\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 293, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/293\hfil Identification of an unknown source term]
{Identification of an unknown source term for a time fractional
fourth-order parabolic equation}

\author[S. Aziz, S. A. Malik\hfil EJDE-2016/293\hfilneg]
{Sara Aziz, Salman A. Malik} 

\address{Sara Aziz \newline
Department of Mathematics,
COMSATS Institute of Information Technology,
Park Road, Chak Shahzad Islamabad, Pakistan}
\email{sara\_aziz\_pk@yahoo.com}

\address{Salman A. Malik \newline
Department of Mathematics,
COMSATS Institute of Information Technology,
Park Road, Chak Shahzad Islamabad, Pakistan}
\email{salman\_amin@comsats.edu.pk, salman.amin.malik@gmail.com}

\thanks{Submitted June 8, 2016. Published November 11, 2016.}
\subjclass[2010]{80A23, 65N21, 26A33, 45J05, 34K37, 42A16}
\keywords{Inverse problem; fractional derivative; integral equation;
 Riesz basis;
\hfill\break\indent  bi-orthogonal system of functions; Fourier series} 

\begin{abstract}
 In this article, we considered two inverse source problems for fourth-order
 parabolic differential equation with fractional derivative in time.
 Determination of a space dependent source term from the data given at
 some time $t=T$  is considered in one problem while other addresses the
 recovery of a time dependent source term from the integral type
 over-determination condition. Existence and uniqueness of the solution
 of both inverse source problems are proved. The stability results for
 the inverse problems are presented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}\label{intr}

We are concerned with the fourth-order parabolic equation
\begin{equation}\label{ProblemEq1}
D^{\alpha,\gamma}_{0_+}u(x,t)+ u_{xxxx}(x,t)= F(x,t),\quad (x,t)\in
\Omega:=[0,1]\times (0,T],
\end{equation}
with initial condition
\begin{equation}
I^{1-\gamma}_{0_+}u(x,t)|_{t=0}=\varphi (x),  \quad
x\in [0,1],\label{ProblemEq2}
\end{equation}
and nonlocal boundary conditions
\begin{gather}
u_x(0,t)=u_x(1,t),\quad u(0,t)=0,\label{ProblemEq3}\\
u_{xxx}(0,t)= u_{xxx}(1,t),\quad u_{xx}(1,t)=0,\quad t\in (0,T],
 \label{ProblemEq4}
\end{gather}
where $D^{\alpha,\gamma}_{0_+}(\cdot)$ stands for the generalized
left sided fractional derivative of order $\alpha$ and type $\gamma$ in the
time variable (also known as Hilfer fractional derivative),
introduced by Hilfer \cite{Hilfer} and is given by
\begin{equation}\label{GFD}
D^{\alpha,\gamma}_{0_+} w(t) := \Bigl[I^{(\gamma-\alpha)}_{0_+}
\frac{d}{dt} \Bigl(I^{(1-\gamma)}_{0_+}\Bigr)\Bigr] w(t),\quad
0<\alpha\leq\gamma<1.
\end{equation}
The  left sided fractional integral is defined by
\begin{equation}
I^\beta_{0_+} w(t)=\frac{1}{\Gamma(\beta)} \int_0 ^t
(t-\tau)^{\beta -1}w(\tau)d\tau, \quad t>0,\quad \beta>0,
\end{equation}
where $w\in L^1_{loc}[0,T]$, $0< t< T\leq\infty,$ is a locally
integrable real-valued function and $\Gamma(\cdot)$ is the Euler
gamma function. The fractional derivative in \eqref{ProblemEq1}
 interpolates the Riemann-Liouville
fractional derivative and Caputo fractional derivative for
$\gamma = \alpha$ and $\gamma = 1$, respectively.
The Riemann-Liouville fractional derivative may has singularity
at $t=0$ and usually has initial conditions in terms of fractional
integral whereas Caputo fractional derivative are used more frequently
in the literature because with Caputo derivative the initial conditions
are more natural \cite{lukashchuk}. Both Riemann-Liouville and Caputo fractional
derivatives can be used in the modelling of anomalous diffusion and
the fractional derivative $D^{\alpha,\gamma}_{0_+} (\cdot)$ has the
properties of both of these fractional derivatives.

The nonlocal boundary conditions such as in \eqref{ProblemEq3}-\eqref{ProblemEq4}
arise  when we cannot measure data directly at the boundary. Such type of
boundary conditions usually known as Samarskii-Ionkin boundary conditions
which arise from particle diffusion in turbulent plasma and in heat
propagation where the law of variation of total quantity of the heat
is given \cite{Ionkin}. For applications of more general nonlocal boundary
conditions see \cite{Cannon-1, Sapago-1, Skub-nonclassicalBCs}.

The direct problem for \eqref{ProblemEq1}-\eqref{ProblemEq4} is the
unique determination of $u(x,t)$  in $\bar{\Omega}$ such that
$u(\cdot,t)\in C^4[0,1]$, $D^{\alpha,\gamma}_{0_+}u(x,\cdot)\in
C(0,T]$, when the initial condition $\varphi(x)$ and the source term
$F(x,t)$ are given and continuous.
The direct problem with $\gamma = 1$ of homogenous
equation \eqref{ProblemEq1}, i.e., $F(x,t)=0$ with initial condition
$u(x,0)=au(x,1)+\phi(x)$ and boundary conditions
\eqref{ProblemEq3}-\eqref{ProblemEq4} was considered by Berdyshev et
al. in \cite{berdyshev}. They proved existence and uniqueness of the
regular solution of the direct problem. The main concern of this
paper are the following inverse problems related to
\eqref{ProblemEq1}-\eqref{ProblemEq4}.
\smallskip

\noindent\textbf{Inverse source problem I (ISP-I)}: For the first
problem, we suppose the source term $F(x,t)$ depends only on the
space variable, i.e., $F(x,t) = f(x)$. The inverse problem is to
determine the source term $f(x)$ and  $u(x,t)$ such that $u(x,t)$
satisfies the equation \eqref{ProblemEq1}-\eqref{ProblemEq4} from
$u(x,T)= \psi(x)$. Indeed, we are looking for the map
$$
\psi(x)\to \{f(x), u(x,t)\},\quad t<T.
$$
By a regular solution of the ISP-I we mean a pair of functions
$\{u(x,t), f(x)\}$ such that $u(\cdot,t)\in C^4[0,1]$,
 $D^{\alpha,\gamma}_{0_+}u(x,\cdot)\in C(0,T]$ and $f(x)\in C[0,1]$.
\smallskip

\noindent\textbf{Inverse source problem II (ISP-II)}: For the
second problem, we consider the source term as $F(x,t) = a(t)f(x,t)$.
We are interested in recovering the time dependent source term $a(t)$
and $u(x,t)$. The inverse source problems of
determination of a time dependent source term was considered by
many, for example see
\cite{Slodicka, Akbar, Wu-TDSOURCE}. Physically, such type of source; that is,
$a(t)f(x,t)$ arise in microwave heating process, in which the
external energy is supplied to a target at a controlled level,
represented by $a(t)$ and $f(x,t)$ is the local  conversion rate of
the microwave energy.

 For problem \eqref{ProblemEq1}-\eqref{ProblemEq4} the ISP-II is
not uniquely solvable an over-determination condition of integral
type given by
\begin{equation}\label{ODC}
\int_0^1xu(x,t) dx= g(t),\quad t\in[0,T],
\end{equation}
is considered, where $g(t)\in AC[0,T]$, the space of absolutely
continuous functions. The integral type condition arise naturally as
over-determination condition for recovering the time dependent source
term, in chemical engineering \cite{cannon}, fluid flow in porous
medium \cite{Ewing} and in some other applications see for example
\cite{prilepko-1, Kamynin}. A regular solution for the ISP-II is a
pair of functions
$\{u(x,t),a(t)\}$ such that $u(\cdot,t)\in C^4[0,1]$,
$D^{\alpha,\gamma}_{0_+}u(x,\cdot)\in C(0,T]$ and $a(t)\in C[0,T]$.

The spectral problem for \eqref{ProblemEq1}-\eqref{ProblemEq4} is  not
self-adjoint and a bi-orthogonal system of functions is constructed
from eigenfunctions of spectral and its adjoint problem.
We proved that both inverse problems are well posed in the sense of
Hadamard (see Section 3 and 4).

It is well known that the inverse problems for the parabolic
equations are ill-posed apart from this the inverse problems
considered here are not easy to handle due to the nonlocal boundary
conditions \eqref{ProblemEq3}-\eqref{ProblemEq4} and the presence of
generalized fractional derivative in time. The fourth order parabolic
differential equations have been considered in applications to
combustion theory \cite{Bebernes-fourthorder3}, image smoothing and
denoising \cite{Lysaker-fourthorder1, patrick-fourthorder5},
incompressible elasticity problem, phase transition and surface
tension problem \cite{Bleher-fourthorder4}, thin film theory,
lubrication theory \cite{Ancona-fourthorder2}.

The calculus of arbitrary order integrals and derivative usually
known as fractional calculus could be considered as old as integer
order calculus. For the history of the subject the interested
readers are referred to \cite{mainardi-histFC}. Fractional calculus
got considerable attention in mathematics and other fields of
science, because fractional integrals and derivatives were used in
the modeling of many physical, chemical, biological process (see
the monographs \cite{mainardi-book,tarasov-book}).


Let us dwell with some of the articles which considered the inverse
problems related to time fractional parabolic equations. A stable
algorithm using mollification techniques has been proposed by Murio
\cite{MurioFracInvCMA} for the inverse problem of boundary function
for time fractional diffusion equation from a given noisy
temperature distribution.



Kirane et al \cite{malik-inv2D} considered two dimensional inverse
source problem for time fractional diffusion equation and prove the
well posedness of the inverse source problem. Jin and Rundell
\cite{JinRundell2012} consider the problem of recovering a spatially
varying potential for a one dimensional time fractional diffusion equation
from the flux measurements at a particular time. Li et al
\cite{YamamotoCite2013} propose algorithms for simultaneous inversion of
order of fractional derivative and a space dependent diffusion coefficient
for a one dimensional time fractional diffusion equation.
Li and Yamamoto \cite{YamamotoCite2015} considered the recovery of orders
of fractional derivatives for a multi term time fractional diffusion equation.
The determination of orders of space and time fractional derivatives for
space-time fractional diffusion equation was considered by Tatar et
al \cite{Tatar4}. Furati et al \cite{FURATI} proved existence and uniqueness
results for the solution of the inverse source problem posed for the heat
equation involving generalized fractional derivative given by \eqref{GFD}.
 Direct and inverse problems for fourth order parabolic equation with
fractional derivative in time was considered in \cite{Berd2016}.
For time fractional diffusion equation, determination of a time dependent
source was considered in \cite{Ismailov2016}. Liu et al \cite{Yamamoto2016}
considered reconstruction of time dependent boundary sources for time
fractional diffusion equation. The inverse problems of recovering the
space dependent sources for time fractional diffusion equations were
considered in \cite{Halyna2016}, \cite{Wei2016}.

The rest of the paper is organized as follows: in Section \ref{PRE},
we recall some basic definitions needed in the sequel and provide
the statements of our main results. Section \ref{main results},
presents our results concerning the existence, uniqueness and
continuous dependence of the solution of ISP-I. In Section 4 we give
the solution of ISP-II. In the last section we provide some examples.

\section{Preliminaries and statements of the main results}\label{PRE}

In this section, we provide some basic definitions, notations from fractional
calculus (for more details see
\cite{samko-book}) and statements of our main results.

The left sided Riemann-Liouville fractional derivative of order
$0<\alpha<1$ is defined by
\begin{equation}\label{RL-Derivative}
D^{\alpha}_{0_+}f(t):= \frac{d}{dt}I^{1-\alpha}_{0_+}f(t)
=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int^t_0
\frac{f(\tau)}{(t-\tau)^{\alpha}}d\tau.
\end{equation}
The Riemann-Liouville fractional derivative of a constant is not
equal to zero.

For $f\in AC[0,T]$ the left-hand sided Caputo fractional derivative of
order $0<\alpha<1$ is defined by
\begin{equation}\label{Caputo-Derivative}
{}^CD^{\alpha}_{0_+}f(t):= I^{1-\alpha}_{0_+}\frac{d}{dt}f(t)
=\frac{1}{\Gamma(1-\alpha)}\int^t_0\frac{f'(\tau)}{(t-\tau)^{\alpha}}d\tau.
\end{equation}
Notice that the generalized fractional derivative
$D^{\alpha, \gamma}_{0_+}$ reduces to the Riemann-Liouville fractional
derivative and Caputo fractional derivative for $\gamma =\alpha$ and
$\gamma=1$, respectively,
 $$
 D^{\alpha,\alpha}_{0_+}w(t) :=D^\alpha_{0_+} w(t), \quad
D^{\alpha,1}_{0_+}w(t):= {}^CD^\alpha_{0_+} w(t),
 $$
 where $D^\alpha_{0_+} w(t)$ and $ {}^CD^\alpha_{0_+} w(t) $
are the left sided Riemann-Liouville and Caputo fractional derivatives of
order $0<\alpha<1$   given by \eqref{RL-Derivative} and
\eqref{Caputo-Derivative}, respectively.
The Laplace transform of the generalized fractional derivative
\eqref{GFD} is given by \cite{Hilfer},
\begin{equation}\label{LAPLACE-GFD}
\mathcal{L}\{D^{\alpha,\gamma}_{0_+} f(t)\} = s^\alpha
\mathcal{L}\{f(t)\} - s^{\alpha-\gamma} I^{1-\gamma}_{0_+}
f(t)\Bigr|_{t=0},\quad 0<\alpha\leq\gamma<1.
\end{equation}

Let $\mathcal{H}$ be a Hilbert space with the inner product
$\langle\cdot ,\cdot\rangle$. A set of functions $\mathfrak{F}$ in
$\mathcal{H}$ is called complete in the interval $I$ if there exists
no function $f$ in $\mathcal{H}$, essentially different from zero,
which is orthogonal to all the functions of the set $\mathfrak{F}$
in the interval $I$. Two sets $S_1$ and $S_2$ of functions of
$\mathcal{H}$ form a bi-orthogonal system of functions if a
one-to-one correspondence can be established between them such that
the scalar product of two corresponding functions is equal to unity
and the scalar product of two non-corresponding functions is equal
to zero, i.e.,
\[
\langle f_i,g_j \rangle = \delta_{ij}
=\begin{cases} 1 & i=j\\
0&i\neq j
\end{cases}
\]
where $f_i\in S_1$, $g_i\in S_2$ and $\delta_{ij}$ is the Kronecker
symbol. The bi-orthogonal system is complete in $\mathcal{H}$ if the
sets $S_1$ and $S_2$ forming bi-orthogonal system are complete in
$\mathcal{H}$.

The Mittag-Leffler function for any $z \in \mathbb{C}$ with parameter
$\xi$ is given by
\begin{equation}\label{e:Mittag-Leffler-1P}
E_{\xi}(z)=\sum  ^{+\infty}_{k=0}\frac{z^{k}}{\Gamma(\xi
k+1)}\quad \operatorname{Re} \xi>0.
\end{equation}
Notice that for $\xi=1$, we have $E_{1}(z)=e^{z}.$

The Mittag-Leffler type function of two parameters
$E_{\xi,\beta}(z)$ which is a generalization of
\eqref{e:Mittag-Leffler-1P} is defined by
\begin{equation}\label{e:Mittag-Leffler-2P}
E_{\xi,\beta}(z)=\sum ^{+\infty}_{k=0}\frac{z^{k}}{\Gamma(\xi k+\beta)}, \quad
z, \beta \in \mathbb{C};\quad \operatorname{Re} \xi>0.
\end{equation}

The Mittag-Leffler type functions $ E_{\xi}(-\mu t^{\xi})$ and
$t^{\beta-1} E_{\xi,\beta}(-\mu t^\xi)$ for $\mu>0$,
$0<\xi\leq \beta\leq 1$ are \emph{completely
monotone} functions, i.e.,
\begin{equation}\label{monotonicity}
(-1)^n[E_{\xi}(-\mu t^{\xi})]^{(n)}\geq0,\quad
(-1)^n[t^{\beta-1}E_{\xi,\beta}(-\mu t^\xi)]^{(n)}\geq0,\quad
n\in\mathbb{N}\cup\{0\}.
\end{equation}

The function $E_{\xi,\beta}$ is an entire function
\cite{prabhakar} and thus is bounded in any finite interval, that is
$$
E_{\xi,\beta}(\mu t^{\xi})\leq M, \quad t\in [b,c],\; b\geq 0,
$$
for some positive constant $M$ and furthermore, we have
\begin{equation}\label{MITTAGLEFFLER-B}
\int^t_0 \tau^{\beta-1}E_{\xi,\beta}(\mu \tau^{\xi}) d\tau<\infty,\quad
\text{for } t\in[b,c],
\end{equation}
(see \cite[page 9]{prabhakar}). The Mittag-Leffler type function
$t^{\beta-1}E_{\xi,\beta}(z)$ whose fractional integral is
\begin{equation}
I^{1-\gamma}_{0_+}[t^{\beta-1}E_{\xi,\beta}(\lambda
t^\xi)]=t^{\beta-\gamma}E_{\xi,\beta-\gamma+1}(\lambda
t^\xi), \quad 0\leq\gamma\leq 1,\; \xi,\; \beta >0,
\; \lambda \in \mathbb{R},\label{fractional integral}
\end{equation}
plays an important role in the forthcoming sections.

The Laplace transform of $t^{\beta-1}E_{\xi,\beta}(\lambda t^\xi)$ is
\begin{equation}\label{LAPLACE-MLF}
\mathcal{L}\{t^{\beta-1}E_{\xi,\beta}(\lambda
t^\xi)\}=\frac{s^{\xi-\beta}}{(s^\xi -\lambda)}, \quad \operatorname{Re}
s>0, \; |\lambda s^{-\xi}| <1,
\end{equation}
where $\xi, \beta,\lambda\in \mathbb{C}$, $\operatorname{Re} \xi>0$ and
$\operatorname{Re} \beta>0$.
Also from \cite{podlubny}, we have
\begin{equation} \label{MLF Estimate}
\lambda t^\xi| E_{\xi,\beta}(-\lambda t^\xi)|
\leq \mathcal{M}, \quad 0<\xi<2,\; \beta \in\mathbb{C},\;
t\geq 0,\; \lambda\geq 0,
\end{equation}
for some constant $\mathcal{M}>0$.

For ISP-I we have the following results:

\begin{theorem} \label{thm2.1} % main
Suppose following conditions hold:
\begin{itemize}
\item[(1)] $\varphi(x)\in C^5[0,1]$ be such that $\varphi(0)=0$,
$\varphi'(0)=\varphi'(1)$, $\varphi''(1)=0=\varphi^{iv}(0)$ and
$\varphi'''(0)=\varphi'''(1)$.

\item[(2)]  $\psi(x)\in C^5[0,1]$ be such
that $\psi(0)=0$, $\psi'(0)=\psi'(1)$, $\psi''(1)=0=\psi^{iv}(0)$ and
$\psi'''(0)=\psi'''(1)$.
\end{itemize}
Then, there exist a regular solution of the ISP-I.
\end{theorem}


\begin{theorem} \label{thm2.2}
A regular solution of the ISP-I (if it exists) is unique.
\end{theorem}


\begin{theorem} \label{thm2.3} % stability
The solution of the ISP-I, under the assumptions of Theorem \ref{thm2.1},
depends continuously on the given data.
\end{theorem}

For second inverse problem (ISP-II), we have the following results:

\begin{theorem} \label{thm2.4} %\label{THEOREM}
Suppose the following conditions hold:
\begin{itemize}

\item[(1)] $\varphi(x)\in C^4[0,1]$ be such that $\varphi(0)=0$,
$\varphi'(0)=\varphi'(1)$, $\varphi''(1)=0$ and
$\varphi'''(0)=\varphi'''(1)$.


\item[(2)] $f(\cdot,t)\in C^4[0,1]$ be such
that $f(0,t)=0$, $f_x(0,t)=f_x(1,t)$, $f_{xx}(1,t)=0$ and
$f_{xxx}(0,t)=f_{xxx}(1,t)$. Furthermore
$\int_0^1xf(x,t)\,dx\neq0$ and
$$
0<\frac{1}{M^*}\leq |\int_0^1
xf(x,t)\,dx|, \quad\text{where } M^*>0.
$$

\item[(3)] $g(t)\in AC[0,T]$ and $g(t)$ satisfies the
consistency condition $\int_0^1 x \varphi(x)\,dx=I^{1-\gamma}_{0_+}g(t)|_{t=0}$.
Then, the ISP-II has a regular solution, furthermore the regular solution
of the ISP-II is unique.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm2.5}
  A regular solution of the ISP-II (under the assumptions of Theorem \ref{thm2.4})
is unique.
\end{theorem}

\begin{theorem}  \label{thm2.6} % stabilityI I
The solution of the ISP-II, under the assumptions of Theorem \ref{thm2.4},
 depends continuously on the given data.
\end{theorem}

\section{Inverse Source Problem I}\label{main results}

In this section, we present proofs of our main results.
Before we proceed further let us construct a bi-orthogonal system of
functions consisting of eigenfunctions of the spectral problem
\eqref{ProblemEq1}--\eqref{ProblemEq4} and its adjoint problem.


\subsection{Construction of two Riesz basis for the space $L^2(0,1)$}

The spectral problem for the initial boundary value problem
\eqref{ProblemEq1}--\eqref{ProblemEq4} given by
\begin{gather}
X^{{iv}}(x)= \lambda X(x),\quad\quad x\in (0,1),\label{SProblemEq1}\\
X(0)=X''(1)=0,\quad X'(0)=X'(1),\quad
X'''(0)=X'''(1).\label{SProblemEq2}
\end{gather}
is non-self-adjoint and the adjoint problem of the spectral problem
\eqref{SProblemEq1}--\eqref{SProblemEq2} is
\begin{gather}
Y^{iv}(x)= \lambda Y(x),\quad\quad x\in (0,1),\label{ASProblemEq1}\\
Y(0)=Y(1),\quad Y''(0)=Y''(1),\quad Y'(0)=Y'''(1)=0.\label{ASProblemEq2}
\end{gather}

The set of eigenfunctions for the boundary value problem
\eqref{SProblemEq1}--\eqref{SProblemEq2}, corresponding to
eigenvalues $\lambda_0=0$ and $\lambda_n=(2\pi n)^4$, is
$$
\{X_0(x)= 2x,\; X_{2n-1}(x) = 2\sin 2\pi n x,\;
 X_{2n}(x)=\frac{e^{2\pi n x} -e^{2\pi n(1-x)}}{e^{2\pi n}-1}
 +\cos 2\pi n x\}
$$
for $n\in\mathbb{N}$ and is a complete set of functions in
$L^2(0,1)$. Furthermore, this set forms a Riesz basis for the space
$L^2(0,1)$ (see \cite[ Lemma 2, and Proposition 1]{berdyshev}). The
set of eigenfunctions is not orthogonal as
$$
\int_0^1 X_0(x) X_{2n-1}dx\neq 0.
$$
For the adjoint problem \eqref{ASProblemEq1}--\eqref{ASProblemEq2},
the eigenfunctions corresponding to eigenvalues $\lambda_0=0$ and
$\lambda_n=(2\pi n)^4$ are given by
$$
\{Y_0(x)= 1,\; Y_{2n-1}(x)= \frac{e^{2\pi n x} +e^{2\pi
n(1-x)}}{e^{2\pi n}-1}+\sin 2\pi n x,\; Y_{2n}(x)= 2\cos 2\pi
nx\}.
$$


The set of functions form a bi-orthogonal system of functions under
the following one-to-one correspondence
\begin{alignat*}{3}
\{&\underbrace{X_0(x)}_\downarrow,& \quad
      &\underbrace{X_{2n-1}(x)}_\downarrow,& \quad
      &\underbrace{X_{2n}(x)}_\downarrow \},\\
\{&Y_0(x), &\quad &Y_{2n-1}(x),&\quad &Y_{2n}(x)\},
\end{alignat*}
i.e., $\langle X_i,Y_j \rangle= \delta_{ij}$ for
$i,j=0,2n-1,2n$, for $n\in\mathbb{N}$, where
$$
\langle g_1,g_2\rangle:=\int_0^1g_1(x)g_2(x)\,dx.
$$
We are in a position to present the proof of the Theorem \ref{thm2.1}.



\begin{proof}[Proof of Theorem \ref{thm2.1}]
 Expanding $u(x,t)$ and $f(x)$ using bi-orthogonal system of functions, we have
\begin{gather}\label{u(x,t)1}
u(x,t)= u_0(t)X_0(x)+\sum^{\infty}_{n=1}
u_{2n-1}(t)X_{2n-1}(x)
+\sum^{\infty}_{n=1}u_{2n}(t)X_{2n}(x),\\
f(x)= f_0X_0(x)+\sum^{\infty}_{n=1} f_{2n-1}X_{2n-1}(x)
+\sum^{\infty}_{n=1}f_{2n}X_{2n}(x), \label{f(x)}
\end{gather}
where $u_0(t), u_{2n-1}(t)$, $u_{2n}(t),$ $f_0, f_{2n-1}$, and
$f_{2n}$ for $n\in\mathbb{N}$, are unknowns to be determined.

From the expansion of $u(x,t)$ given by \eqref{u(x,t)1} and using
properties of the bi-orthogonal system of functions, we have
\begin{gather*}
u_0(t) = \langle u(x,t),Y_0(x)\rangle,\quad
u_{2n-1}(t) = \langle u(x,t),Y_{2n-1}(x)\rangle,\\
u_{2n}(t) = \langle u(x,t),Y_{2n}(x)\rangle.
\end{gather*}
Consider
$$
u_{2n-1}(t) = \langle u(x,t),Y_{2n-1}(x)\rangle:=\int_0^1 u(x,t) Y_{2n-1}\;dx\,.
$$
Taking the fractional derivative under the integral and using
\eqref{ProblemEq1} with $F(x,t)=f(x)$, we have
$$
D_{0_+}^{\alpha,\gamma}u_{2n-1}(t)
=-\int_0^1 u_{xxxx} Y_{2n-1}(x)\;dx+\int_0^1f(x) Y_{2n-1}(x)\;dx.
$$
Integrating by parts and using the boundary conditions
\eqref{ProblemEq3}--\eqref{ProblemEq4}, we obtain
\begin{equation}\label{LinearFDE1}
D_{0_+}^{\alpha,\gamma}u_{2n-1}(t)+\lambda_nu_{2n-1}(t)=f_{2n-1}.
\end{equation}
Similarly, we have the  linear fractional differential equations
\begin{gather}
D_{0_+}^{\alpha,\gamma}u_0(t)=f_0,\label{LinearFDE2}\\
D_{0_+}^{\alpha,\gamma}u_{2n}(t)+\lambda_n
u_{2n}(t)=f_{2n}.\label{LinearFDE3}
\end{gather}
Taking Laplace transform of \eqref{LinearFDE1} and using formula
\eqref{LAPLACE-GFD}, we obtain
$$
\mathcal{L}\{u_{2n-1}(t)\}=I^{1-\gamma}_{0_+}u_{2n-1}(t)r|_{t=0}
\Bigl(\frac {s^{\alpha-\gamma}}{s^\alpha+\lambda_n}\Bigr)
+\frac {f_{2n-1}}{s(s^\alpha +\lambda_n)}.
$$
The solution of \eqref{LinearFDE1} is obtained by applying inverse Laplace
transform, formula \eqref{LAPLACE-MLF} and
$\mathcal{L}^{-1}(\mathcal{L}\{f_1(t)\} \mathcal{L}\{f_2(t)\}) = (f_1 * f_2)(t)$,
\begin{equation}
\begin{aligned}
u_{2n-1}(t)
&= I^{1-\gamma}_{0_+}u_{2n-1}(t)\Bigr|_{t=0}t^{\gamma-1}E_{\alpha,\gamma}
 (-\lambda_nt^{\alpha})  \\
&\quad +f_{2n-1}\int_0^t \tau^{\alpha-1}E_{\alpha,\alpha}
 (-\lambda_n\tau^{\alpha})d\tau.
\end{aligned} \label{u-solution1}
\end{equation}
Similarly, the solutions of \eqref{LinearFDE2} and \eqref{LinearFDE3}
are given by
\begin{gather}
u_0(t)=I^{1-\gamma}_{0_+}u_0(t)\Bigr|_{t=0}\frac{t^{\gamma-1}}{\Gamma(\gamma)}
 +f_0\frac{t^{\alpha}}{\Gamma{(\alpha+1)}},\label{u-solution2}\\
u_{2n}(t)=I^{1-\gamma}_{0_+}u_{2n}(t)\Bigr|_{t=0}t^{\gamma-1}E_{\alpha,\gamma}
(-\lambda_nt^{\alpha})+ f_{2n}\int_0^t \tau^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n\tau^{\alpha})d\tau, \label{u-solution3}
\end{gather}
respectively. By the initial condition \eqref{ProblemEq2}, we have
$$
I^{1-\gamma}_{0_+}u_0(t)r|_{t=0} = \varphi_0,\quad
I^{1-\gamma}_{0_+}u_{2n-1}(t)r|_{t=0} = \varphi_{2n-1},\quad
I^{1-\gamma}_{0_+}u_{2n}(t)r|_{t=0}=\varphi_{2n},
$$
where $\varphi_0,\varphi_{2n-1}$ and $\varphi_{2n}$ are the coefficients
of series expansion of $\varphi(x)$ when expanded using the bi-orthogonal
system and are given by
\begin{equation}  \label{Coefficents-bi-orthogonal}
\begin{gathered}
\varphi_0 =\int_0^1\varphi(x)Y_0(x)\;dx,\quad
\varphi_{2n-1} =\int_0^1\varphi(x)Y_{2n-1}(x)\;dx,  \\
\varphi_{2n} =\int_0^1\varphi(x)Y_{2n}(x)\;dx.
\end{gathered}
\end{equation}
Alike, using the condition $u(x,T)=\psi(x),$ we have
\begin{equation}\label{OverdetCondition}
u_0(T) = \psi_0,\quad u_{2n-1}(T) = \psi_{2n-1},\quad u_{2n}(T)=\psi_{2n},
\end{equation}
where $\psi_0, \psi_{2n-1}$ and $\psi_{2n}$ are the coefficients of
series expansion of the function $\psi(x)$ in terms of the
bi-orthogonal system of functions.
\end{proof}

Before we proceed further let us fix some notation
$$
\mathcal{E}^{(1)}_{n}(t):=t^{\gamma-1}E_{\alpha,\gamma}(-\lambda_nt^{\alpha}),\quad
\mathcal{E}^{(2)}_{n}(t):=\int_0^t \tau^{\alpha-1}
E_{\alpha,\alpha}(-\lambda_n\tau^{\alpha})d\tau.
$$
By using these notation and taking
\eqref{u-solution1}--\eqref{u-solution3} into account we can write
\begin{gather*}
  {u}_0(t)
= \varphi_0\frac{t^{\gamma-1}}{\Gamma(\gamma)}
 +f_0\frac{t^{\alpha}}{\Gamma{(\alpha+1)}},\label{uSeries-1} \\
\quad{u}_{2n-1}(t)=\varphi_{2n-1}\mathcal{E}^{(1)}_{n}(t)+f_{2n-1}
 \mathcal{E}^{(2)}_{n}(t), \label{uSeries-2}\\
{u}_{2n}(t)=\varphi_{2n}\mathcal{E}^{(1)}_{n}(t)+f_{2n}\
mathcal{E}^{(2)}_{n}(t)\label{uSeries-3}.
\end{gather*}
Due to \eqref{OverdetCondition}--\eqref{uSeries-3} the unknowns
$f_0,f_{2n-1},f_{2n}$ are determined as
\begin{gather}
f_0=\Bigl(\psi_0-\frac{\varphi_0T^{\gamma-1}}{\Gamma(\gamma)}\Bigr)
\frac{\Gamma(1+\alpha)}{T^\alpha},\label{f-solution1}\\
 f_{2n-1}=\frac{\psi_{2n-1}-\varphi_{2n-1}\mathcal{E}^{(1)}_{n}(T)}
{\mathcal{E}^{(2)}_{n}(T)},\quad\label{f-solution2}\\
f_{2n}=\frac{\psi_{2n}-\varphi_{2n}
\mathcal{E}^{(1)}_{n}(T)}{\mathcal{E}^{(2)}_{n}(T)}\label{f-solution3}.
\end{gather}
The solution of the ISP-I is given by the series
\eqref{u(x,t)1} and \eqref{f(x)}, where $u_0(t)$, $u_{2n-1}(t)$,
$u_{2n}(t)$, $ f_0, f_{2n-1}$ and  $f_{2n}$ given by
\eqref{uSeries-1}--\eqref{f-solution3}, respectively.


Before proceeding further, we recall \cite[Lemma 5 on page 89]{Kesel'man}.

\begin{lemma} \label{berdyshev}
 Let $f\in L^2(0,1)$ and
$$
a_n=\int_0^1f(x)e^{\mu n(x-1)}dx,\quad b_n=\int_0^1f(x)e^{-\mu  nx}dx,
$$
 where $\mu$ is any complex number such that $\operatorname{Re}  \mu>0$.
Then the  series
$$
\sum_{n=1}^\infty |a_n|^2,\quad
\sum_{n=1}^\infty |b_n|^2
$$
 are convergent.
\end{lemma}

\noindent \textbf{Existence of the solution of the ISP-I}:
To show that the solution of the inverse problem represented by the series
\eqref{u(x,t)1} and \eqref{f(x)} is a regular solution we need to show that
\begin{itemize}
\item The series corresponding to $u(x,t)$,
$u_x(x,t), u_{xx}(x,t), u_{xxx}(x,t), u_{xxxx}(x,t)$, and
$D^{\alpha,\gamma}_{0_+}u(x,t)$ represent continuous functions.

\item The series corresponding to $f(x)$ is continuous on $[0,1]$.
\end{itemize}
Let \begin{equation}
u(x,t) = \mathcal{W}_0+\sum_{n=1}^{\infty}\mathcal{W}_{2n-1}
+\sum_{n=1}^{\infty}\mathcal{W}_{2n}\label{u(x,t)},
\end{equation}
where $\mathcal{W}_0 =u_0(t)X_0(x)$,
$\mathcal{W}_{2n-1}={u}_{2n-1}(t)X_{2n-1}(x)$,
$\mathcal{W}_{2n}=u_{2n}(t)X_{2n}(x)$,
 and $u_0(t),{u}_{2n-1}(t)$
and ${u}_{2n}(t)$ are given by \eqref{uSeries-1}--\eqref{uSeries-3}.

We shall show that all the series involved in \eqref{u(x,t)} represents
a continuous function on
 $\Omega_\epsilon:=[0,1]\times[\epsilon,T]$ for $\epsilon>0$. By using \eqref{MLF
Estimate} the bound for $\mathcal{E}^{(1)}_{n}(t)$ is obtained as
\begin{equation}\label{mlf E1}\mathcal{E}^{(1)}_{n}(t)\leq
\frac{C_1}{t^{1+\alpha-\gamma}\lambda_n},\quad t\in
[\epsilon,T],\end{equation} and using \eqref{MITTAGLEFFLER-B}, we can have
$$
\mathcal{E}^{(2)}_{n}(t)\leq C_2, \quad t\in [\epsilon,T],
$$
where $C_1$ and $C_2$ are constants.
For some fixed time (say) $T$, using above estimates together with
\eqref{monotonicity}--\eqref{MITTAGLEFFLER-B},
we can choose
$\mathcal{M}_1$, and $\mathcal{M}_2$, independent of $n$, such that
$$
|\mathcal{E}^{(1)}_{n}(T)|\leq\mathcal{M}_1, \quad
|\mathcal{E}^{(2)}_{n}(T)|^{-1}\leq\mathcal{M}_2,\quad n\in\mathbb{N}.
$$

From \eqref{Coefficents-bi-orthogonal} and integration by parts, we
have
$$
|\varphi_{2n-1}|=\frac{1}{\lambda_n}\langle \varphi^{iv}(x),Y_{2n-1}(x)\rangle,
\quad
|\varphi_{2n}|=\frac{\sqrt{2}}{(2\pi n)}
\langle\varphi'(x),\,\sqrt{2}\sin 2\pi n x\rangle,
$$
using elementary inequality $ab\leq 1/2(a^2+b^2)$  for all $a,b
\in \mathbb{R}$, we obtain
$$
|\varphi_{2n-1}|\leq\frac{1}{2}(\frac{1}{\lambda_n^2}+\mathcal{I}_n^2),
\quad |\varphi_{2n}|\leq\frac{1}{\sqrt{2}}
\{\frac{1}{(2\pi n)^2} +(\langle\varphi'(x),\,
\sqrt{2}\sin 2\pi n x\rangle)^2\},
$$
where $\mathcal{I}_n=\langle \varphi^{iv}(x),Y_{2n-1}(x)\rangle$. By
Lemma \ref{berdyshev} we conclude that the series
$\sum_{n=1}^{\infty}\mathcal{I}_n^2$ converges absolutely.
The sequence $\{\sqrt{2}\sin 2\pi n x\}_{n=1}^{\infty}$ is an
orthonormal sequence in $L^2(0,1)$, hence by Bessel's inequality, we have
$$
\sum_{n=1}^{\infty}|\varphi_{2n}|\leq\frac{1}{\sqrt{2}}
\{\sum_{n=1}^{\infty}\frac{1}{(2\pi n)^2}
+\|\varphi'(x)\|_{L^2(0,1)}^2\}.
$$
Also, we have
$$
|\varphi_0|=\langle\varphi(x), Y_0(x)\rangle
\leq 2\|\varphi(x)\|_{L^2(0,1)}.
$$
  Similarly, the estimates for $\psi_0,\psi_{2n-1}$ and $\psi_{2n}$
are obtained as
\begin{gather*}
|\psi_0|\leq 2\|\psi(x)\|_{L^2(0,1)},\quad
\sum_{n=1}^{\infty}|\psi_{2n-1}|
\leq \frac{1}{2}\Big(\frac{1}{\lambda_n^2}+\mathcal{J}_n^2\Big), \\
\sum_{n=1}^{\infty}|\psi_{2n}| \leq \frac{1}{\sqrt{2}}
\Big\{\sum_{n=1}^{\infty}\frac{1}{(2\pi n)^2}
 +\|\psi'(x)\|_{L^2(0,1)}^2\Big\},
\end{gather*}
where $\mathcal{J}_n=\langle \psi^{iv}(x),Y_{2n-1}(x)\rangle$. Consequently, from
\eqref{f-solution1}--\eqref{f-solution3}, we obtained the following estimates
\begin{gather}
T^{1+\alpha-\gamma}|f_0|
\leq 2C_3\left(\|\psi(x)\|_{L^2(0,1)}+\|\varphi(x)\|_{L^2(0,1)}\right),
\label{fseries1} \\
\sum_{n=1}^{\infty}|f_{2n-1}|
\leq \frac{\mathcal{M}_2}{2}\Big\{\frac{1}{\lambda_n^2}+\mathcal{J}_n^2
+\mathcal{M}_1\Big(\frac{1}{\lambda_n^2}+\mathcal{I}_n^2\Big)\Big\},
 \label{fseries2}  \\
\begin{aligned}
\sum_{n=1}^{\infty}|f_{2n}|
&\leq \frac{\mathcal{M}_2}{\sqrt{2}}
\Big\{\sum_{n=1}^{\infty}\frac{1}{(2\pi n)^2} +\|\psi'(x)\|_{L^2(0,1)}^2 \\
&\quad +\mathcal{M}_1\Big(\sum_{n=1}^{\infty}\frac{1}{(2\pi n)^2}
 +\|\varphi'(x)\|_{L^2(0,1)}^2\Big)\Big\}\label{fseries3},
\end{aligned}
\end{gather}
where
$$
C_3=\max\Big\{\frac
{\Gamma(1+\alpha)}{\Gamma(\gamma)},t^{1-\gamma}\Gamma(1+\alpha),
\frac{t^{\alpha}}{\Gamma(\gamma)},
\frac{t^{1+2\alpha-\gamma}}{\Gamma(1+\alpha)}\Big\},
$$
for all $t\in [\epsilon,T]$. From estimates
\eqref{fseries1}--\eqref{fseries3} the series expansion of $f(x)$
given by \eqref{f(x)} represents a continuous function on
$\Omega_{\epsilon}$.

Using \eqref{fseries1}--\eqref{fseries3} and $|X_n(x)|\leq2$
for $n\in\mathbb{N}\cup\{0\},$ we have the following estimates for
the series involved in \eqref{u(x,t)},
\begin{gather*}
t^{1+\alpha-\gamma}|\mathcal{W}_0|
\leq 4C_3\{\|\varphi(x)\|_{L^2(0,1)}+C_3(\|\psi(x)\|_{L^2(0,1)}
 +\|\varphi(x)\|_{L^2(0,1)})\},
\\
t^{1+\alpha-\gamma}\sum_{n=1}^{\infty}|\mathcal{W}_{2n-1}|
\leq 2\Big[\frac{C_1C_4}{\lambda_n}+
\frac{t^{1+\alpha-\gamma}C_2\mathcal{M}_2}{2}
\Big\{\frac{1}{\lambda_n^2}+\mathcal{J}_n^2
+\mathcal{M}_1 \Big(\frac{1}{\lambda_n^2}+\mathcal{I}_n^2\Big)\Big\}\Big],
\\
\begin{aligned}
t^{1+\alpha-\gamma}\sum_{n=1}^{\infty}|\mathcal{W}_{2n}|
&\leq 2\Big[\frac{C_1C_5}{\lambda_n}+\frac{t^{1+\alpha-\gamma}C_2\mathcal{M}_2}
{\sqrt{2}}\Big\{\sum_{n=1}^{\infty}\frac{1}{(2\pi n)^2}
+\|\psi'(x)\|_{L^2(0,1)}^2 \\
&\quad +\mathcal{M}_1(\sum_{n=1}^{\infty}\frac{1}{(2\pi n)^2}
 +\|\varphi'(x)\|_{L^2(0,1)}^2)\Big\}\Big],
\end{aligned}
\end{gather*}
where $C_4$ and $C_5$ are positive constants such that
$$
\sum_{n=1}^{\infty}|\varphi_{2n-1}|\leq C_4,\quad \text{and}\quad
\sum_{n=1}^{\infty}|\varphi_{2n}|\leq C_5.
$$
Thus all the series in \eqref{u(x,t)} are bounded above by uniformly
convergent numerical series. Consequently, by Weierstrass M-test the
series expansion of $u(x,t)$ given by \eqref{u(x,t)} is uniformly
convergent in $\Omega_\epsilon$.

Notice that
$$
X^{iv}_0(x)=0, \quad X^{iv}_{2n-1}(x)=\lambda_n X_{2n-1}(x),\quad
X^{iv}_{2n}(x)=\lambda_nX_{2n}(x).
$$
Let us show that the series representation of $u_{xxxx}(x,t)$
obtained from \eqref{u(x,t)} is uniformly convergent series.

Integration by parts leads us to the following estimates
\begin{gather}
|\varphi_{2n-1}|=\frac{1}{(2\pi n)^5}\langle\varphi^{v}(x), \quad
\frac{e^{2\pi n x}  -e^{2\pi n(1-x)}}{e^{2\pi n}-1}-\cos 2\pi n x\rangle
=\frac{\mathcal{I}^*_n}{(2\pi n)^5},\label{varphi1}
\\
|\varphi_{2n}|=\frac{1}{(2\pi n)^5}\langle\varphi^{v}(x),2\sin(2\pi
nx)\rangle
\leq\frac{\sqrt{2}}{(2\pi n)^5}\|\varphi^{v}(x)\|_{L^2(0,1)}\label{varphi2},
\\
|\psi_{2n-1}|=\frac{1}{(2\pi n)^5}\langle\psi^{v}(x),\frac{e^{2\pi n x}
-e^{2\pi n(1-x)}}{e^{2\pi n}-1}-\cos 2\pi n x\rangle
=\frac{\mathcal{J}^*_n}{(2\pi n)^5},\label{psi1}
\\
|\psi_{2n}|=\frac{1}{(2\pi n)^5}\langle\psi^{v}(x),2\sin(2\pi nx)\rangle
\leq\frac{\sqrt{2}}{(2\pi n)^5}\|\psi^{v}(x)\|_{L^2(0,1)}\label{psi2},
\end{gather}
where $\mathcal{I}_n^*=\langle\varphi^{v}(x)$,
$(e^{2\pi n x} -e^{2\pi n(1-x)})/(e^{2\pi n}-1)-\cos 2\pi n x\rangle$ and
\[
\mathcal{J}_n^*=\langle\psi^{v}(x),(e^{2\pi n x} -e^{2\pi n(1-x)})/(e^{2\pi n}-1)
-\cos 2\pi n x\rangle.
\]
 Using \eqref{varphi1}--\eqref{psi2} in
\eqref{f-solution1}--\eqref{f-solution3} the estimates for $f_{2n-1}$
and $f_{2n}$, are
\begin{gather}
|f_{2n-1}| \leq \mathcal{M}_2\{\frac{1}{(2\pi n)^5}\mathcal{J}^*_n
+\frac{\mathcal{M}_1}{\lambda_n}\mathcal{I}^*_n\},\label{f1}\\
|f_{2n}| \leq \mathcal{M}_2\{\frac{2}{(2\pi n)^5}\|\psi^{v}(x)\|_{L^2(0,1)}
+\frac{2\mathcal{M}_1}{(2\pi n)^5}\|\varphi^{v}(x)\|_{L^2(0,1)}\}\label{f2}.
\end{gather}
From \eqref{varphi1}--\eqref{f2} we have
\begin{gather}
t^{1+\alpha-\gamma}\sum_{n=1}^{\infty}
|\frac{\partial^4\mathcal{W}_{2n-1}}{\partial x^4}|
\leq \sum_{n=1}^{\infty}2\lambda_n
\Big\{\frac{C_1\mathcal{I}^*_n}{\lambda_n(2\pi n)^5}
+t^{1+\alpha-\gamma}\mathcal{M}_2C_2(\frac{\mathcal{J}^*_n}{(2\pi n)^5}
+\frac{\mathcal{M}_1\mathcal{I}^*_n}{(2\pi n)^5}) \Big\},
 \label{derivativeseries-1}\\
\begin{aligned}
t^{1+\alpha-\gamma}\sum_{n=1}^{\infty} |\frac{\partial^4\mathcal{W}_{2n}}{\partial
x^4}|
&\leq \sum_{n=1}^{\infty}2\lambda_n
\Big\{\frac{C_1\|\varphi^{v}(x)\|_{L^2(0,1)}}{\lambda_n(2\pi n)^5}
+t^{1+\alpha-\gamma}\mathcal{M}_2C_2  \\
&\times \Big(\frac{\|\varphi^{v}(x)\|_{L^2(0,1)}}{(2\pi n)^5}
+\frac{\mathcal{M}_1\|\psi^{v}(x)\|_{L^2(0,1)}}{(2\pi n)^5}\Big)
\Big\}.\label{derivativeseries-2}
\end{aligned}
\end{gather}
By using the inequality $2ab\leq (a^2+b^2)$ and Lemma
\ref{berdyshev} the series involved in
\eqref{derivativeseries-1}--\eqref{derivativeseries-2} are uniformly
convergent. Moreover by the assumptions on $\varphi(x)$ and
$\psi(x)$ it can be concluded that the series expansion of
$u_{xxxx}(x,t)$ is bounded above by convergent series and
represents a continuous function.

Next we show that the series corresponding to fractional
derivative $D^{\alpha,\gamma}_{0_+} u(x,t)$ is uniformly convergent,
i.e.,
$$
D^{\alpha,\gamma}_{0_+}\sum_{n=1}^{\infty} \mathcal{W}_{2n-1}(t),\quad
D^{\alpha,\gamma}_{0_+} \sum_{n=1}^{\infty}\mathcal{W}_{2n}(t),
$$
are uniformly convergent. From
\eqref{LinearFDE1}--\eqref{LinearFDE3}, we have
\begin{gather}
D^{\alpha,\gamma}_{0_+}\mathcal{W}_0 = f_0X_0(x),\\
\sum_{n=1}^{\infty}D^{\alpha,\gamma}_{0_+}\mathcal{W}_{2n-1}
= \sum_{n=1}^{\infty}\left[\lambda_n u_{2n-1}(t)+f_{2n-1}\right]X_{2n-1}(x),\\
\sum_{n=1}^{\infty}D^{\alpha,\gamma}_{0_+}\mathcal{W}_{2n}
= \sum_{n=1}^{\infty}\left[\lambda_nu_{2n}(t)+f_{2n}\right]X_{2n}(x).
\end{gather}
Using estimates \eqref{varphi1}--\eqref{f2} and Weierstrass M-test,
the series $\sum_{n=1}^{\infty}D^{\alpha,\gamma}_{0_+}\mathcal{W}_{2n-1}$
 and  $\sum_{n=1}^{\infty}D^{\alpha,\gamma}_{0_+}\mathcal{W}_{2n}$ are
uniformly convergent on $\Omega_\epsilon$.


At this stage let us recall the \cite[Lemma 15.2, page 278]{samko-book}.

\begin{lemma} \label{kilbas}
Let the fractional derivative $D^{\alpha,\gamma}_{0_+}f_n$ exists
for all $n\in\mathbb{N}$ and the series
$\sum_{n=1}^{\infty}f_n$ and
$\sum_{n=1}^{\infty}D^{\alpha,\gamma}_{0_+}f_{n}$ are
uniformly convergent on every subinterval $[\epsilon, b]$ for
$\epsilon > 0$ then
$$
D^{\alpha,\gamma}_{0_+}\Big(\sum_{n=1}^{\infty}f_{n}(x)\Big)
=\sum_{n=1}^{\infty}D^{\alpha,\gamma}_{0_+}f_{n}(x), \quad
0<\alpha\leq\gamma<1,\; 0<x<b.
$$
\end{lemma}

By the estimates \eqref{varphi1}--\eqref{f2} and Lemmas
\ref{kilbas} and \ref{lem3.3} it can be deduced that
 the series involved in $D^{\alpha,\gamma}_{0_+} u(x,t)$
are bounded above by uniformly convergent numerical series and hence
by Weierstrass M-test $D^{\alpha,\gamma}_{0_+} u(x,t)$ is uniformly
convergent.


\begin{proof}[Proof of Theorem \ref{thm2.2}]
 (Uniqueness of the solution of the ISP-I) \\
Suppose $\{u_1(x,t), f_1(x)\}$ and $\{u_2(x,t), f_2(x)\}$ are two solution sets
of the ISP-I, then $\bar{u}(x,t) = u_1(x,t)-u_2(x,t)$ and
$\bar{f}(x) = f_1(x)-f_2(x)$ satisfy
\begin{gather}
D_{0_+}^{\alpha,\gamma}\bar{u}(x,t)+\bar{u}_{xxxx}(x,t)
= \bar{f}(x),\hspace{19mm} (x,t)\in \Omega,\label{U-Problem-I2D1}\\
I_{0_+}^{1-\gamma}\bar{u}(x,t)\Bigr|_{t=0}=0,\quad \bar{u}(x,T)=0,
\quad x\in [0,1],\label{U-Problem-I2D2}\\
\bar{u}_x(0,t)=\bar{u}_x(1,t),\quad \bar{u}(0,t)=0,\quad
 t\in [0,T],\label{U-Problem-I2D3}\\
\bar{u}_{xxx}(0,t)=\bar{u}_{xxx}(1,t),\quad
\bar{u}_{xx}(1,t)=0,\quad t\in [0,T],\label{U-Problem-I2D4}
\end{gather}
Following the strategy in \cite{moiseev}, we consider the functions
\begin{equation} \label{u-eqn1}
\begin{gathered}
\bar {u}_0(t)=\int_0^{1}\bar u(x,t)Y_0(x)dx, \\
\bar {u}_{2n-1}(t)=\int_0^{1}\bar {u}(x,t)Y_{2n-1}(x)dx,\\
\bar {u}_{2n}(t)=\int_0^{1}\bar {u}(x,t)Y_{2n}(x)dx,
\end{gathered}
\end{equation}
and
\begin{equation} \label{u-eqn2}
\begin{gathered}
\bar{f}_0=\int_0^{1}\bar f(x)Y_0(x)dx, \\
\bar {f}_{2n-1}=\int_0^{1}\bar {f}(x)Y_{2n-1}(x)dx, \\
\bar {f}_{2n}=\int_0^{1}\bar {f}(x)Y_{2n}(x)dx.
\end{gathered}
\end{equation}
Applying the time fractional derivative $D_{0_+}^{\alpha,\gamma}(\cdot)$
to both sides of each equation in \eqref{u-eqn1}, we obtain
\begin{equation} \label{UFDE}
\begin{gathered}
D_{0_+}^{\alpha,\gamma}\bar {u}_0(t)
 =\int_0^{1}D_{0_+}^{\alpha,\gamma}\bar u(x,t)Y_0(x)dx, \\
D_{0_+}^{\alpha,\gamma}\bar {u}_{2n-1}(t)
 =\int_0^{1}D_{0_+}^{\alpha,\gamma}\bar {u}(x,t)Y_{2n-1}(x)dx,\\
D_{0_+}^{\alpha,\gamma}\bar {u}_{2n}(t)=\int_0^{1}D_{0_+}^{\alpha,\gamma}
 \bar {u}(x,t)Y_{2n}(x)dx.
\end{gathered}
\end{equation}
Let us take the third equation in \eqref{UFDE}. Using
\eqref{U-Problem-I2D1} together with the conditions
\eqref{U-Problem-I2D3}-\eqref{U-Problem-I2D4}, we obtain the
fractional differential equation
\begin{equation}
D_{0_+}^{\alpha,\gamma}\bar{u}_{2n}(t)+\lambda_n \bar{u}_{2n}(t)
=\bar{f}_{2n}.\label{u-FDE1}
\end{equation}
By using Laplace transform technique the solution of \eqref{u-FDE1} is
\begin{equation}\label{SolutionUnique}
\bar{u}_{2n}(t)=I^{1-\gamma}_{0_+}\bar{u}_{2n}(t)\bigr|_{t=0}t^{\gamma-1}
E_{\alpha,\gamma}(-\lambda_nt^{\alpha})+
\bar{f}_{2n}\int_0^t \tau^{\alpha-1}E_{\alpha,\alpha}
(-\lambda_n\tau^{\alpha})d\tau.
\end{equation}
Since
$$
\bar {u}_{2n}(t)=\int_0^{1}\bar {u}(x,t)Y_{2n}(x)dx\;\Rightarrow\;
  I^{1-\gamma}_{0_+} \bar {u}_{2n}(t)\Bigl|_{t=0}
=\int_0^{1}I^{1-\gamma}_{0_+} \bar{u}(x,t)\Bigl|_{t=0}Y_{2n}(x)dx
$$
and by using the initial condition from \eqref{U-Problem-I2D2} the solution
\eqref{SolutionUnique} takes the form
\begin{equation}
\bar{u}_{2n}(t)=\bar{f}_{2n}\int_0^t \tau^{\alpha-1}E_{\alpha,\alpha}
(-\lambda_n\tau^{\alpha})d\tau.
\end{equation}
By using the final temperature condition from \eqref{U-Problem-I2D2}, we obtain
$\bar{f}_{2n}=0$ and consequently $\bar{u}_{2n}(t)=0$ for all $t\in[0,T]$.

Similarly, we can shaow that for all $t\in[0,T]$,
\begin{equation}
\bar{u}_0(t)=0, \quad \bar{u}_{2n-1}(t)=0,\quad
\bar{f}_0=0,\quad  \bar{f}_{2n-1}=0.
\end{equation}
The uniqueness of the regular solution of the ISP-I follows from the
completeness of the set $\{Y_0(x),\; Y_{2n-1}(x),\; Y_{2n}(x)\}$,
$n\in\mathbb{N}$ (see \cite[Lemma 2]{berdyshev}).

It remains to show that  $u(x,t)$ given by \eqref{u(x,t)} agrees with
the initial and final data. We have
\begin{gather*}
I^{1-\gamma}_{0_+}\mathcal{W}_0
= \big\{\varphi_0+ \frac{t^{1+\alpha-\gamma}}{\Gamma{(2+\alpha-\gamma)}}
f_0\big\}X_0(x),\\
I^{1-\gamma}_{0_+}\mathcal{W}_{2n-1}
 \big\{E_{\alpha, 1}(-\lambda_{n}t^{\alpha})\varphi_{2n-1}+
t^{1+\alpha-\gamma}E_{\alpha,2+\alpha-\gamma}(-\lambda_{n}t^{\alpha})f_{2n-1}
 \big\}X_{2n-1}(x),\\
I^{1-\gamma}_{0_+}\mathcal{W}_{2n}
= \big\{E_{\alpha, 1}(-\lambda_{n}t^{\alpha})\varphi_{2n}+
t^{1+\alpha-\gamma}E_{\alpha,2+\alpha-\gamma}(-\lambda_{n}t^{\alpha})f_{2n}\big\}
X_{2n}(x).
\end{gather*}
 The term by term fractional integral of \eqref{u(x,t)} converges to
$I^{1-\gamma}_{0_+}u(x,t)$ and it
is uniformly convergent on $[\epsilon,T]$. For $t=0$ we have,
\begin{gather*}
I^{1-\gamma}_{0_+}\mathcal{W}_0|_{t=0}=\varphi_0X_0(x),\quad
 I^{1-\gamma}_{0_+}\mathcal{W}_{2n-1}\big|_{t=0}=\varphi_{2n-1}X_{2n-1}(x),\\
 I^{1-\gamma}_{0_+}\mathcal{W}_{2n}\big|_{t=0} = \varphi_{2n}X_{2n}.
 \end{gather*}
Therefore,
\[
I^{1-\gamma}_{0_+}u(x,t)\big|_{t=0}
=\varphi_0X_0+\sum_{n=1}^{\infty}\varphi_{2n-1}X_{2n-1}+\sum_{n=1}^{\infty}
\varphi_{2n}X_{2n},
\]
which is the series expansion of $\varphi(x)$, when expanded using
bi-orthogonal system.

Similarly, we can show that for $u(x,t)$ given by \eqref{u(x,t)} the
over-determination is also satisfied, that is, $u(x,T)=\psi(x)$.
\end{proof}

Before providing the proof of our stability result, i.e., Theorem \ref{thm2.3}
let us mention the following result from
\cite{ionkin}.

\begin{lemma} \label{lem3.3}
For any function $f\in L^2(0,1)$ the  inequality
\begin{equation}\label{IonkinLem-Estimates}
 r_1\|f\|_{L^2(0,1)}^2\leq  \sum_{n=0}^{\infty} f_{n}^2\leq R_1\|f\|_
 {L^2(0,1)}^2,
\end{equation}
is valid, where $r_1$ and $R_1$ are constants and $f_{n}$ are
coefficients of the bi-orthogonal expansion of the function $f$ in
any Riesz basis $\{\mathcal{R}_n(x)\}$ given by
$$
f_{n} = \langle f,\mathcal {W}_{n}\rangle,\quad n \in\mathbb{N}\cup \{0\},
$$
where $\{\mathcal {W}_{n}(x)\}$ is corresponding bi-orthogonal set
of Riesz basis $\{\mathcal{R}_n(x)\}$.
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm2.3}]
Let $\{u(x,t),f(x)\},\{\tilde{u}(x,t),\tilde{f}(x)\}$ be two solution sets
of the ISP-I corresponding to the data
$\{\varphi,\psi\},\{\tilde{\varphi},\tilde{\psi}\}$ respectively.
By Lemma \ref{lem3.3}, we have
\[
{\| f-\tilde{f}\|_{L^2(0,1)}}^2
\leq \frac{1}{r_1}\sum ^{\infty}_{n=0}
(f_{n}-\tilde{f}_{n})^2.
\]
Consider
\begin{equation}
\begin{aligned}
(f_0-\tilde{f}_0)^2
&= \Big(\frac{\Gamma(1+\alpha)}{T^\alpha}\Big)^2
\Bigl[\Big(\psi_0-\frac{T^{\gamma-1}}{\Gamma(\gamma)}\varphi_0\Big)
-\Big(\tilde{\psi}_0-\frac
{T^{\gamma-1}}{\Gamma(\gamma)}\tilde{\varphi}_0\Big)\Bigr]^2 \\
&\leq  2C^2_3\Bigl[(\psi_0 -\tilde{\psi}_0)^2+
C_3^2(\varphi_0-\tilde{\varphi}_0)^2\Bigr],
\end{aligned}\label{F0}
\end{equation}
where we have used $(a\pm b)^2\leq 2a^2+2b^2$. Similarly, we have
\begin{gather}
\begin{aligned}
&\sum  ^{\infty}_{n=1}(f_{2n-1}-\tilde{f}_{2n-1})^2 \\
&\leq \sum ^{\infty}_{n=1}2{(\mathcal{M}_2)^2}
\Bigl[\Big(\psi_{2n-1}-\tilde{\psi}_{2n-1}\Big)^2
+(\mathcal{M}_1)^2 (\varphi_{2n-1}-\tilde{\varphi}_{2n-1})^2\Bigr],
\end{aligned} \label{F2n-1} \\
\sum  ^{\infty}_{n=1}(f_{2n}-\tilde{f}_{2n})^2
\leq \sum ^{\infty}_{n=1} 2{(\mathcal{M}_2)^2}
\Bigl[\Big(\psi_{2n}-\tilde{\psi}_{2n}\Big)^2
 +(\mathcal{M}_1)^2 (\varphi_{2n}-\tilde{\varphi}_{2n})^2\Bigr].
\label{F2n}
\end{gather}
Setting
$$
N=\max\left\{2C_3^2,\,2C_3^4,\,2{(\mathcal{M}_1)^2}{(\mathcal{M}_2)^2},\,
2{(\mathcal{M}_2)^2}\right\},
$$
and using the estimates
\eqref{F0}-\eqref{F2n} we have
\begin{equation}
\begin{aligned}
&\sum  ^{\infty}_{n=0}(f_{n}-\tilde{f}_{n})^2 \\
&\leq 3N \Bigl[(\varphi_0-\tilde{\varphi}_0)^2+ \sum
^{\infty}_{n=1}(\varphi_{2n-1}-\tilde{\varphi}_{2n-1})^2
+\sum ^{\infty}_{n=1}(\varphi_{2n}
-\tilde{\varphi}_{2n})^2 \\
&\quad +(\psi_0-\tilde{\psi}_0)^2
+\sum  ^{\infty}_{n=1}(\psi_{2n-1}-\tilde{\psi}_{2n-1})^2
+\sum ^{\infty}_{n=1}(\psi_{n} -\tilde{\psi}_{2n})^2\Bigr]  \\
&\leq 3NR_1\Big(\|\varphi-\tilde\varphi\|^2_{L^2(0,1)}
+\|\psi-\tilde{\psi}\|^2_{L^2(0,1)}\Big).
\end{aligned}
\end{equation}
By  Lemma \ref{lem3.3}, we have
\begin{gather*}
\|f-\tilde{f}\|^2_{L^2(0,1)}
\leq \frac{1}{r_1}\sum ^{\infty}_{n=0}(f_{n}-\tilde{f}_{n})^2
\leq\frac{3NR_1}{r_1}\Bigl(\|\varphi-\tilde{\varphi}\|^2_{L^2(0,1)}
 +\|\psi-\tilde{\psi}\|^2_{L^2(0,1)}\Bigr),\\
\| f-\tilde{f}\|_{L^2(0,1)}
\leq  \sqrt{\frac{3NR_1}{r_1}}\Bigl(\|\varphi-\tilde{\varphi}\|_{L^2(0,1)}
+\|\psi-\tilde{\psi}\|_{L^2(0,1)}\Bigr).
\end{gather*}
Similarly we can obtain a stability result for $u(x,t)$.
\end{proof}

\section{Inverse source problem II}

In this section, we shall deal with ISP-II for
\eqref{ProblemEq1}--\eqref{ProblemEq4}, with $F(x,t)=a(t)f(x,t)$,
where $f(x,t)$ is known and a pair of functions $\{u(x,t), a(t)\}$
is to be determined.


\begin{proof}[Proof of Theorem \ref{thm2.4}]
 To determine the solution of ISP-II, i.e., the pair of functions
$\{u(x,t), a(t)\}$, we expand $u(x,t)$ and $f(x,t)$ using
bi-orthogonal system functions
\begin{gather}
u(x,t)=\sum_{n=1}^{\infty}u_0(t)X_{\,0}(x)+\sum_{n=1}^{\infty}
u_{2n-1}(t)X_{2n-1}(x)+\sum_{n=1}^{\infty}u_{2n}(t)X_{2n}(x),
\label{usolution ISP-II}\\
f(x,t)=\sum_{n=1}^{\infty}f_0(t)X_{\,0}(x)+\sum_{n=1}^{\infty}
f_{2n-1}(t)X_{2n-1}(x)+\sum_{n=1}^{\infty}f_{2n}(t)X_{2n}(x),
\label{fsolution ISP-II}
\end{gather}
where $u_0(t),u_{2n-1}(t)$ and  $u_{2n}(t)$ are to be determined,
$f_0(t),f_{2n-1}(t)$ and $f_{2n}(t)$ are coefficients of $f(x,t)$,
when expanded by using bi-orthogonal system. The following linear
fractional differential equations are obtained
\begin{gather}
 D^{\alpha,\gamma}_{0_+}u_{\,0}(t)=  a(t)f_0(t),\label{ISP-IIFDE1}\\
D^{\alpha,\gamma}_{0_+}u_{2n-1}(t)
=-\lambda_nu_{2n-1}(t)+a(t)f_{2n-1}(t),\label{ISP-IIFDE2}\\
D^{\alpha,\gamma}_{0_+}u_{2n}(t)
=-\lambda_n u_{2n}(t)+a(t)f_{2n}(t)\label{ISP-IIFDE3}, \quad n\in \mathbb{N}.
\end{gather}
The solutions of the fractional differential equations
\eqref{ISP-IIFDE1}--\eqref{ISP-IIFDE3} are
\begin{gather}
u_0(t)=\varphi_0\frac{t^{\gamma-1}}{\Gamma(\gamma)}
+a(t)f_0(t)*\frac{t^{\alpha-1}}{\Gamma(\alpha)},\label{ISP-IIu1}\\
u_{2n-1}(t)= \varphi_{2n-1}\mathcal{E}^{(1)}_n(t)+a(t)f_{2n-1}(t)*
\mathcal{E}^{(3)}_n(t),\label{ISP-IIu2}\\
u_{2n}(t)=\varphi_{2n}\mathcal{E}^{(1)}_n(t)+a(t)f_{2n}(t)*
\mathcal{E}^{(3)}_n(t),\label{ISP-IIu3}
\end{gather}
where * is the integral convolution operator and
$$
\mathcal{E}^{(3)}_n(t)=t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_nt^{\alpha}).
$$
Taking the generalized fractional derivative $D^{\alpha,\gamma}_{0_+}$, under
the integral sign of the over-determination condition \eqref{ODC} and
using \eqref{ProblemEq1} along with  $F(x,t)=a(t)f(x,t),$ we obtain
\begin{equation}\label{a(t)}
a(t)=\Big(\int_0^1 xf(x,t)dx \Big)^{-1}
\Big(D^{\alpha,\gamma}_{0_+}g(t)+\int_0^1 x u_{xxxx}(x,t)dx\Big).
\end{equation}

From the conditions of Theorem \ref{thm2.4}, we have $\int_0^1 xf(x,t)dx\neq0$ and is
given by
\begin{equation}  \label{int f}
\begin{aligned}
&\int_0^1 x f(x,t)dx \\
&=\frac{2}{3}f_0(t)-\sum_{n=1}^{\infty}\frac{1}{\pi n}f_{2n-1}(t)
+\sum_{n=1}^{\infty}\Big(\frac{-1}{2\pi^2n^2}+\frac{1+e^{2\pi n}}
{2\pi n(e^{2\pi n}-1)}\Big)f_{2n}(t),
\end{aligned}
\end{equation}
and
\begin{equation} \label{int u}
\begin{aligned}
\int_0^1 xu_{xxxx}\,dx
&= \sum_{n=1}^\infty \lambda_n \Big\{-\frac{1}{\pi n}
\Big(\mathcal{E}^{(1)}_n(t)\varphi_{2n-1}(t)+a(t)f_{2n-1}(t)*
 \mathcal{E}^{(3)}_n(t)\Big) \\
&\quad +\Big(\frac{-1}{2\pi^2n^2}+\frac{1+e^{2\pi n}}{2\pi n(e^{2\pi
n}-1)}\Big) \\
&\quad\times \big(\mathcal{E}^{(1)}_n(t)\varphi_{2n}(t)+a(t)f_{2n}(t)*\mathcal{E}^{(3)}_n(t)
\big)\Big\}.
\end{aligned}
\end{equation}
By \eqref{int f}--\eqref{int u}, we have the following
linear Volterra type integral equation of second kind
\begin{equation} \label{DProblem-I2D12}
a(t)=\Big(\int_0^1 x
f(x,t)dx\Big)^{-1}\Big(D^{\alpha,\gamma}_{0_+}g(t)
+\mathcal{T}(t)+ \int_0^tK(t,\,\tau)a(\tau)\,d\tau\Big),
\end{equation}
where
\begin{equation} \label{DProblem-I2D13}
\begin{aligned}
\mathcal{T}(t)
&= \sum_{n=1}^\infty\;\lambda_n\Big\{-\frac{1}{\pi n}
 \mathcal{E}_n^{(1)}(t)\varphi_{2n-1}(t) \\
&\quad +\Big(\frac{-1}{2\pi^2n^2}+\frac{1+e^{2\pi n}}{2\pi n(e^{2\pi
n}-1)}\Big)\mathcal{E}_n^{(1)}(t)\varphi_{2n}(t)\Big\},
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
K(t,\tau)
&= \sum_{n=1}^\infty\;\lambda_n\Big\{-\frac{1}{\pi n}
\Big(f_{2n-1}(\tau)\mathcal{E}_n^{(3)}(t-\tau)\Big) \\
&\quad +\Big(\frac{-1}{2\pi^2n^2}+\frac{1+e^{2\pi n}}{2\pi n(e^{2\pi
n}-1)}\Big)
\Big(f_{2n}(\tau)\mathcal{E}_n^{(3)}(t-\tau)\Big)\Big\}.
\end{aligned}\label{kseries}
\end{equation}
\end{proof}

Let us consider the space of continuous functions $C[0,T]$, equipped with
the Chebyshev norm
$$
\|f\|_{C[0,T]}:=\smash{ \max_{0 \leq t \leq T}}|f(t)|.
$$
Define an operator $\mathcal{B}(a(t)):=a(t)$, where the operator
$\mathcal{B}$ is
\begin{equation}\label{B(a(t))}
\mathcal{B}(a(t))=\Big(\int_0^1 x f(x,t)dx\Big)^{-1}
\Bigl(D^{\alpha,\gamma}_{0_+}g(t)
+\mathcal{T}(t)+ \int_0^tK(t,\,\tau)a(\tau)\,d\tau\Bigr).
\end{equation}
To show that the mapping $\mathcal{B}$:$\,C[0,T]\to C[0,T]$
is a contraction map. First of all, we shall show that
 $a(t)\in C[0,T]$ implies that $\mathcal{B}(a(t))\in C[0,T]$.

By using \eqref{MLF Estimate} there exists a constant $C_{6}$  such
that \begin{equation}\label{mlf E3}
       t\mathcal{E}_n^{(3)}(t)\leq \frac{C_{6}}{\lambda_n}\quad t\in[\epsilon,T] .
     \end{equation}

Using \eqref{Coefficents-bi-orthogonal}, integration by parts and Bessel's
inequality, we obtained the  inequalities
$$
|\varphi_{2n-1}|\leq\frac{1}{\lambda_n}\mathcal{I}_n,\quad\text{and}\quad
|\varphi_{2n}|\leq\frac{\sqrt{2}}{\lambda_n}\|\varphi^{iv}(x)\|_{L^2(0,1)}.
$$
Similarly we obtain
$$
|f_{2n-1}|\leq\frac{1}{\lambda_n}\mathcal{H}_n,\quad\text{and}\quad
|f_{2n}|\leq\frac{\sqrt{2}}{\lambda_n}\|f^{iv}(x)\|_{L^2(0,1)},
$$
where $\mathcal{H}_n=\langle f^{iv},Y_{2n-1}\rangle$.
From estimates \eqref{mlf E1}, \eqref{mlf E3} and using above relations
we have
\begin{gather*}
t^{1+\alpha-\gamma}|\mathcal{T}(t)|
\leq \sum_{n=1}^{\infty}C_1\Big\{\frac{\mathcal{I}_n}{\pi n
\lambda_n}+\Big(\frac{1}{\pi^2n^2}+\frac{1}{\pi
n}\Big)\frac{\sqrt{2}}{\lambda_n}\|\varphi^{iv}(x)\|_{L^2(0,1)}\Big\},
\\
t|K(t,\tau)| \leq \sum_{n=1}^{\infty}C_{6}\Big\{\frac{\mathcal{H}_n}{\pi n
\lambda_n}+\Big(\frac{1}{\pi^2n^2}+\frac{1}{\pi
n}\Big)\frac{\sqrt{2}}{\lambda_n}\|f^{iv}(x)\|_{L^2(0,1)}\Big\}.
\end{gather*}
Hence, the series \eqref{DProblem-I2D13} and
\eqref{kseries} are uniformly convergent by Weierstrass M-test. The
uniform convergence of the series \eqref{kseries} allow us to write
$$
\|K(t,\tau)\|_{C[0,T]}\leq K_1, \quad t\in(0,T],
$$
where $K_1$ is a constant,  consequently $\mathcal{B}(a(t))\in C[0,\,T]$.

Without loss of generality we set $T$ such that $T< 1/{K_1{M^*}}$.

Let us show that the mapping $\mathcal{B}:C[0,\,T]\to
C[0,\,T]$ is contraction, for this we take
\begin{equation}\label{DProblem-I2D19}
\begin{gathered}
|\mathcal{B}(a)-\mathcal{B}(c)|
\leq {M^*}\int_0^t |a(\tau)-c(\tau)||K(t,\,\tau)|d\tau
\leq TK_1{M^*}\smash{ \max_{0 \leq t \leq T}} |a(\tau)-c(\tau)|,  \\
\|\mathcal{B}(a)-\mathcal{B}(c)\|_{C[0,T]}  \leq TK_1M^*\|a-c\|_{C[0,T]},
\end{gathered}
\end{equation}
thus, the mapping $\mathcal{B}(\cdot)$ is a contraction which assures the
unique determination of $a\in C[0,T]$ by Banach fixed point theorem.

The solution $u(x,t)$ is formally given
by the series \eqref{usolution ISP-II}; the uniform convergence of
the series involved in  $u(x,t)$,$u_x(x,t)$, $u_{xx}(x,t)$,
$u_{xxx}(x,t)$, $u_{xxxx}(x,y)$ and $D_{0_+}^{\alpha,\gamma}u(x,t)$
directly follows from the estimates obtained in the previous section.

\begin{proof}[Proof of Theorem \ref{thm2.5}]
 (Uniqueness of the solution of the ISP-II)
We have already proved uniqueness of the source term $a(t)$ in Theorem \ref{thm2.4},
it remains to prove uniqueness of $u(x,t)$.

Let $u(x,t)$ and $v(x,t)$ be two solutions, and let
$\bar{u}(x,t)=u(x,t)-v(x,t)$.
Then $\bar{u}(x,t)$ satisfy the  equation
\begin{equation}
 D^{\alpha,\gamma}_{0_+}\bar{u}(x,t)=\bar{u}_{xxxx}(x,t) ,\quad (x,t) \in \Omega,
 \end{equation}
with initial condition
 \begin{equation}
 I^{1-\gamma}_{0_+}\bar{u}(x,t)|_{t=0}=0,\quad  x\in[0,1],
 \end{equation}
and nonlocal boundary conditions
 \begin{gather}
\bar{u}_{x}(0,t)=\bar{u}_{x}(1,t),\quad \bar{u}(0,t)=0\quad t\in [0,T],\\
\bar{u}_{xxx}(0,t)=0=\bar{u}_{xxx}(1,t)\quad \bar{u}_{xx}(1,t)=0,
\quad t\in [0,T].
\end{gather}
Consider the functions
\begin{gather*} %\label{uISP2-eqn1}
\bar {u}_0(t)=\int_0^{1}\bar u(x,t)Y_0(x)dx, \\
\bar {u}_{2n-1}(t)=\int_0^{1}\bar {u}(x,t)Y_{2n-1}(x)dx,\\
\bar {u}_{2n}(t)=\int_0^{1}\bar {u}(x,t)Y_{2n}(x)dx.
\end{gather*}
Following the same steps as  in the proof of Theorem \ref{thm2.2}, we can show that
$$
\bar {u}_0(t) = 0,\quad \bar {u}_{2n-1}(t)=0,\quad \bar {u}_{2n}(t)=0,\quad
t\in[0,T].
$$
Consequently, the uniqueness of the solution follows from the completeness
 of the set of function $\{Y_0(x), Y_{2n-1}(x), Y_{2n}(x)\}$, $n\in\mathbb{N}$.
\end{proof}

The proof of Theorem \ref{thm2.6}, the stability result is similar to
the proof of Theorem \ref{thm2.3}. Theefore, we omit it.

\section{Examples}

In this section, we provide some examples for ISP-I and ISP-II.

\begin{example} \label{examp1} \rm
 Consider the ISP-I with initial and final temperatures
$$
\varphi(x)=\sin2\pi x,\quad \psi(x)=(1+T^\alpha)\sin2\pi x.
$$
The coefficients of the series expansions of $\phi (x)$ and $\psi(x)$
using bi-orthogonal system of functions are
$$
\varphi_0=0,\quad \varphi_{2n}=0,\quad
\varphi_{2n-1}=\begin{cases} 1/2, & n=1,\\
0, & n\neq 1,
\end{cases}
$$
and
$$
\psi_0=0,\quad \psi_{2n}=0,\quad
 \psi_{2n-1}=\begin{cases} (1+T^\alpha)/2, & n=1,\\
 0,& n\neq 1.
\end{cases}
$$
Using \eqref{f-solution1}--\eqref{f-solution3}, we have
$$
f_0=0,\quad f_{2n}=0,\quad
f_{2n-1}=\begin{cases}
 \frac{(1+T^\alpha)-\mathcal{E}_1^{(1)}(T)}
{2\mathcal{E}_1^{(2)}(T)}, & n=1,\\
0,& n\neq 1.
\end{cases}
$$
Substituting the series coefficient of $f(x)$ in
\eqref{uSeries-1}--\eqref{uSeries-3} we obtain
\begin{gather*}
u_0=0,\quad u_{2n}=0,\\
u_{2n-1}=\begin{cases}
\frac{\mathcal{E}_1^{(1)}(t)}{2}+
\frac{(1+T^\alpha)-\mathcal{E}_1^{(1)}(T)}
{2\mathcal{E}_1^{(2)}(T)}\mathcal{E}_1^{(2)}(t), & n=1,\\
0, & n\neq 1.
\end{cases}
\end{gather*}
Hence the solution of ISP-I is
\begin{gather*}
f(x)=\Big(\frac{(1+T^\alpha)-\mathcal{E}_1^{(1)}(T)}
{\mathcal{E}_1^{(2)}(T)}\Big)\sin(2\pi x),\\
u(x,t)=\Big(\mathcal{E}_1^{(1)}(t)+\frac{(1+T^\alpha)
-\mathcal{E}_1^{(1)}(T)}{\mathcal{E}_1^{(2)}(T)}
\mathcal{E}_1^{(2)}(t)\Big)\sin (2\pi x).
\end{gather*}
\end{example}


\begin{example} \label{examp2} \rm
Consider the ISP-II with given the data
\begin{gather*}
\varphi(x)=0,\quad
g(t)=\frac{2}{3}\Big(\frac{t^{\gamma+1}}{\Gamma(\gamma)}
+\frac{t^{\alpha+2}}{\Gamma(\alpha+1)}\Big), \\
f(x,t)=2\Big(\frac{\Gamma(\gamma+2)}{\Gamma(\gamma)\Gamma(\gamma-\alpha+2)}
 t^{\gamma-\alpha}
+\frac{\Gamma(\alpha+3)}{\Gamma(3)\Gamma(\alpha+1)}t\Big)x.
\end{gather*}
By using the bi-orthogonal system, the coefficients of the series expansion
are
\[
\varphi_0=0,\quad \varphi_{2n}=0,\quad \varphi_{2n-1}=0,
\]
 and
\[
f_0=\frac{\Gamma(\gamma+2)}{\Gamma(\gamma)\Gamma(\gamma-\alpha+2)}
t^{\gamma-\alpha}
+\frac{\Gamma(\alpha+3)}{\Gamma(3)\Gamma(\alpha+1)}t,\quad
f_{2n}=0,\quad  f_{2n-1}=0.
\]
The solution is
\[
u(x,t)=2\Big(\frac{t^{\gamma+1}}{\Gamma(\gamma)}
+\frac{t^{\alpha+2}}{\Gamma(\alpha+1)}\Big)x,
\]
which satisfies the initial condition \eqref{ProblemEq2}
and the over-determination condition \eqref{ODC}.
By using the value of $u(x,t)$ in \eqref{a(t)}, we obtained the source
term as $a(t)=t$.

Hence $\{u(x,t), a(t)\}$ forms the solution set for the ISP-II.
\end{example}

\subsection*{Acknowledgments}
The authors would like to express their gratitude to the reviewers
for their insightful comments which ultimately improve the quality of the paper.
S. A. Malik was partially supported by COMSATS and ISESCO.


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\end{thebibliography}

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