# Time-dependent source identification problem for a fractional Schrödinger
equation with the Riemann-Liouville derivative

Ravshan Ashurov  and  Marjona Shakarova  Institute of Mathematics, Uzbekistan
Academy of Science, Tashkent, Student Town str. 100174<EMAIL_ADDRESS>National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Student
Town str. 100174<EMAIL_ADDRESS>
#  Time-dependent source identification problem for a fractional Schrödinger
equation with the Riemann-Liouville derivative

Ravshan Ashurov  and  Marjona Shakarova  Institute of Mathematics, Uzbekistan
Academy of Science, Tashkent, Student Town str. 100174<EMAIL_ADDRESS>National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Student
Town str. 100174<EMAIL_ADDRESS>
###### Abstract.

The Schrödinger equation
$i\partial_{t}^{\rho}u(x,t)-u_{xx}(x,t)=p(t)q(x)+f(x,t)$ ( $0<t\leq
T,\,0<\rho<1$), with the Riemann-Liouville derivative is considered. An
inverse problem is investigated in which, along with $u(x,t)$, also a time-
dependent factor $p(t)$ of the source function is unknown. To solve this
inverse problem, we take the additional condition $B[u(\cdot,t)]=\psi(t)$ with
an arbitrary bounded linear functional $B$. Existence and uniqueness theorem
for the solution to the problem under consideration is proved. Inequalities of
stability are obtained. The applied method allows us to study a similar
problem by taking instead of $d^{2}/dx^{2}$ an arbitrary elliptic differential
operator $A(x,D)$, having a compact inverse.

AMS 2000 Mathematics Subject Classifications : Primary 35R11; Secondary 34A12.
Key words: Schrödinger type equations, the Riemann-Liouville derivatives,
time-dependent source identification problem.

##  1\. Introduction

The fractional integration of order $\sigma<0$ of function $h(t)$ defined on
$[0,\infty)$ has the form (see e.g. [1], p.14, [2], Chapter 3)

$J_{t}^{\sigma}h(t)=\frac{1}{\Gamma(-\sigma)}\int\limits_{0}^{t}\frac{h(\xi)}{(t-\xi)^{\sigma+1}}d\xi,\quad
t>0,$

provided the right-hand side exists. Here $\Gamma(\sigma)$ is Euler’s gamma
function. Using this definition one can define the Riemann-Liouville
fractional derivative of order $\rho$:

$\partial_{t}^{\rho}h(t)=\frac{d}{dt}J_{t}^{\rho-1}h(t).$

Note that if $\rho=1$, then the fractional derivative coincides with the
ordinary classical derivative of the first order:
$\partial_{t}h(t)=(d/dt)h(t)$.

Let $\rho\in(0,1)$ be a fixed number and $\Omega=(0,\pi)\times(0,T]$. Consider
the following initial-boundary value problem for the Shrödinger equation

(1.1) $\left\\{\begin{aligned}
&i\partial_{t}^{\rho}u(x,t)-u_{xx}(x,t)=p(t)q(x)+f(x,t),\quad(x,t)\in\Omega;\\\
&u(0,t)=u(\pi,t)=0,\quad 0\leq t\leq T;\\\ &\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}u(x,t)=\varphi(x),\quad 0\leq x\leq\pi,\end{aligned}\right.$

where $t^{1-\rho}p(t)$, $t^{1-\rho}f(x,t)$ and $\varphi(x),q(x)$ are
continuous functions in the closed domain $\overline{\Omega}$. This problem is
also called the forward problem.

If $p(t)$ is a known function, then under certain conditions on the given
functions a solution to problem (1.1) exists and it is unique (see, e.g.,
[3]).

We note the following property of the Riemann-Liouville integrals, which
simplifies the verification of the initial condition in problem (1.1) (see
e.g. [1], p.104):

(1.2)
$\lim\limits_{t\rightarrow+0}J_{t}^{\alpha-1}h(t)=\Gamma(\alpha)\lim\limits_{t\rightarrow+0}t^{1-\alpha}h(t).$

From here, in particular, it follows that the solution of the forward problem
can have a singularity at zero $t=0$ of order $t^{\rho-1}$.

Let $C[0,l]$ be the set of continuous functions defined on $[0,l]$ with the
standard max-norm $||\cdot||_{C[0,l]}$. The purpose of this paper is not only
to find a solution $u(x,t)$, but also to determine the time-dependent part
$p(t)$ of the source function. To solve this time-dependent source
identification problem one needs an extra condition. Following the papers of
A. Ashyralyev et al. [4]-[6] we consider the additional condition in a rather
general form:

(1.3) $B[u(\cdot,t)]=\psi(t),\quad 0\leq t\leq T,$

where $B:C[0,\pi]\rightarrow R$ is a given bounded linear functional:
$||B[h(\cdot,t)]||_{C[0,T]}\leq b||h(x,t)||_{C(\overline{\Omega})}$, and
$\psi(t)$ is a given continuous function. For example, as the functional $B$
one can take $B[u(\cdot,t)]=u(x_{0},t)$, $x_{0}\in[0,\pi]$, or
$B[u(\cdot,t)]=\int_{0}^{\pi}u(x,t)dx$, or a linear combination of these two
functionals.

We call the initial-boundary value problem (1.1) together with additional
condition (1.3) _the inverse problem_.

When solving the inverse problem, we will investigate the Cauchy and initial-
boundary value problems for various differential equations. In this case, by
the solution of the problem we mean the classical solution, i.e. we will
assume that all derivatives and functions involved in the equation are
continuous with respect to the variable $x$ and $t$ in a open set. As an
example, let us give the definition of the solution to the inverse problem.

######  Definition 1.1.

A pair of functions $\\{u(x,t),p(t)\\}$ with the properties

  1. (1)

$\partial_{t}^{\rho}u(x,t),u_{xx}(x,t)\in C(\Omega)$,

  2. (2)

$t^{1-\rho}u(x,t)\in C(\overline{\Omega})$,

  3. (3)

$t^{1-\rho}p(t)\in C[0,T]$,

and satisfying conditions (1.1), (1.3) is called the solution of the inverse
problem.

Note that condition 3 in this definition is taken in order to cover a wider
class of functions, as function $p(t)$. In this regard, it should be noted
that, to the best of our knowledge, the time-dependent source identification
problem for equations with the Riemann-Liouville derivative is being studied
for the first time.

Taking into account the boundary conditions in problem (1.1), it is convenient
for us to introduce the Hölder classes as follows. Let $\omega_{g}(\delta)$ be
the modulus of continuity of function $g(x)\in C[0,\pi]$, i.e.

$\omega_{g}(\delta)=\sup\limits_{|x_{1}-x_{2}|\leq\delta}|g(x_{1})-g(x_{2})|,\quad
x_{1},x_{2}\in[0,\pi].$

If $\omega_{g}(\delta)\leq C\delta^{a}$ is true for some $a>0$, where $C$ does
not depend on $\delta$ and $g(0)=g(\pi)=0$, then $g(x)$ is said to belong to
the Hölder class $C^{a}[0,\pi]$. Let us denote the smallest of all such
constants $C$ by $||g||_{C^{a}[0,\pi]}$. Similarly, if the continuous function
$h(x,t)$ is defined on $[0,\pi]\times[0,T]$, then the value

$\omega_{h}(\delta;t)=\sup\limits_{|x_{1}-x_{2}|\leq\delta}|h(x_{1},t)-h(x_{2},t)|,\quad
x_{1},x_{2}\in[0,\pi]$

is the modulus of continuity of function $h(x,t)$ with respect to the variable
$x$. In case when $\omega_{h}(\delta;t)\leq C\delta^{a}$, where $C$ does not
depend on $t$ and $\delta$ and $h(0,t)=h(\pi,t)=0,\,\,t\in[0,T]$, then we say
that $h(x,t)$ belongs to the Hölder class $C_{x}^{a}(\overline{\Omega})$.
Similarly, we denote the smallest constant $C$ by
$||h||_{C_{x}^{a}(\overline{\Omega})}$.

Let $C_{2,x}^{a}(\overline{\Omega})$ denote the class of functions $h(x,t)$
such that $h_{xx}(x,t)\in C_{x}^{a}(\overline{\Omega})$ and
$h(0,t)=h(\pi,t)=0,\,t\in[0,T]$. Note that condition $h_{xx}(x,t)\in
C_{x}^{a}(\overline{\Omega})$ implies that
$h_{xx}(0,t)=h_{xx}(\pi,t)=0,\,\,t\in[0,T]$. For a function of one variable
$g(x)$, we introduce classes $C_{2}^{a}[0,\pi]$ in a similar way.

######  Theorem 1.2.

Let $a>\frac{1}{2}$ and the following conditions be satisfied

  1. (1)

$t^{1-\rho}f(x,t)\in C^{a}_{x}(\overline{\Omega})$,

  2. (2)

$\varphi\in C^{a}[0,\pi]$,

  3. (3)

$t^{1-\rho}\psi(t),\,\,t^{1-\rho}\partial_{t}^{\rho}\psi(t)\in C[0,T]$,

  4. (4)

$q\in C_{2}^{a}[0,\pi],\,\,B[q(x)]\neq 0$.

Then the inverse problem has a unique solution $\\{u(x,t),p(t)\\}$.

Everywhere below we denote by $a$ an arbitrary number greater than $1/2$:
$a>1/2$.

If we additionally require that the initial function $\varphi\in
C_{2}^{a}[0,\pi]$, then we can establish the following result on the stability
of the solution of the inverse problem.

######  Theorem 1.3.

Let assumptions of Theorem 1.2 be satisfied and let $\varphi\in
C_{2}^{a}[0,\pi]$. Then the solution to the invest problem obeys the stability
estimate

$||t^{1-\rho}\partial_{t}^{\rho}u||_{C(\overline{\Omega})}+||t^{1-\rho}u_{xx}||_{C(\overline{\Omega})}+||t^{1-\rho}p||_{C[0,T]}$
(1.4) $\leq
C_{\rho,q,B}\bigg{[}||\varphi_{xx}||_{C^{a}[0,\pi]}+||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}\partial_{t}^{\rho}\psi||_{C[0,T]}+||t^{1-\rho}f(x,t)||_{C_{x}^{a}(\overline{\Omega})}\bigg{]},$

where $C_{\rho,q,B}$ is a constant, depending only on $\rho,q$ and $B$.

It should be noted that the method proposed here, based on the Fourier method,
is applicable to the equation in (1.1) with an arbitrary elliptic differential
operator $A(x,D)$ instead of $d^{2}/dx^{2}$, if only the corresponding
spectral problem has a complete system of orthonormal eigenfunctions in
$L_{2}(G),\,G\subset R^{N}$.

The interest in the study of source (right-hand side of the equation $F(x,t)$)
identification inverse problems is caused primarily in connection with
practical requirements in various branches of mechanics, seismology, medical
tomography, and geophysics (see e.g. the survey paper Y. Liu et al. [7]). The
identification of $F(x,t)=h(t)$ is appropriate, for example, in cases of
accidents at nuclear power plants, when it can be assumed that the location of
the source is known, but the decay of the radiation power over time is unknown
and it is important to estimate it. On the other hand, one example of the
identification of $F(x,t)=g(x)$ can be the detection of illegal wastewater
discharges, which is a serious problem in some countries.

The inverse problem of determining the source function $F$ with the final time
observation have been well studied and many theoretical researches have been
published for classical partial differential equations (see, e.g. monographs
Kabanikhin [8] and Prilepko, Orlovsky, and Vasin [9]). As for fractional
differential equations, it is possible to construct theories parallel to [8],
[9], and the work is now ongoing. Let us mention only some of these works (a
detailed review can be found in [7]).

It should be noted right away that for the abstract case of the source
function $F(x,t)$ there is currently no general closed theory. Known results
deal with separated source term $F(x,t)=h(t)g(x)$. The appropriate choice of
the overdetermination depends on the choice whether the unknown is $h(t)$ or
$g(x)$.

Relatively fewer works are devoted to the case when the unknown is the
function $h(t)$ (see the survey work [7] and [10] for the case of subdiffusion
equations, and, for example, [4]-[6] for the classical heat equation).

Uniqueness questions in the inverse problem of finding a function $g(x)$ in
fractional diffusion equations with the sourse function $g(x)h(t)$ has been
studied in, e.g. [11], [12], [13].

In many papers, authors have considered an equation, in which $h(t)\equiv 1$
and $g(x)$ is unknown (see, e.g. [14] \- [20]). The case of subdiffusion
equations whose elliptic part is an ordinary differential expression is
considered in [14] \- [19]. The authors of the articles [21] \- [25] studied
subdiffusion equations in which the elliptic part is either a Laplace operator
or a second-order selfadjoint operator. The paper [26] studied the inverse
problem for the abstract subdiffution equation. In this article [26] and most
other articles, including [21] \- [24], the Caputo derivative is used as a
fractional derivative. The subdiffusion equation considered in the recent
papers [3], [27] contains the fractional Riemann-Liouville derivative, and the
elliptical part is an arbitrary elliptic expression of order $m$. In [28] and
[25], the fractional derivative in the subdiffusion equation is a two-
parameter generalized Hilfer fractional derivative. Note also that the papers
[28], [21], [24] contain a survey of papers dealing with inverse problems of
determining the right-hand side of the subdiffusion equation.

In [25], [29], [30], non-self-adjoint differential operators (with nonlocal
boundary conditions) were taken as elliptical part of the equation, and
solutions to the inverse problem were found in the form of biortagonal series.

In [20], the authors considered an inverse problem for simultaneously
determining the order of the Riemann-Liouville fractional derivative and the
source function in the subdiffusion equations. Using the classical Fourier
method, the authors proved the uniqueness and existence of a solution to this
inverse problem.

It should be noted that in all of the listed works, the Cauchy conditions in
time are considered (an exception is work [31], where the integral condition
is set with respect to the variable $t$). In the recent paper [32], for the
best of our knowledge, an inverse problem for subdiffusion equation with a
nonlocal condition in time is considered for the first time.

The papers [33] \- [34] deal with the inverse problem of determining the order
of the fractional derivative in the subdiffusion equation and in the wave
equation, respectively.

Time-dependent source identification problem (1.1) for classical Schrödinger
type equations (i.e. $\rho=1$) with additional condition (1.3) was for the
first time investigated in papers of A. Ashyralyev at al. [4]-[6]. To
investigate the inverse problem (1.1), (1.3) we borrow some original ideas
from these papers.

##  2\. Preliminaries

In this section, we recall some information about Mittag-Leffler functions,
differential and integral equations, which we will use in the following
sections.

For $0<\rho<1$ and an arbitrary complex number $\mu$, by $E_{\rho,\mu}(z)$ we
denote the Mittag-Leffler function of complex argument $z$ with two
parameters:

(2.1) $E_{\rho,\mu}(z)=\sum\limits_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\rho
k+\mu)}.$

If the parameter $\mu=1$, then we have the classical Mittag-Leffler function:
$E_{\rho}(z)=E_{\rho,1}(z)$.

Since $E_{\rho,\mu}(z)$ is an analytic function of $z$, then it is bounded for
$|z|\leq 1$. On the other hand the well known asymptotic estimate of the
Mittag-Leffler function has the form (see, e.g., [35], p. 133):

######  Lemma 2.1.

Let $\mu$ be an arbitrary complex number. Further let $\alpha$ be a fixed
number, such that $\frac{\pi}{2}\rho<\alpha<\pi\rho$, and $\alpha\leq|\arg
z|\leq\pi$. Then the following asymptotic estimate holds

$E_{\rho,\mu}(z)=-\sum\limits_{k=1}^{2}\frac{z^{-k}}{\Gamma(\rho-k\mu)}+O(|z|^{-3}),\,\,|z|>1.$

We can choose the parameter $\alpha$ so that the following estimate is valid:

######  Corollary 2.2.

For any $t\geq 0$ one has

$|E_{\rho,\mu}(it)|\leq\frac{C}{1+t},$

where constant $C$ does not depend on $t$ and $\mu$.

We will also use a coarser estimate with positive number $\lambda$ and
$0<\varepsilon<1$:

(2.2) $|t^{\rho-1}E_{\rho,\rho}(-i\lambda
t^{\rho})|\leq\frac{Ct^{\rho-1}}{1+\lambda t^{\rho}}\leq
C\lambda^{\varepsilon-1}t^{\varepsilon\rho-1},\quad t>0,$

which is easy to verify. Indeed, let $t^{\rho}\lambda<1$, then
$t<\lambda^{-1/\rho}$ and

$t^{\rho-1}=t^{\rho-\varepsilon\rho}t^{\varepsilon\rho-1}<\lambda^{\varepsilon-1}t^{\varepsilon\rho-1}.$

If $t^{\rho}\lambda\geq 1$, then $\lambda^{-1}\leq t^{\rho}$ and

$\lambda^{-1}t^{-1}=\lambda^{-1+\varepsilon}\lambda^{-\varepsilon}t^{-1}\leq\lambda^{\varepsilon-1}t^{\varepsilon\rho-1}.$

######  Lemma 2.3.

Let $t^{1-\rho}g(t)\in C[0,T]$. Then the unique solution of the Cauchy problem

(2.3) $\left\\{\begin{aligned} &i\partial_{t}^{\rho}y(t)+\lambda
y(t)=g(t),\quad 0<t\leq T;\\\ &\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}y(t)=y_{0}\end{aligned}\right.$

has the form

$y(t)=t^{\rho-1}E_{\rho,\rho}(i\lambda
t^{\rho})y_{0}-i\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i\lambda(t-s)^{\rho})g(s)ds.$

###### Proof.

Multiply equation (2.3) by $(-i)$ and then apply formula (7.2.16) of [36], p.
174 (see also [37], [38]).

∎

Let us denote by $A$ the operator $-d^{2}/dx^{2}$ with the domain
$D(A)=\\{v(x)\in W_{2}^{2}(0,\pi):v(0)=v(\pi)=0\\}$, where $W_{2}^{2}(0,\pi)$
\- the standard Sobolev space. Operator $A$ is selfadjoint in $L_{2}(0,\pi)$
and has the complete in $L_{2}(0,\pi)$ set of eigenfunctions $\\{v_{k}(x)=\sin
kx\\}$ and eigenvalues $\lambda_{k}=k^{2}$, $k=1,2,...$.

Consider the operator $E_{\rho,\mu}(itA)$, defined by the spectral theorem of
J. von Neumann:

$E_{\rho,\mu}(itA)h(x,t)=\sum\limits_{k=1}^{\infty}E_{\rho,\mu}(it\lambda_{k})h_{k}(t)v_{k}(x),$

here and everywhere below, by $h_{k}(t)$ we will denote the Fourier
coefficients of a function $h(x,t)$: $h_{k}(t)=(h(x,t),v_{k})$,
$(\cdot,\cdot)$ stands for the scalar product in $L_{2}(0,\pi)$. This series
converges in the $L_{2}(0,\pi)$ norm. But we need to investigate the uniform
convergence of this series in $\Omega$. To do this, we recall the following
statement.

######  Lemma 2.4.

Let $g\in C^{a}[0,\pi]$. Then for any $\sigma\in[0,a-1/2)$ one has

$\sum\limits_{k=1}^{\infty}k^{\sigma}|g_{k}|<\infty.$

For $\sigma=0$ this assertion coincides with the well-known theorem of S. N.
Bernshtein on the absolute convergence of trigonometric series and is proved
in exactly the same way as this theorem. For the convenience of readers, we
recall the main points of the proof (see, e.g. [39], p. 384).

###### Proof.

In theorem (3.1) of A. Zygmund [39], p. 384, it is proved that for an
arbitrary function $g(x)\in C[0,\pi]$, with the properties $g(0)=g(\pi)=0$,
one has the estimate

$\sum\limits_{k=2^{n-1}+1}^{2^{n}}|g_{k}|^{2}\leq\omega_{g}^{2}\bigg{(}\frac{1}{2^{n+1}}\bigg{)}.$

Therefore, if $\sigma\geq 0$, then by the Cauchy-Bunyakovsky inequality

$\sum\limits_{k=2^{n-1}+1}^{2^{n}}k^{\sigma}|g_{k}|\leq\bigg{(}\sum\limits_{k=2^{n-1}+1}^{2^{n}}|g_{k}|^{2}\bigg{)}^{\frac{1}{2}}\bigg{(}\sum\limits_{k=2^{n-1}+1}^{2^{n}}k^{2\sigma}\bigg{)}^{\frac{1}{2}}\leq
C2^{n(\frac{1}{2}+\sigma)}\omega_{g}\bigg{(}\frac{1}{2^{n+1}}\bigg{)},$

and finally

$\sum\limits_{k=2}^{\infty}k^{\sigma}|g_{k}|=\sum\limits_{n=1}^{\infty}\sum\limits_{k=2^{n-1}+1}^{2^{n}}k^{\sigma}|g_{k}|\leq
C\sum\limits_{n=1}^{\infty}2^{n(\frac{1}{2}+\sigma)}\omega_{g}\bigg{(}\frac{1}{2^{n+1}}\bigg{)}.$

Obviously, if $\omega_{g}(\delta)\leq C\delta^{a}$, $a>1/2$ and
$0<\sigma<a-1/2$, then the last series converges:

$\sum\limits_{k=2}^{\infty}k^{\sigma}|g_{k}|\leq C||g||_{C^{a}[0,\pi]}.$

∎

######  Lemma 2.5.

Let $h(x,t)\in C_{x}^{a}(\overline{\Omega})$ Then $E_{\rho,\mu}(itA)h(x,t)\in
C(\overline{\Omega})$ and $\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\mu}(itA)h(x,t)\in C([0,\pi]\times(0,T])$. Moreover, the
following estimates hold

(2.4) $||E_{\rho,\mu}(itA)h(x,t)||_{C(\overline{\Omega})}\leq
C||h||_{C_{x}^{a}(\overline{\Omega})};$ (2.5)
$\bigg{|}\bigg{|}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\mu}(itA)h(x,t)\bigg{|}\bigg{|}_{C[0,\pi]}\leq
Ct^{-1}||h||_{C_{x}^{a}(\overline{\Omega})},\,\,t>0.$

If $h(x,t)\in C_{2,x}^{a}(\overline{\Omega})$, then

(2.6) $\bigg{|}\bigg{|}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\mu}(itA)h(x,t)\bigg{|}\bigg{|}_{C(\overline{\Omega})}\leq
C||h_{xx}||_{C_{x}^{a}(\overline{\Omega})}.$

###### Proof.

By definition one has

$|E_{\rho,\mu}(itA)h(x,t)|=\bigg{|}\sum\limits_{k=1}^{\infty}E_{\rho,\mu}(it\lambda_{k})h_{k}(t)v_{k}(x)\bigg{|}\leq\sum\limits_{k=1}^{\infty}|E_{\rho,\mu}(it\lambda_{k})h_{k}(t)|.$

Corollary 2.2 and Lemma 2.4 imply

$|E_{\rho,\mu}(itA)h(x,t)|\leq
C\sum\limits_{k=1}^{\infty}\bigg{|}\frac{h_{k}(t)}{1+t\lambda_{k}}\bigg{|}\leq
C||h||_{C_{x}^{a}(\overline{\Omega})}.$

On the other hand,

$\bigg{|}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\mu}(itA)h(x,t)\bigg{|}\leq
C\sum\limits_{k=1}^{\infty}\bigg{|}\frac{\lambda_{k}h_{k}(t)}{1+t\lambda_{k}}\bigg{|}\leq
Ct^{-1}||h||_{C_{x}^{a}(\overline{\Omega})},\,\,t>0.$

If $h(x,t)\in C_{2,x}^{a}(\overline{\Omega})$, then
$h_{k}(t)=-\lambda_{k}^{-1}(h_{xx})_{k}(t)$. Therefore,

$\bigg{|}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\mu}(itA)h(x,t)\bigg{|}\leq
C||h_{xx}||_{C_{x}^{a}(\overline{\Omega})},\,\,0\leq t\leq T.$

∎

######  Lemma 2.6.

Let $t^{1-\rho}g(x,t)\in C_{x}^{a}(\overline{\Omega})$. Then there is a
positive constant $c_{1}$ such that

(2.7)
$\bigg{|}t^{1-\rho}\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x,s)ds\bigg{|}\leq
c_{1}\frac{t^{\rho}}{\rho}||t^{1-\rho}g||_{C_{x}^{a}(\overline{\Omega})}.$

###### Proof.

Apply estimate (2.4) to get

$\bigg{|}t^{1-\rho}\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x,s)ds\bigg{|}\leq
Ct^{1-\rho}\int_{0}^{t}(t-s)^{\rho-1}s^{\rho-1}ds\cdot||t^{1-\rho}g||_{C_{x}^{a}(\overline{\Omega})}.$

For the integral one has

(2.8)
$\int_{0}^{t}(t-s)^{\rho-1}s^{\rho-1}ds=\int\limits_{0}^{\frac{t}{2}}\cdot+\int\limits_{\frac{t}{2}}^{t}\cdot\leq\frac{2^{2(1-\rho)}}{\rho}t^{2\rho-1}.$

Denoting $c_{1}=4C$ we obtain the assertion of the lemma. ∎

######  Corollary 2.7.

If function $g(x,t)$ can be represented in the form $g_{1}(x)g_{2}(t)$, then
the right-hand side of estimate (2.7) has the form:

$c_{1}\frac{t^{\rho}}{\rho}||g_{1}||_{C^{a}[0,\pi]}||t^{1-\rho}g_{2}||_{C[0,T]}.$

######  Lemma 2.8.

Let $t^{1-\rho}g(x,t)\in C_{x}^{a}(\overline{\Omega})$. Then

$\bigg{|}\bigg{|}\int\limits_{0}^{t}(t-s)^{\rho-1}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x,s)ds\bigg{|}\bigg{|}_{C(\overline{\Omega})}\leq
C||t^{1-\rho}g||_{C_{x}^{a}(\overline{\Omega})}.$

###### Proof.

Let

$S_{j}(x,t)=\sum\limits_{k=1}^{j}\left[\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i\lambda_{k}(t-s)^{\rho})g_{k}(s)ds\right]\lambda_{k}v_{k}(x).$

Choose $\varepsilon$ so that $0<\varepsilon<a-1/2$ and apply the inequality
(2.2) to get

$|S_{j}(t)|\leq
C\sum\limits_{k=1}^{j}\int\limits_{0}^{t}(t-s)^{\varepsilon\rho-1}s^{\rho-1}\lambda_{k}^{\varepsilon}|s^{1-\rho}g_{k}(s)|ds.$

By Lemma 2.4 we have

$|S_{j}(t)|\leq C||t^{1-\rho}g||_{C_{x}^{a}(\overline{\Omega})}$

and since

$\int\limits_{0}^{t}(t-s)^{\rho-1}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\rho}(i(t-s)^{\rho}A)h(s)ds=\sum\limits_{j=1}^{\infty}S_{j}(t),$

the last inequality implies the assertion of the lemma.

∎

######  Lemma 2.9.

Let $t^{1-\rho}G(x,t)\in C_{x}^{a}(\overline{\Omega})$ and $\varphi\in
C^{a}[0,\pi]$. Then the unique solution of the following initial-boundary
value problem

(2.9) $\left\\{\begin{aligned}
&i\partial_{t}^{\rho}w(x,t)-w_{xx}(x,t)=G(x,t),\,\,0<t\leq T;\\\
&w(0,t)=w(\pi,t)=0,\,\,0<t\leq T;\\\ &\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}w(x,t)=\varphi(x),\,\,0\leq x\leq\pi,\end{aligned}\right.$

has the form

$w(x,t)=t^{\rho-1}E_{\rho}(it^{\rho}A)\varphi(x)-i\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)G(x,s)ds.$

###### Proof.

According to the Fourier method, we will seek the solution to this problem in
the form

$w(x,t)=\sum\limits_{k=1}^{\infty}T_{k}(t)v_{k}(x),$

where $T_{k}(t)$ are the unique solutions of the problems

(2.10) $\left\\{\begin{aligned}
&i\partial_{t}^{\rho}T_{k}+\lambda_{k}T_{k}(t)=G_{k}(t),\quad 0<t\leq T;\\\
&\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}T_{k}(t)=\varphi_{k},\end{aligned}\right.$

Lemma 2.3 implies

$T_{k}(t)=t^{\rho-1}E_{\rho}(i\lambda_{k}t^{\rho})\varphi_{k}-i\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i\lambda_{k}(t-s)^{\rho})G_{k}(s)ds.$

Hence the solution to problem (3.1) has the form

$w(x,t)=t^{\rho-1}E_{\rho}(it^{\rho}A)\varphi(x)-i\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)G(x,s)ds.$

Note that the existence of the first term follows from estimate (2.4), and the
existence of the second term follows from Lemma 2.6.

By Lemma 2.8 and estimate (2.5), we obtain: $w_{xx}(x,t)\in C(\Omega)$. Since
$i\partial_{t}^{\rho}w(x,t)=-w_{xx}(x,t)+G(x,t)$, then
$\partial_{t}^{\rho}w(x,t)\in C(\Omega)$.

The uniqueness of the solution can be proved by the standard technique based
on completeness of the set of eigenfunctions $\\{v_{k}(x)\\}$ in
$L_{2}(0,\pi)$ (see, e.g., [3]).

∎

Let $t^{1-\rho}F(x,t)\in C(\overline{\Omega})$ and $g(x)\in C^{a}[0,\pi]$.
Consider the Volterra integral equation

(2.11)
$w(x,t)=F(x,t)+\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x)B[w(\cdot,s)]ds.$

######  Lemma 2.10.

There is a unique solution $t^{1-\rho}w\in C(\overline{\Omega})$ to the
integral equation (2.11).

###### Proof.

Equation (2.11) is similar to the equations considered in the book [40], p.
199, Eq. (3.5.4) and it is solved in essentially the same way. Let us remind
the main points.

Equation (2.11) makes sense in any interval $[0,t_{1}]\in[0,T]$,
$(0<t_{1}<T)$. Choose $t_{1}$ such that

(2.12) $c_{1}b||g||_{C^{a}[0,\pi]}\frac{t_{1}^{\rho}}{\rho}<1$

and prove the existence of a unique solution $t^{1-\rho}w(x,t)\in
C([0,\pi]\times[0,t_{1}])$ to the equation (2.11) on the interval $[0,t_{1}]$
(here the constant $c_{1}$ is taken from estimate (2.7), see Corollary 2.7).
For this we use the Banach fixed point theorem for the space
$C([0,\pi]\times[0,t_{1}])$ with the weight function $t^{1-\rho}$ (see, e.g.,
[40], theorem 1.9, p. 68), where the distance is given by

$d(w_{1},w_{2})=||t^{1-\rho}[w_{1}(x,t)-w_{2}(x,t)]||_{C([0,\pi]\times[0,t_{1}])}.$

Let us denote the right-hand side of equation (2.11) by $Pw(x,t)$, where $P$
is the corresponding linear operator. To apply the Banach fixed point theorem
we have to prove the following:

(a) if $t^{1-\rho}w(x,t)\in C([0,\pi]\times[0,t_{1}])$, then
$t^{1-\rho}Pw(x,t)\in C([0,\pi]\times[0,t_{1}])$;

(b) for any $t^{1-\rho}w_{1},t^{1-\rho}w_{2}\in C([0,\pi]\times[0,t_{1}])$ one
has

$d(Pw_{1},Pw_{2})\leq\delta\cdot d(w_{1},w_{2}),\,\,\delta<1.$

Lemmas 2.5 and 2.6 imply condition (a). On the other hand, thanks to (2.7)
(see Corollary 2.7) we arrive at

$\bigg{|}\bigg{|}t^{1-\rho}\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x)B[w_{1}(\cdot,s)-w_{2}(\cdot,s)]ds\bigg{|}\bigg{|}_{C([0,\pi]\times[0,t_{1}])}\leq\delta\cdot
d(w_{1},w_{2}),$

where $\delta=c_{1}b||g||_{C^{a}[0,\pi]}\frac{t_{1}^{\rho}}{\rho}<1$ since
condition (2.12).

Hence by the Banach fixed point theorem, there exists a unique solution
$t^{1-\rho}w^{\star}(x,t)\in C([0,\pi]\times[0,t_{1}])$ to equation (2.11) on
the interval $[0,t_{1}]$ and this solution is a limit of the convergent
sequence $w_{n}(x,t)=P^{n}F(x,t)=PP^{n-1}F(x,t)$:

$\lim\limits_{n\rightarrow\infty}d(w_{n}(x,t),w^{\star}(x,t))=0.$

Next we consider the interval $[t_{1},t_{2}]$, where $t_{2}=t_{1}+l_{1}<T$,
and $l_{1}>0$. Rewrite the equation (2.11) in the form

$w(x,t)=F_{1}(x,t)+\int\limits_{t_{1}}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x)B[w(\cdot,s)]ds,$

where

$F_{1}(x,t)=F(x,t)+\int\limits_{0}^{t_{1}}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)g(x)B[w(\cdot,s)]ds,$

is a known function, since the function $w(x,t)$ is uniquely defined on the
interval $[0,t_{1}]$. Using the same arguments as above, we derive that there
exists a unique solution $t^{1-\rho}w^{\star}(x,t)\in
C([0,\pi]\times[t_{1},t_{2}])$ to equation (2.11) on the interval
$[t_{1},t_{2}].$ Taking the next interval $[t_{2},t_{3}]$, where
$t_{3}=t_{2}+l_{2}<T$, and $l_{2}>0$, and repeating this process (obviously,
$l_{n}>l_{0}>0$), we conclude that there exists a unique solution
$t^{1-\rho}w^{\star}(x,t)\in C([0,\pi]\times[0,T])$ to equation (2.11) on the
interval $[0,T]$, and this solution is a limit of the convergent sequence
$t^{1-\rho}w_{n}(x,t)\in C([0,\pi]\times[0,T])$:

(2.13)
$\lim\limits_{n\rightarrow\infty}||t^{1-\rho}[w_{n}(x,t)-w^{\star}(x,t)]||_{C(\overline{\Omega})}=0,$

with the choice of certain $w_{n}$ on each $[0,t_{1}],\cdots[t_{L-1},T]$. ∎

We need the following kind of Gronwall’s inequality:

######  Lemma 2.11.

Let $0<\rho<1$. Assume that the non-negative function $h(t)\in C[0,T]$ and the
positive constants $K_{0}$ and $K_{1}$ satisfy

$h(t)\leq K_{0}+K_{1}\int\limits_{0}^{t}(t-s)^{\rho-1}s^{\rho-1}h(s)ds$

for all $t\in[0,T]$. Then there exists a positive constant $C_{\rho,T}$,
depending only on $\rho$, $K_{2}$ and $T$ such that

(2.14) $h(t)\leq K_{0}\cdot C_{\rho,T}.$

Usually Gronwall’s inequality is formulated with a continuous function $k(s)$
instead of $K_{1}(t-s)^{\rho-1}s^{\rho-1}$. However, estimate (2.14) is proved
in a similar way to the well-known Gronwall inequality. For the convenience of
the reader, we present a proof of estimate (2.14).

###### Proof.

Iterating the hypothesis of Gronwall’s inequality gives

$h(t)\leq
K_{0}+K_{0}K_{1}\int\limits_{0}^{t}(t-s)^{\rho-1}s^{\rho-1}ds+K_{1}^{2}\int\limits_{0}^{t}(t-s)^{\rho-1}s^{\rho-1}\int\limits_{0}^{s}(s-\xi)^{\rho-1}\xi^{\rho-1}h(\xi)d\xi
ds$ $\leq
K_{\rho,T}+K_{1}^{2}\int\limits_{0}^{t}u(\xi)\xi^{\rho-1}\int\limits_{\xi}^{t}(t-s)^{\rho-1}(s-\xi)^{\rho-1}s^{\rho-1}dsd\xi,$

where

$K_{\rho,T}=K_{0}+K_{0}K_{1}\int\limits_{0}^{T}(t-s)^{\rho-1}s^{\rho-1}ds.$

For the inner integral we have (see (2.8))

$\int\limits_{\xi}^{t}(s-\xi)^{\rho-1}(t-s)^{\rho-1}ds=\int\limits_{0}^{t-\xi}y^{\rho-1}(t-\xi-y)^{\rho-1}dy\leq\frac{2^{2(1-\rho)}}{\rho}(t-\xi)^{2\rho-1}.$

Now the hypothesis is

$h(t)\leq
K_{0}+K_{1}^{2}\frac{2^{2(1-\rho)}}{\rho}\int\limits_{0}^{t}(t-s)^{2\rho-1}h(s)ds.$

By repeating this process so many times that $k\rho>1$, we make sure that
there is a positive constant $C_{\rho}=C(\rho,K_{2},T)>0$ such that

$h(t)\leq K_{0}+C_{\rho}\int\limits_{0}^{t}h(s)ds,$

or

$\frac{h(\xi)}{K_{0}+C_{\rho}\int\limits_{0}^{\xi}h(s)ds}\leq 1.$

Multiply this by $C_{\rho}$ to get

$\frac{d}{d\xi}\ln\bigg{(}K_{0}+C_{\rho}\int\limits_{0}^{\xi}h(s)ds\bigg{)}\leq
C_{\rho}.$

Integrate from $\xi=0$ to $\xi=t$ and exponentiate to obtain

$K_{0}+C_{\rho}\int\limits_{0}^{t}h(s)ds\leq K_{0}e^{C_{\rho}t}.$

Finally note that the left side is $\geq h(t)$. ∎

##  3\. Auxiliary problem and proof of Theorem 1.2

Let us consider the following auxiliary initial-boundary value problem

(3.1) $\left\\{\begin{aligned}
&i\partial_{t}^{\rho}\omega(x,t)-\omega_{xx}(x,t)=-i\mu(t)q^{\prime\prime}(x)+f(x,t),\quad(x,t)\in\Omega;\\\
&\omega(0,t)=\omega(\pi,t)=0,\quad 0\leq t\leq T;\\\
&\lim\limits_{t\rightarrow 0}J_{t}^{\rho-1}\omega(x,t)=\varphi(x),\quad 0\leq
x\leq\pi,\end{aligned}\right.$

where function $\mu(t)$ is the unique solution of the Cauchy problem:

(3.2) $\left\\{\begin{aligned} &\partial_{t}^{\rho}\mu(t)=p(t),\quad 0<t\leq
T;\\\ &\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}\mu(t)=0.\end{aligned}\right.$

Note that the solution to the Cauchy problem (3.2) has the form (see, e.g.,
[37]):

$\mu(t)=\frac{1}{\Gamma(\rho)}\int\limits_{0}^{t}(t-s)^{\rho-1}p(s)ds,$

######  Definition 3.1.

A functions $\omega(x,t)$ with the properties

  1. (1)

$\partial_{t}^{\rho}\omega(x,t),\omega_{xx}(x,t)\in C(\Omega)$,

  2. (2)

$t^{1-\rho}\omega_{xx}(x,t)\in C((0,\pi)\times[0,T])$,

  3. (3)

$t^{1-\rho}\omega(x,t)\in C(\overline{\Omega})$,

and satisfying conditions (3.2) is called the solution of problem (3.2).

######  Lemma 3.2.

Let $\omega(x,t)$ be a solution of problem (3.1). Then the unique solution
$\\{u(x,t),p(t)\\}$ to the inverse problem (1.1), (1.3) has the form

(3.3) $u(x,t)=\omega(x,t)-i\mu(t)q(x),$ (3.4)
$p(t)=\frac{i}{B[q(x)]}\\{\partial^{\rho}_{t}\psi(t)-B[\partial^{\rho}_{t}\omega(\cdot,t)]\\},$

where

(3.5) $\mu(t)=\frac{i}{B[q(x)]}[\psi(t)-B[\omega(\cdot,t)]].$

###### Proof.

Substitute the function $u(x,t)$, defined by equality (3.3), into the equation
in (1.1). Then

$i\partial_{t}^{\rho}\omega(x,t)+\partial_{t}^{\rho}\mu(t)q(x)-\omega_{xx}(x,t)+i\mu(t)q^{\prime\prime}(x)=p(t)q(x)+f(x,t).$

Since $\partial_{t}^{\rho}\mu(t)=p(t)$ (see (3.2)), we obtain equation (3.1),
i.e. function $u(x,t)$, defined by (3.3) is a solution of the equation in
(1.1). As for the initial condition, again by virtue of (3.2) we get

$\lim\limits_{t\rightarrow 0}J_{t}^{\rho-1}u(x,t)=\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}\omega(x,t)-i\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}\mu(t)q(x)=\lim\limits_{t\rightarrow
0}J_{t}^{\rho-1}\omega(x,t)=\varphi(x).$

On the other hand, conditions $q(0)=q(\pi)=0$ imply $u(0,t)=u(\pi,t)=0$,
$0\leq t\leq T$.

From Definition LABEL:def2 of solution $\omega(x,t)$ and the property of the
functions $\mu(t)$ and $q(x)$ it immediately follows that the function
$u(x,t)$ satisfies the requirements: $\partial_{t}^{\rho}u(x,t),u_{xx}(x,t)\in
C(\Omega)$, $t^{1-\rho}u(x,t)\in C(\overline{\Omega})$.

Thus function $u(x,t)$, defined as (3.3) is a solution of the initial-boundary
value problem (1.1).

Let us prove equation (3.4). Rewrite (3.3) as

$iq(x)\mu(t)=\omega(x,t)-u(x,t).$

Apply (1.3) to obtain

$i\mu(t)B[q(x)]=B[\omega(\cdot,t)]-\psi(t),$

or, since $B[q(x)]\neq 0$, we get (3.5). Finally, using equality
$\partial_{t}^{\rho}\mu(t)=p(t)$, we have

$p(t)=\frac{i}{B[q(x)]}[\partial_{t}^{\rho}\psi(t)-B[\partial_{t}^{\rho}\omega(\cdot,t)]],$

which coincides with (3.4). Moreover, from the definition of solution
$\omega(x,t)$ of problem (3.1) and the property of function $\psi(t)$ one has
$t^{1-\rho}p(t)\in C[0,T]$. ∎

Thus, to solve the inverse problem (1.1), (1.3), it is sufficient to solve the
initial-boundary value problem (3.1).

######  Theorem 3.3.

Under the assumptions of Theorem 1.2, problem (3.1) has a unique solution.

###### Proof.

Let

(3.6)
$G(x,s)=\frac{i}{B[q(x)]}(B[\omega(\cdot,s)]-\psi(s))q^{\prime\prime}(x)+f(x,s)$

and suppose that $s^{1-\rho}G(x,s)\in C_{x}^{a}(\overline{\Omega})$. Then by
Lemma 2.9 problem (3.1) is equivalent to the integral equation

$\omega(x,t)=t^{\rho-1}E_{\rho}(it^{\rho}A)\varphi(x)-i\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)G(x,s)ds.$

Rewrite this equation as

(3.7)
$\omega(x,t)=F(x,t)+\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)\frac{q^{\prime\prime}(x)}{B[q(x)]}B[\omega(\cdot,s)]ds,$

where

$F(x,t)=t^{\rho-1}E_{\rho}(it^{\rho}A)\varphi(x)-i\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)\bigg{[}-\frac{iq^{\prime\prime}(x)}{B[q(x)]}\psi(s)+f(x,s)\bigg{]}ds.$

In order to apply Lemma 2.10 to equation (3.7), we show that
$t^{1-\rho}F(x,t)\in C(\overline{\Omega})$. Indeed, by estimate (2.4) one has
$E_{\rho}(it^{\rho}A)\varphi(x)\in C(\overline{\Omega})$. According to the
conditions of the Theorem 1.2
$h(x,s)=s^{1-\rho}[-\frac{iq^{\prime\prime}(x)}{B[q(x)]}\psi(s)+f(x,s)]\in
C_{x}^{a}(\overline{\Omega})$. Therefore, by virtue of estimate (2.7), the
second term of function $t^{1-\rho}F(x,t)$ also belongs to the class
$C(\overline{\Omega})$. Hence, by virtue of Lemma 2.10, the Volterra equation
(3.7) has a unique solution $t^{1-\rho}\omega(x,t)\in C(\overline{\Omega})$.

Let us show that $\partial_{t}^{\rho}\omega(x,t),\,\omega_{xx}(t)\in
C(\Omega)$. First we consider $F_{xx}(x,t)$ and note, that by estimate (2.5)
we have $\frac{\partial^{2}}{\partial x^{2}}E_{\rho}(it^{\rho}A)\varphi(x)\in
C([0,\pi]\times(0,T])$. Since function $h$ defined above, belongs to the class
$C_{x}^{a}(\overline{\Omega})$, then by Lemma 2.8, the second term of function
$F_{xx}(x,t)$ belongs to $C(\Omega)$ and satisfies the estimate:

$\bigg{|}\bigg{|}t^{1-\rho}\int\limits_{0}^{t}(t-s)^{\rho-1}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\rho}(i(t-s)^{\rho}A)\bigg{[}-\frac{iq^{\prime\prime}(x)}{B[q(x)]}\psi(s)+f(x,s)\bigg{]}ds\bigg{|}\bigg{|}_{C(\overline{\Omega})}$

(3.8) $\leq
C\bigg{[}\bigg{|}\bigg{|}t^{1-\rho}\frac{q^{\prime\prime}(x)}{B[q(x)]}\psi(t)\bigg{|}\bigg{|}_{C^{a}_{x}(\overline{\Omega})}+||t^{1-\rho}f(x,t)||_{C^{a}_{x}(\overline{\Omega})}\bigg{]}\leq
C_{a,q,B}\big{[}||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f(x,t)||_{C^{a}_{x}(\overline{\Omega})}\big{]}.$

We pass to the second term on the right-hand side of equality (3.7). Since
$t^{1-\rho}\omega(x,t)\in C(\overline{\Omega})$, the conditions of Theorem 1.2
imply that $s^{1-\rho}\frac{q^{\prime\prime}(x)}{B[q(x)]}B[\omega(\cdot,s)]\in
C_{x}^{a}(\overline{\Omega})$. Then again by Lemma 2.8, this term belongs to
$C(\overline{\Omega})$ and satisfies the estimate:

$\bigg{|}\bigg{|}t^{1-\rho}\int\limits_{0}^{t}(t-s)^{\rho-1}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho,\rho}(i(t-s)^{\rho}A)\frac{q^{\prime\prime}(x)}{B[q(x)]}B[\omega(\cdot,s)]ds\bigg{|}\bigg{|}_{C(H)}$
(3.9) $\leq
C\bigg{|}\bigg{|}\frac{q^{\prime\prime}(x)}{B[q(x)]}B[t^{1-\rho}\omega(\cdot,t)]\bigg{|}\bigg{|}_{C(\overline{\Omega})}\leq
C_{a,q,B}||t^{1-\rho}\omega(x,t)||_{C(\overline{\Omega})}.$

Thus, $\omega_{xx}(x,t)\in C((0,T];H)$. On the other hand, by virtue of
equation (3.1) and the conditions of Theorem 1.2, we will have

$\partial_{t}^{\rho}\omega(x,t)=\omega_{xx}(x,t)-i\mu(t)q^{\prime\prime}(x)+f(x,t)\in
C(\overline{\Omega})).$

The fact that here $\mu\in C[0,T]$ follows again from the conditions of the
Theorem 1.2 and equality (3.5).

It remains to show that $t^{1-\rho}G(x,t)\in C_{x}^{a}(\overline{\Omega})$.
But this fact follows from the conditions of Theorem 1.2 and the already
proven assertion: $t^{1-\rho}\omega(x,t)\in C(\overline{\Omega})$.

∎

As noted above Theorem 1.2 is an immediate consequence of Lemma 3.2 and
Theorem 3.3.

##  4\. Proof of Theorem 1.3

First we prove the following statement on the stability of the solution to
problem (3.1), (3.2).

######  Theorem 4.1.

Let assumptions of Theorem 1.3 be satisfied. Then the solution to problem
(3.1), (3.2) obeys the stability estimate

(4.1) $||t^{1-\rho}\partial_{t}^{\rho}\omega||_{C(\overline{\Omega})}\leq
C_{\rho,q,B}\big{[}||\varphi_{xx}||_{C^{a}[0,\pi]}+||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f(x,t)||_{C^{a}_{x}(\overline{\Omega})}\big{]},$

where $C_{\rho,q,B,\epsilon}$ is a constant, depending only on $\rho,q$ and
$B$.

###### Proof.

Let us begin the proof of the inequality (4.1) by establishing an estimate for
$\omega_{xx}(x,t)$ and then use it with equation (3.1). To this end we have
from (2.6)

$\bigg{|}\bigg{|}\frac{\partial^{2}}{\partial
x^{2}}E_{\rho}(it^{\rho}A)\varphi\bigg{|}\bigg{|}_{C(\overline{\Omega})}\leq
C||\varphi_{xx}||_{C^{a}[0,\pi]}.$

This estimate together with (3.8) implies

$||t^{1-\rho}F_{xx}(x,t)||_{C(\overline{\Omega})}\leq
C||\varphi_{xx}||_{C^{a}[0,\pi]}+C_{a,q,B}\big{[}||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f(x,t)||_{C^{a}_{x}(\overline{\Omega})}\big{]}.$

Then, using equality (3.7) and inequality (3.9), we get

(4.2) $||t^{1-\rho}\omega_{xx}(x,t)||_{C(\overline{\Omega})}\leq
C||\varphi_{xx}||_{C^{a}[0,\pi]}+C_{a,q,B}\big{[}||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f(x,t)||_{C^{a}_{x}(\overline{\Omega})}+||t^{1-\rho}\omega(x,t)||_{C(\overline{\Omega})}\big{]}.$

As a result, we have obtained an estimate for $\omega_{xx}(x,t)$ through
$\omega(x,t)$. To estimate $||t^{1-\rho}\omega(x,t)||_{C(\overline{\Omega})}$,
we will proceed as follows. Apply estimates (2.4) and (2.7) to get

$||t^{1-\rho}F(x,t)||_{C(\overline{\Omega})}\leq||\varphi||_{C^{a}[0,\pi]}+\frac{T^{\rho}}{\rho}\big{[}C_{q,B}||q^{\prime\prime}||_{C^{a}[0,\pi]}||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f||_{C^{a}_{x}(\overline{\Omega})}\big{]}$

Again by estimate (2.4) we have

$\bigg{|}\bigg{|}t^{1-\rho}\int\limits_{0}^{t}(t-s)^{\rho-1}E_{\rho,\rho}(i(t-s)^{\rho}A)\frac{q^{\prime\prime}(x)}{B[q(x)]}B[\omega(\cdot,s)]ds\bigg{|}\bigg{|}_{C[0,\pi]}\leq$
$C_{q,B}||q^{\prime\prime}||_{C^{a}[0,\pi]}\int\limits_{0}^{t}(t-s)^{\rho-1}||\omega(x,s)||_{C[0,\pi]}.$

Therefore, from equation (3.7) we obtain an estimate

$||t^{1-\rho}\omega(x,t)||_{C[0,\pi]}\leq||\varphi||_{C^{a}[0,\pi]}+C_{q,\rho,B}[||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f||_{C^{a}_{x}(\overline{\Omega})}]+$
$C_{q,B}\int\limits_{0}^{t}(t-s)^{\rho-1}s^{\rho-1}||s^{1-\rho}\omega(x,s)||_{C[0,\pi]}ds$

for all $t\in[0,T]$. Finally, the Gronwall inequality (2.14) implies

$||t^{1-\rho}\omega(x,t)||_{C(\overline{\Omega})}\leq
C_{q,\rho,B}\big{[}||\varphi||_{C^{a}[0,\pi]}+||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f||_{C^{a}_{x}(\overline{\Omega})}\big{]}.$

We substitute this estimate in (4.2) and apply $||\varphi||_{C^{a}[0,\pi]}\leq
C||\varphi_{xx}||_{C^{a}[0,\pi]}$ to get

$||t^{1-\rho}\omega_{xx}||_{C(\overline{\Omega})}\leq
C_{\rho,q,B}\big{[}||\varphi_{xx}||_{C^{a}[0,\pi]}+||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}f||_{C^{a}_{x}(\overline{\Omega})}\big{]},$

To obtain estimate (4.1), it remains to note that

$\partial_{t}^{\rho}\omega(x,t)=\omega_{xx}(x,t)-i\mu(t)q^{\prime\prime}(x)+f(x,t)$

and use the estimate

$||t^{1-\rho}\mu||_{C[0,T]}\leq
C_{q,B}\big{[}||t^{1-\rho}\psi||_{C[0,T]}+||t^{1-\rho}\omega||_{C(\overline{\Omega})}\big{]},$

which follows from definition (3.5) and the conditions of Theorem 1.2. ∎

Proof of Theorem 1.3.

Apply (3.4) to get

$||t^{1-\rho}p(t)||_{C[0,T]}\leq
C_{q,B}\big{[}||t^{1-\rho}\partial_{t}^{\rho}\omega||_{C(\overline{\Omega})}+||t^{1-\rho}\partial_{t}^{\rho}\psi||_{C[0,T]}\big{]}.$

Equations (3.3) and (3.2) imply

$\partial^{\rho}_{t}u(x,t)=\partial^{\rho}_{t}\omega(x,t)+p(t)q(x).$

Hence, from estimates of $\partial^{\rho}_{t}\omega(x,t)$ and $p(t)$, we
obtain an estimate for $\partial^{\rho}_{t}u(x,t)$. On the other hand, by
virtue of equation (1.1), we will have

$-u_{xx}(x,t)=-i\partial_{t}^{\rho}u(x,t)+p(t)q(x)+f(x,t).$

Now, to establish estimate (4.1), it suffices to use the statement of Theorem
4.1.

##  5\. Acknowledgement

The authors are grateful to A. O. Ashyralyev for posing the problem and they
convey thanks to Sh. A. Alimov for discussions of these results. The authors
acknowledge financial support from the Ministry of Innovative Development of
the Republic of Uzbekistan, Grant No F-FA-2021-424.

## References

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