equation

involving the bi-ordinal Hilfer fractional differential operators} Uzbek Mathematical Journal 2021, Volume 65,
Issue 2, pp.61-77. DOI: 10.29229/uzmj.2021-2-5

\noindent 32. \textbf{ A. V. Pskhu}, {\it Uravneniya v chastnykh proizvodnykh drobnogo poryadka

(Fractional Partial Differential Equations)}, Moscow: Nauka. 2005.

\noindent 33. \textbf{ Berkolaiko and P. Kuchment },

{\it Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths.},
In Spectral Geometry, volume 84 of Proceedings of Symposia in Pure Mathematics. American Math. Soc., Providence 117-137. 2012.

\noindent 34. \textbf{G. Berkolaiko and P. Kuchment}, {\it Introduction to Quantum Graphs},

Mathematical Surveys and Monographs.volume 186. AMS, 2013.

\noindent 35. \textbf{S. Gnutzmann, J. P. Keating, F. Piotet}, {\it Quantum ergodicity on graphs},

PHYSICAL REVIEW LETTERS. PRL 101, 264102 (2008) DOI: 10.1103/PhysRevLett.101.264102

\noindent 36. \textbf{G. Berkolaiko }, {\it An elementary introduction to quantum graphs}, arXiv: 1603.07356v2 [math-ph] 17 dec. 2016.

\noindent 37. \textbf{Kadirkulov B. J., Jalilov M. A.} {On a nonlocal problem for fourth-order mixed type equation
with the Hilfer operator.} Bulletin of the Institute of Mathematics, 2020, No.1, pp.59-67 (in Russian).

\noindent 38. \textbf{ A. A. Alikhanov},

{\it A priori estimate for solutions of boundary value problems for fractional-order equations},
Differential equations. 2010.V.46.Issue 5.pp 660-666.

\noindent 39. \textbf{ Fatma Karo\c{c}} \emph{Existence and uniqueness for fractional order functional differensial equations with

Hilfer derivative.} \\ Differential equations and Applications.Valume 12, Number 4(2020), 323-336. doi:10.7153/dea-2020-12-21.

\noindent 40. \textbf{Joel Friedman and Jean-Pierre Tillich} \emph{Wave equations for graphs and the edge-based Laplacian.},

PACIFIC JOURNAL OF MATHEMATICS, Vol. 216, No. 2, 2004.
(https://msp.org/pjm/2004/216-2/pjm-v216-n2-p03-s.pdf).

\noindent 41. \textbf{Ç. À. Ñîáèðîâ, Ì. Ð. Ýøèìáåòîâ} \emph{Ìåòîä Ôîêàñà äëÿ óðàâíåíèÿ òåïëîïðîâîäíîñòè íà ìåòðè÷åñêèõ ãðàôàõ.},

Ñîâðåìåííàÿ ìàòåìàòèêà. Ôóíäàìåíòàëüíûå íàïðàâëåíèÿ, 2021, òîì 67, âûïóñê 4, 766-782,
(DOI: https://doi.org/10.22363/2413-3639-2021-67-4-766-782).

\noindent 42. \textbf{Ravshan Ashurov and Yusuf Fayziev} \emph{On the Nonlocal Problems in Time for Time-Fractional

Subdiffusion Equations},\\ Fractal and Fractional , 2022, 6(1), 41; (https://doi.org/10.3390/fractalfract6010041 ).
\end{document}