# Closed systems refuting quantum-speed-limit hypotheses

Niklas Hörnedal  0000-0002-2005-8694<EMAIL_ADDRESS>Department of
Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg,
Luxembourg  Ole Sönnerborn  0000-0002-1726-4892<EMAIL_ADDRESS>Department of Mathematics and Computer Science, Karlstad University, SE-651 88
Karlstad, Sweden  Department of Physics, Stockholm University, SE-106 91
Stockholm, Sweden

###### Abstract

Many quantum speed limits for isolated systems can be generalized to also
apply to closed systems. This is, for example, the case with the well-known
Mandelstam-Tamm quantum speed limit. Margolus and Levitin derived an equally
well-known and ostensibly related quantum speed limit, and it seems to be
widely believed that the Margolus–Levitin quantum speed limit can be similarly
generalized to closed systems. However, a recent geometrical examination of
this limit reveals that it differs significantly from most known quantum speed
limits. In this paper, we show that, contrary to the common belief, the
Margolus-Levitin quantum speed limit does not extend to closed systems in an
obvious way. More precisely, we show that for every hypothetical bound of
Margolus-Levitin type, there are closed systems that evolve with a conserved
normalized expected energy between states with any given fidelity in a time
shorter than the bound. We also show that for isolated systems, the
Mandelstam-Tamm quantum speed limit and a slightly weakened version of this
limit that we call the Bhatia-Davies quantum speed limit always saturate
simultaneously. Both of these evolution time estimates extend
straightforwardly to closed systems. We demonstrate that there are closed
systems that saturate the Mandelstam-Tamm but not the Bhatia-Davies quantum
speed limit.

##  I Introduction

Many quantum speed limits (QSLs) for isolated systems can be generalized to
also apply to closed systems [1, 2, 3]. By an isolated system we mean one
whose Hamiltonian does not change over time, while a closed system may have a
time-varying Hamiltonian. The famous Mandelstam-Tamm QSL is of this kind [4,
5]. The Mandelstam-Tamm QSL states that it takes at least the time

$\tau_{\textsc{mt}}=\frac{\pi}{2\Delta H}$ (1)

for an isolated system to evolve between two fully distinguishable
states.111“State” will always refer to a pure quantum state, that is, a state
that can be represented by a density operator of rank $1$. Two states
$\rho_{1}$ and $\rho_{2}$ are fully distinguishable if their fidelity
vanishes.,222All quantities are expressed in units such that $\hbar=1$. Here,
$\Delta H$ is the energy uncertainty. More generally, it takes at least the
time

$\tau_{\textsc{mt}}(\delta)=\frac{\arccos\sqrt{\delta}}{\Delta H}$ (2)

for an isolated system to evolve between two states with fidelity
$\delta$.333The fidelity between two states $\rho_{1}$ and $\rho_{2}$ is
$\operatorname{tr}(\rho_{1}\rho_{2})$.  The originators of the estimate
$\tau_{\textsc{mt}}(\delta)$ are also Mandelstam and Tamm, but it was
rediscovered and formulated in a more concise way in Ref. [6].

The Mandelstam-Tamm QSL can be extended to closed systems by replacing the
denominator in (2) with the corresponding time average. Thus, the evolution
time of a closed system evolving between two states with fidelity $\delta$ is
bounded from below by

$\bar{\tau}_{\textsc{mt}}(\delta)=\frac{\arccos\sqrt{\delta}}{\langle\langle\Delta
H_{t}\rangle\rangle},$ (3)

with $\langle\langle\Delta H_{t}\rangle\rangle$ being the time average of the
energy uncertainty. Since the Fubini-Study distance between two states with
fidelity $\delta$ is $\arccos\sqrt{\delta}$, and the Fubini-Study speed with
which a state evolves is $\Delta H_{t}$ [5], the Mandelstam-Tamm QSL is
saturated if and only if the state follows a Fubini-Study geodesic in the
projective Hilbert space. Mandelstam and Tamm’s QSL has also been extended to
systems in mixed states [8, 9, 7].

Margolus and Levitin [10] derived a seemingly similar evolution time estimate.
The Margolus-Levitin QSL states that the time it takes for an isolated system
to evolve between two fully distinguishable states is greater than or equal to

$\tau_{\textsc{ml}}=\frac{\pi}{2\langle H-\epsilon_{\mathrm{min}}\rangle},$
(4)

where $\langle H-\epsilon_{\mathrm{min}}\rangle$ is the expected energy
$\langle H\rangle$ shifted by the smallest occupied energy
$\epsilon_{\mathrm{min}}$.444An energy $\epsilon$ is occupied by $\rho$ if
$\langle\epsilon|\rho|\epsilon\rangle>0$ for an energy eigenstate
$|\epsilon\rangle$ with eigenvalue $\epsilon$. The quantity $\langle
H-\epsilon_{\mathrm{min}}\rangle$ is hereafter referred to as the normalized
expected energy. A more general result states that the time it takes for an
isolated system to evolve between two states with fidelity $\delta$ is lower
bounded by

$\tau_{\textsc{ml}}(\delta)=\frac{\alpha(\delta)}{\langle
H-\epsilon_{\mathrm{min}}\rangle},$ (5)

where

$\alpha(\delta)=\min_{z^{2}\leq\delta}\bigg{\\{}\frac{1+z}{2}\arccos\bigg{(}\frac{2\delta-1-z^{2}}{1-z^{2}}\bigg{)}\bigg{\\}}.$
(6)

As $\tau_{\textsc{mt}}(\delta)$, the bound $\tau_{\textsc{ml}}(\delta)$ is
tight, and $\tau_{\textsc{ml}}(0)=\tau_{\textsc{ml}}$. The bound
$\tau_{\textsc{ml}}(\delta)$ was established numerically in Ref. [11] and
derived analytically in Ref. [12]. Reference [12] also contains a geometric
interpretation of $\tau_{\textsc{ml}}(\delta)$ and a complete description of
the systems that reach the bound.

A natural guess is that the Margolus-Levitin QSL is also valid for closed
systems provided the time average of the normalized expected energy is placed
in the denominator. More generally, one might expect that there is a QSL for
closed systems of the form $\mathcal{L}(\delta)/\langle\langle
H_{t}-\epsilon_{\text{min};t}\rangle\rangle$ where $\mathcal{L}$ is some non-
negative function that depends only on the fidelity $\delta$ between the
initial and final states and $\langle\langle
H_{t}-\epsilon_{\text{min};t}\rangle\rangle$ is the time average of the
instantaneous normalized expected energy.555To be interesting, $\mathcal{L}$
in such a hypothetical QSL should satisfy $\mathcal{L}(1)=0$ and
$\mathcal{L}(\delta)>0$ for $0\leq\delta<1$. It has been argued that one can
take $\mathcal{L}(\delta)=\arccos\sqrt{\delta}$ [13, 14, 15]. However, for
$0<\delta<1$, such a hypothetical QSL is violated by an isolated system that
evolves between two states with fidelity $\delta$ in time
$\tau_{\textsc{ml}}(\delta)$. This is because $\alpha(\delta)$ is strictly
smaller than $\arccos\sqrt{\delta}$ for $0<\delta<1$ [11, 12]. For $\delta=0$,
such a hypothetical QSL can be violated by a closed system in the family of
systems constructed in the next section.666Reference [13] has been criticized
before [16]. However, the modified Margolus-Levitin estimate in Ref. [16] is
also incorrect because it is violated by any isolated system that evolves
between two states with fidelity $0<\delta<1$ in time
$\tau_{\textsc{ml}}(\delta)$ [12]. It has also been argued that one can take
$\mathcal{L}(\delta)=\sin^{2}(\arccos\sqrt{\delta})/2=(1-\delta)/2$ [14]. Nor
is that possible as we show below.

The main result of this paper is as follows: For each state $\rho$, each
fidelity $\delta$, and each positive $\mathcal{L}(\delta)$, there exists a
time-dependent Hamiltonian $H_{t}$ that evolves $\rho$ to a state with
fidelity $\delta$ relative to $\rho$ in a time less than
$\mathcal{L}(\delta)/\langle\langle
H_{t}-\epsilon_{\text{min};t}\rangle\rangle$. The Hamiltonian can be chosen
such that the normalized expected energy remains fixed at an arbitrary
predetermined value.

Lui _et al._ [17] used the Bhatia–Davies inequality to transform the
Mandelstam-Tamm QSL into an upper bound for an operationally defined QSL [18].
This upper bound is a new QSL that we call the Bhatia-Davies QSL, although we
should rightly attribute it to the authors of Ref. [17]. The Bhatia-Davies QSL
states that the time it takes for an isolated system to evolve between two
states with fidelity $\delta$ is bounded from below by

$\tau_{\textsc{bd}}(\delta)=\frac{\arccos\sqrt{\delta}}{\sqrt{\langle\epsilon_{\text{max}}-H\rangle\langle
H-\epsilon_{\mathrm{min}}\rangle}},$ (7)

where $\epsilon_{\text{max}}$ is the largest and $\epsilon_{\mathrm{min}}$ is
the smallest occupied energy. The Bhatia-Davies QSL also extends
straightforwardly to closed systems:

$\bar{\tau}_{\textsc{bd}}(\delta)=\frac{\arccos\sqrt{\delta}}{\langle\langle\sqrt{\langle\epsilon_{\text{max};t}-H_{t}\rangle\langle
H_{t}-\epsilon_{\text{min};t}\rangle}\,\rangle\rangle}.$ (8)

The Bhatia-Davies QSL is weaker than that of Mandelstam and Tamm in the sense
that $\bar{\tau}_{\textsc{mt}}(\delta)\geq\bar{\tau}_{\textsc{bd}}(\delta)$
with a strict inequality in general for both isolated and closed systems. We
show that the Mandelstam-Tamm and the Bhatia-Davies QSLs are always saturated
simultaneously for isolated systems, but that this need not be the case for
closed systems: We give examples of closed systems that saturate the
Mandelstam-Tamm QSL but not the Bhatia-Davies QSL.

##  II Time-dependent systems that disprove common belief

One obtains a relatively simple type of time-dependent Hamiltonian if one
conjugates a time-independent Hermitian operator $H$ with a one-parameter
group of unitaries generated by a Hermitian operator $A$:

$H_{t}=e^{-iAt}He^{iAt}.$ (9)

Such a group action will preserve the eigenvalues but rotate the eigenvectors
of $H$. If a state $\rho$ evolves under the influence of $H_{t}$, that is, if

$\dot{\rho}_{t}=-i[H_{t},\rho_{t}],\qquad\rho_{0}=\rho,$ (10)

the state in the rotating frame picture,

$\rho^{\textsc{rf}}_{t}=e^{iAt}\rho_{t}e^{-iAt},$ (11)

evolves as if $H-A$ governs the dynamics:

$\dot{\rho}^{\textsc{rf}}_{t}=-i[H-A,\rho^{\textsc{rf}}_{t}],\qquad\rho^{\textsc{rf}}_{0}=\rho.$
(12)

As a consequence, in the Schrödinger picture,

$\rho_{t}=e^{-iAt}e^{-i(H-A)t}\rho e^{i(H-A)t}e^{iAt}.$ (13)

The behavior of $\rho_{t}$ can be quite complex even though $H_{t}$ has a
relatively simple time dependence. However, Eq. (13) tells us that if the
initial state $\rho$ commutes with $H-A$, the evolving state will behave as if
the time-independent “effective Hamiltonian” $A$ generates it:

$\rho_{t}=e^{-iAt}\rho e^{iAt}.$ (14)

This observation will be of central importance below.

The eigenvectors of $H_{t}$ will also evolve with $A$ as an effective
Hamiltonian: If $|j\rangle$ is an eigenvector of $H$ with the eigenvalue
$\epsilon_{j}$, then $|j;t\rangle=e^{-iAt}|j\rangle$ is an eigenvector of
$H_{t}$ with the eigenvalue $\epsilon_{j}$. As a result, the occupations of
the energy levels are constant over time:

$\langle j;t|\rho_{t}|j;t\rangle=\langle j|\rho|j\rangle.$ (15)

This means that the expected energy $\langle H_{t}\rangle$, the energy
uncertainty $\Delta H_{t}$, the normalized expected energy $\langle
H_{t}-\epsilon_{\text{min};t}\rangle$, and the “dual” normalized expected
energy $\langle\epsilon_{\text{max};t}-H_{t}\rangle$ are conserved quantities;
see Refs. [19, 12] for a QSL involving the dual of the normalized expected
energy.

Another important fact is that if the initial state $\rho$ satisfies
$A\rho+\rho A=A$, the evolving state $\rho_{t}$ is a Fubini-Study geodesic;
see Appendix A in Ref. [7]. If such is the case, the Mandelstam-Tamm QSL is
saturated, and the system evolves between two states with fidelity $\delta$ in
time $\bar{\tau}_{\textsc{mt}}(\delta)$. Interestingly, given any initial
state $\rho$ and any Hermitian operator $H$, there is an elegant way to
construct an $A$ such that $[H-A,\rho]=0$ and $A\rho+\rho A=A$: Write
$\rho=|u\rangle\langle u|$ and define

$A=(H-\langle u|H|u\rangle)|u\rangle\langle u|+|u\rangle\langle u|(H-\langle
u|H|u\rangle).$ (16)

In the following, we show how to disprove two hypotheses about QSLs with
appropriate choices of $\rho$ and $H$, and $A$ defined as in (16).

###  II.1 The nonexistence of a time-dependent Margolus-Levitin quantum speed
limit

The Mandelstam-Tamm and Margolus-Levitin QSLs tell us that an isolated system
with large energy uncertainty that evolves along a Fubini-Study geodesic has a
correspondingly large normalized expected energy [12]. Interestingly, this
need not hold for systems with time-dependent Hamiltonians. Below we give an
example of a family of closed systems evolving along geodesics, whose
normalized expected energies are conserved with an arbitrary common
predetermined value and whose energy uncertainties are conserved and form an
unbounded set.

Fix an $E>0$ and consider a quantum system in a state $\rho=|u\rangle\langle
u|$. Let $H_{\theta}$ be a Hermitian operator, to be specified, and define
$A_{\theta}$ as in Eq. (16):

$A_{\theta}=(H_{\theta}-\langle u|H_{\theta}|u\rangle)|u\rangle\langle
u|+|u\rangle\langle u|(H_{\theta}-\langle u|H_{\theta}|u\rangle).$ (17)

Furthermore, let $H_{\theta;t}=e^{-iA_{\theta}t}H_{\theta}e^{iA_{\theta}t}$
and let $\rho_{\theta;t}$ be the state at time $t$ generated from $\rho$ by
$H_{\theta;t}$. Then $\rho_{\theta;t}$ is a Fubini-Study geodesic.

Figure 1: Graphs illustrating the dependence of the energy uncertainty (blue)
and the normalized expected energy (red) on the angle $\theta$. The
requirement that the normalized expected energy be constant forces the energy
uncertainty to grow toward infinity with decreasing angle.

To specify $H_{\theta}$ let $|v\rangle$ be a unit vector perpendicular to
$|u\rangle$ and define the Pauli operators $X$ and $Z$ as

$\displaystyle X$ $\displaystyle=|u\rangle\langle u|-|v\rangle\langle v|,$
(18) $\displaystyle Z$ $\displaystyle=|u\rangle\langle v|+|v\rangle\langle
u|.$ (19)

Let $\mu(\theta)=E/(1-\cos\theta)$ for $0<\theta<\pi$, and define

$H_{\theta}=\mu(\theta)(\sin\theta Z-\cos\theta X).$ (20)

The largest and the smallest eigenvalues of $H_{\theta}$, and thus of
$H_{\theta;t}$, are $\mu(\theta)$ and $-\mu(\theta)$, respectively, both of
which are occupied by $\rho_{\theta;t}$. Furthermore, the normalized expected
energy and the energy uncertainty are

$\displaystyle\langle
H_{\theta;t}-\epsilon_{\text{min},\theta;t}\rangle=\mu(\theta)(1-\cos\theta)=E,$
(21) $\displaystyle\Delta H_{\theta;t}=\mu(\theta)\sin\theta=E\cot(\theta/2).$
(22)

The blue graph in Fig. 1 illustrates how the energy uncertainty depends on
$\theta$, and the red graph accentuates that the normalized expected energy
does not depend on $\theta$. Note that both quantities are independent of $t$.

Figure 2: Rabi and Bloch vector representations of $H_{\theta}$, $A_{\theta}$,
and $\rho$. The vector representing $\rho$ points along the positive $x$ axis,
and the vector representing $H_{\theta}$ makes the angle $\theta$ with the
negative $x$ axis. The purple circle represents the expected energy level to
which $\rho$ belongs. As time passes, the state and the Hamiltonian rotate
around the $z$ axis with the same angular velocity. The dashed vectors
represent $H_{\theta;t}$ and $\rho_{\theta;t}$ at a time $t>0$. The expected
energy level rotates with the state.

Next, fix a fidelity $0\leq\delta<1$ (for $\delta=1$ there is nothing to
prove), let $\mathcal{L}(\delta)$ be any positive number representing the
numerator in a hypothetical extension of the Margolus-Levitin QSL, and choose
$\theta$ such that

$\cot(\theta/2)>\frac{\arccos{\sqrt{\delta}}}{\mathcal{L}(\delta)}.$ (23)

Furthermore, let $\tau(\delta)$ be the first time $t$ the fidelity between
$\rho_{\theta;t}$ and $\rho$ is $\delta$.777Since $\rho_{\theta,t}$ is a
geodesic, the fidelity between $\rho_{\theta,t}$ and $\rho$ will sooner or
later be $\delta$. Since the state follows a Fubini-Study geodesic, the
Mandelstam-Tamm QSL is saturated:
$\tau(\delta)=\bar{\tau}_{\textsc{mt}}(\delta)$. According to Eqs. (21)–(23),

$\tau(\delta)=\frac{\arccos\sqrt{\delta}}{E\cot(\theta/2)}<\frac{\mathcal{L}(\delta)}{E}=\frac{\mathcal{L}(\delta)}{\langle\langle
H_{\theta;t}-\epsilon_{\text{min},\theta;t}\rangle\rangle}.$ (24)

We conclude that for each state $\rho$ and each fidelity $\delta$, there is a
time-dependent Hamiltonian that evolves $\rho$ to a state with fidelity
$\delta$ relative to $\rho$ in a time less than a hypothetical QSL of the form
given by the rightmost expression in (24). Thus, the Margolus-Levitin QSL does
not straightforwardly extend to closed systems.

In Fig. 2 we have represented $H_{\theta}$, $A_{\theta}$, and $\rho$ as Rabi
and Bloch vectors relative to $X$, $Y$, and $Z$, with $Y=i(|u\rangle\langle
v|-|v\rangle\langle u|)$. The angle between $H_{\theta}$ and the negative $x$
axis is $\theta$. As time passes, the state and the Hamiltonian rotate around
the $z$ axis with the same angular speed. Note that $\rho_{\theta;t}$ moves
along the equator in the Bloch sphere and thus is a Fubini-Study geodesic. The
dotted vectors represent the state and the Hamiltonian at a time $t>0$.

The purple circle formed by intersecting the Bloch sphere with a plane
perpendicular to the extension of the vector representing $H_{\theta}$
represents the expected energy level to which $\rho$ belongs. This circle
rotates together with $H_{\theta;t}$ and always lies in a plane perpendicular
to the vector representing $H_{\theta;t}$. The key observation is that this
circle corresponds to the normalized expected energy $E$ irrespective of the
value of angle $\theta$, and $\rho_{\theta;t}$ will evolve together with that
circle.

Most initial states will not evolve in such a well-behaved manner as those
located on the $x$ axis of the Bloch sphere. In Fig. 3 we have drawn the
evolution curve of a state not on the $x$ axis.

Figure 3: An evolution curve starting from a state not on the equator of the
Bloch sphere. Here, $\theta=30^{\circ}$ and $E=1$. The left figure shows the
evolution curve in the Schrödinger picture, and the right figure shows the
same curve in the rotating frame picture. The warmer colors indicate more
recent times, and the blue arrow represents the state at the final time.

In the rotating frame picture (12), the evolution curve forms a circle around
the $x$ axis. This is since $H_{\theta}-A_{\theta}\propto X$.

###  II.2 The Bhatia-Davies quantum speed limit

In the previous section, we exploited that for some members of a family of
closed systems with a common conserved normalized expected energy, the energy
uncertainty is large enough for the evolution time to violate any hypothetical
straightforward extension of the Margolus-Levitin QSL. For the systems in the
family, we allowed for an arbitrary width of the energy spectrum. If we
require that the spectral width does not exceed a given value, the evolution
time will be uniformly bounded from below by a positive constant. This is
because the energy uncertainty cannot exceed the spectral width.

The Bhatia-Davies inequality [20] provides a tighter bound on the energy
uncertainty than the spectral width. The Bhatia-Davies inequality states that
the variance of any observable $B$ is bounded from above according to

$\Delta^{2}B\leq\langle b_{\mathrm{max}}-B\rangle\langle
B-b_{\mathrm{min}}\rangle,$ (25)

with $b_{\mathrm{max}}$ and $b_{\mathrm{min}}$ being the largest and the
smallest occupied eigenvalue of $B$. Consequently, the evolution time of an
isolated system is bounded by $\tau_{\textsc{bd}}(\delta)$ defined in (7), and
the evolution time of a closed system is bounded by
$\bar{\tau}_{\textsc{bd}}(\delta)$ defined in (8).

Equality holds in the Bhatia-Davies inequality if and only if the state
occupies at most two eigenvalues of $B$. Since the state of an isolated system
saturating the Mandelstam-Tamm QSL occupies only two energy levels [21, 7],
the Mandelstam-Tamm and Bhatia-Davies QSLs are always saturated simultaneously
for isolated systems.

The Mandelstam-Tamm and Bhatia-Davies QSLs generalize to closed systems as in
(3) and (8), respectively, and a natural guess would be that also these QSLs
are always saturated simultaneously. However, as we will see, a time-dependent
Hamiltonian can evolve a state at a constant speed along a Fubini-Study
geodesic in such a way that the state during the entire evolution occupies
more than two energy levels. Such an evolution will saturate the
Mandelstam–Tamm QSL but not the Bhatia–Davies QSL. This is because the Bhatia-
Davies inequality will be strict over the entire evolution time interval,
which means that the denominator in (8) is strictly greater than the
denominator in (3).

###  II.3 A nonsaturation of the Bhatia-Davies quantum speed limit

Let $H$ be a Hermitian operator with at least three distinct eigenvalues, and
let $\rho=|u\rangle\langle u|$ be any state occupying at least three of those.
Define $A$ as in (16), let $H_{t}=e^{-iAt}He^{iAt}$, and let $\rho_{t}$ be the
state at time $t$ generated from $\rho$ by $H_{t}$. Since $[H-A,\rho]=0$ and
$A\rho+\rho A=A$, the system evolves between two states with fidelity $\delta$
in time $\bar{\tau}_{\textsc{mt}}(\delta)$. Furthermore, according to (15),
$\rho_{t}$ always occupies at least three different energy levels. Therefore,

$\Delta^{2}H_{t}<\langle\epsilon_{\text{max};t}-H_{t}\rangle\langle
H_{t}-\epsilon_{\text{min};t}\rangle,$ (26)

and $\bar{\tau}_{\textsc{mt}}(\delta)>\bar{\tau}_{\textsc{bd}}(\delta)$. We
conclude that the Mandelstam-Tamm QSL is saturated but not the Bhatia-Davies
QSL.

##  III Summary

A common view is that the Margolus-Levitin quantum speed limit extends to an
evolution time estimate for closed systems of the form
$\mathcal{L}(\delta)/\langle\langle
H_{t}-\epsilon_{\text{min};t}\rangle\rangle$, where $\mathcal{L}$ is a
positive function that depends only on the fidelity $\delta$ between the
initial and final states and $\langle\langle
H_{t}-\epsilon_{\text{min};t}\rangle\rangle$ is the time average of the
normalized expected energy. We have shown that this is not the case. More
precisely, for any fidelity $\delta$ and any positive numbers $E$ and
$\mathcal{L}(\delta)$, we have constructed a closed system with a conserved
normalized expected energy $E$ that evolves between two states with fidelity
$\delta$ in a time strictly less than $\mathcal{L}(\delta)/E$.

We have also considered a QSL for isolated systems called the Bhatia-Davies
QSL. This QSL extends straightforwardly to closed systems. We have shown that
the Bhatia-Davies and Mandelstam-Tamm QSLs are always simultaneously saturated
for isolated systems but that this need not be the case for closed systems.
The state of a closed system that saturates the Mandelstam-Tamm QSL but not
the Bhatia-Davies QSL must, at some instant, occupy at least three different
energy levels.

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# A new regularisation for time-fractional backward heat conduction problem

M.Thamban Nair and P. Danumjaya  Department of Mathematics, BITS Pilani, K.K.
Birla Goa Campus, Zuarinagar, Goa 403726, INDIA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

It is well-known that the backward heat conduction problem of recovering the
temperature $u(\cdot,t)$ at a time $t\geq 0$ from the knowledge of the
temperature at a later time, namely $g:=u(\cdot,\tau)$ for $\tau>t$, is ill-
posed, in the sense that small error in $g$ can lead to large deviation in
$u(\cdot,t)$. However, in the case of a time fractional backward heat
conduction problem (TFBHCP), the above problem is well-posed for $t>0$ and
ill-posed for $t=0$. We use this observation to obtain stable approximate
solutions for the TFBHCP for $t=0$, and derive error estimates under suitable
source conditions. We shall also provide some numerical examples to illustrate
the approximation properties of the regularized solutions.

##  1\. Introduction

For $0<\alpha<1$, consider the time-fractional heat equation

(1) $\frac{\partial^{\alpha}u}{\partial
t^{\alpha}}=\frac{\partial^{2}u}{\partial x^{2}},\quad 0<x<\pi.$

In the above, we used the $\alpha$-derivative of $u$ with respect to $t$ in
the Caputo sense. That is, if $\varphi$ is a real valued differentiable
function on an open interval of the form $(0,a)$ for some $a>0$,

$\frac{d^{\alpha}\varphi}{dt^{\alpha}}(t)=\frac{1}{\Gamma(1-\alpha)}\int_{0}^{t}\frac{\varphi^{\prime}(s)}{(t-s)^{\alpha}}ds,\quad
0<t<a.$

It is to be observed that for $\alpha=1$, the equation (1) reduces to the
ordinary heat equation

$\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x^{2}},\quad
0<x<\pi,$

and in that case, under the boundary condition

(2) $u(0,t)=0=u(\pi,t),\quad t>0,$

and initial condition

(3) $u(0,x)=f_{0}(x),\quad 0<x<\pi$

for $f_{0}\in L^{2}[0,\pi]$, the solution $u(\cdot,\cdot)$ is given by

(4) $u(\cdot,t)=\sum_{n=1}^{\infty}E(-\lambda_{n}^{2}t)\langle
f_{0},\varphi_{n}\rangle\varphi_{n},$

where $E(\cdot)$ is the exponential function $E(s):=e^{s}$ and

$\lambda_{n}:=n,\quad\varphi_{n}(x)=\sqrt{\frac{2}{\pi}}\sin(kx).$

We know that $\\{\varphi_{n}:n\in{\mathbb{N}}\\}$ is an orthonormal basis of
$L^{2}[0,\pi]$.

It is known (cf. [3]) that, for the time-fractional heat equation (1) along
with the boundary and initial conditions (2)-(3), similar representation for
its solution is possible by replacing $E(-\lambda_{n}^{2}t)$ in (4) by
$E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})$, where $E_{\alpha}(\cdot)$ is the
Mittag-Leffler function defined by

(5) $E_{\alpha}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(k\alpha+1)},\quad
z\in{\mathbb{C}},$

with $\Gamma(\cdot)$ being the gamma function

$\Gamma(s):=\int_{0}^{\infty}e^{-t}t^{s-1}dt,\quad s>0.$

Clearly, $E_{1}(z)=E(z)$ for $z\in{\mathbb{C}}$. In fact, for any $f_{0}\in
L^{2}[0,\pi]$, the solution $u_{\alpha}(\cdot,\cdot)$ of (1) along with the
boundary and initial conditions (2)-(3), is given by

(6)
$u_{\alpha}(\cdot,t)=\sum_{n=1}^{\infty}E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})\langle
f_{0},\varphi_{n}\rangle\varphi_{n}$

with $\langle\cdot,\cdot\rangle$ being the $L^{2}$-inner product (cf. [3,
11]). Throughout, by $\|\cdot\|$ we shall mean the norm induced by the
$L^{2}$-inner product.

With regard to the Mittag-Leffler function defined in (5), we have the
following result which we shall make use of throughout the paper.

######  Lemma 1.1.

(cf. [6, 9]) Given real numbers $\alpha_{0},\alpha_{1}$ such that
$0<\alpha_{0}<\alpha_{1}<1$, there exist $C_{1}>0$ and $C_{2}>0$ such that

$\frac{C_{1}}{\Gamma(1-\alpha)(1+x)}\leq
E_{\alpha}(-x)\leq\frac{C_{2}}{\Gamma(1-\alpha)(1+x)}$

for all $x>0$ and for all $\alpha\in[\alpha_{0},\alpha_{1}]$.

In view of the above lemma, it can be seen that $u_{\alpha}(\cdot,t)$ in (6 )
is a $C^{\infty}$-function for each $t>0$.

Analogous to the well-studied ordinary backward heat conduction problem, let
us consider the following inverse problem, the time-fractional backward heat
conduction problem (TFBHCP) associated with (1) and (2):

Problem ($P_{t}$): Knowing $g:=u_{\alpha}(\cdot,\tau)$ for some $\tau>0$, find
$u_{\alpha}(\cdot,t)$ for $0\leq t<\tau$. Many studies have shown that
fractional diffusion equation model is appropriate for investigating problems
arising in the areas of spatially disordered systems, porous media, fractal
media, turbulent fluids and plasmas, biological media with traps, and stock
price movements and so on (see [9, 11], and the references therein). The
regularization theory for the inverse problems associated with fractional-
order PDEs is still in its infancy.

We shall see that the inverse problem $P_{t}$ is well-posed if $0<t<\tau$ and
ill-posed if $t=0$. This observation and the subsequent analysis lead us to
the conclusion that $\\{P_{t}:0<t<\tau\\}$ gives a regularization family for
obtaining stable approximate solutions for the ill-posed inverse problem
$(P_{0})$. To our knowledge, no study is carried out using the above
observation, though various regularization methods are discussed recently
(see, e.g. [11, 1, 3, 5], and the references there in). We shall also provide
estimates for the error $\|u_{\alpha}(t,\cdot)-u_{\alpha}(0,\cdot)\|_{2}$
under certain a priori source condition.

An outline of this paper is as follows. In section 2, we discuss the ill-
posedness of the time fractional backward heat conduction problem (TFBHCP).
Section 3 deals an operator theoretic formulation of the inverse problems. The
new regularization family for the ill-posed inverse problem $P_{0}$ is
introduced and its convergence is proved in Section 4. In Section 5 and 6, we
derive the error estimates for the noisy data and source conditions,
respectively. Finally, we perform some numerical experiments to validate the
theoretical results in Section 7.

##  2\. Ill-Posednss of the Inverse Problem

Let $g=u_{\alpha}(\cdot,\tau)$. Then, from equation (6), we have

(7) $g=\sum_{n=1}^{\infty}E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\langle
f_{0},\varphi_{n}\rangle\varphi_{n}.$

Hence,

(8) $\langle f_{0},\varphi_{n}\rangle=\frac{\langle
g,\varphi_{n}\rangle}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\quad\forall\,n\in{\mathbb{N}}.$

This shows that $g:=u_{\alpha}(\cdot,\tau)$ must satisfy the Picard condition:

(9) $\sum_{n=1}^{\infty}\frac{|\langle
g,\varphi_{n}\rangle|^{2}}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})^{2}}<\infty.$

By Lemma 1.1, it is to be observed that

$\frac{1}{C_{2}}\Gamma(1-\alpha)(1+\lambda_{n}^{2}\tau^{\alpha})\leq\frac{1}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})^{2}}\leq\frac{1}{C_{1}}\Gamma(1-\alpha)(1+\lambda_{n}^{2}\tau^{\alpha}).$

Therefore, Picard condition (9) on $g$ is equivalent to the requirement

(10) $\sum_{n=1}^{\infty}(1+\lambda_{n}^{2}\tau^{\alpha})|\langle
g,\varphi_{n}\rangle|^{2}<\infty$

which is again equivalent to

$\sum_{n=1}^{\infty}n^{2}|\langle g,\varphi_{n}\rangle|^{2}<\infty.$

Using (8), the representation of $u_{\alpha}(\cdot,t)$ in (6) takes the form

(11)
$u_{\alpha}(\cdot,t)=\sum_{n=1}^{\infty}\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\langle
g,\varphi_{n}\rangle\varphi_{n}$

so that

(12)
$\|u_{\alpha}(\cdot,t)\|^{2}=\sum_{n=1}^{\infty}\Big{|}\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\Big{|}^{2}|\langle
g,\varphi_{n}\rangle|^{2}.$

Again, using Lemma 1.1, we have

(13)
$\frac{C_{1}}{C_{2}}\frac{(1+\lambda_{n}^{2}\tau^{\alpha})}{(1+\lambda_{n}^{2}t^{\alpha})}\leq\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\leq\frac{C_{2}}{C_{1}}\frac{(1+\lambda_{n}^{2}\tau^{\alpha})}{(1+\lambda_{n}^{2}t^{\alpha})}$

Note that,

$\frac{(1+\lambda_{n}^{2}\tau^{\alpha})}{(1+\lambda_{n}^{2}t^{\alpha})}\geq\frac{\lambda_{n}^{2}\tau^{\alpha}}{(\lambda_{n}^{2}+\lambda_{n}^{2}t^{\alpha})}=\frac{\tau^{\alpha}}{(1+t^{\alpha})},$

and for $0<t\leq\tau$,

$\frac{(1+\lambda_{n}^{2}\tau^{\alpha})}{(1+\lambda_{n}^{2}t^{\alpha})}\leq\frac{(\lambda_{n}^{2}+\lambda_{n}^{2}\tau^{\alpha})}{\lambda_{n}^{2}t^{\alpha}}=\frac{(1+\tau^{\alpha})}{t^{\alpha}}.$

Hence, for $0<t\leq\tau$,

(14)
$\frac{C_{1}}{C_{2}}\frac{\tau^{\alpha}}{(1+t^{\alpha})}\leq\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\leq\frac{C_{2}}{C_{1}}\frac{(1+\tau^{\alpha})}{t^{\alpha}}.$

Hence, the representation (12) together with (14) imply that if $0<t\leq\tau$,
then

(15)
$\frac{C_{1}}{C_{2}}\frac{\tau^{\alpha}}{(1+t^{\alpha})}\|g\|\leq\|u_{\alpha}(\cdot,t)\|\leq\frac{C_{2}}{C_{1}}\frac{(1+\tau^{\alpha})}{t^{\alpha}}\|g\|$

and if $t=0$, then (12) and (13) imply

(16)
$\|u_{\alpha}(\cdot,0)\|^{2}\geq\Big{(}\frac{C_{1}}{C_{2}}\Big{)}^{2}\sum_{n=1}^{\infty}(1+\lambda_{n}^{2}\tau^{\alpha})^{2}|\langle
g,\varphi_{n}\rangle|^{2}.$

Note that, by (10) the series on the right hand side of the above inequality
converges. However, corresponding to an initial temperature
$\tilde{u}_{\alpha}(\cdot,0)$, if the temperature at time $\tau$ is
$\tilde{g}$, then the above arguments lead to

$\|u_{\alpha}(\cdot,0)-\tilde{u}_{\alpha}(\cdot,0)\|^{2}\geq\Big{(}\frac{C_{1}}{C_{2}}\Big{)}^{2}\sum_{n=1}^{\infty}(1+\lambda_{n}^{2}\tau^{\alpha})^{2}|\langle
g-\tilde{g},\varphi_{n}\rangle|^{2}.$

From this we obtain

(17)
$\|u_{\alpha}(\cdot,0)-\tilde{u}_{\alpha}(\cdot,0)\|\geq(1+\lambda_{n}^{2}\tau^{\alpha})\Big{(}\frac{C_{1}}{C_{2}}\Big{)}\|g-\tilde{g}\|\quad\forall\,n\in{\mathbb{N}}.$

This shows that small error in $g$ can lead to large deviation in the solution
$u_{\alpha}(\cdot,0)$, even when the the data satisfy the Picard condition as
in (9). Thus, from the inequalities in (15) and (17), we can infer the
following.

######  Theorem 2.1.

Let $C_{1},C_{2}$ and $\alpha$ be as in Lemma 1.1. Then the TFBHCP $(P_{t})$
is well-posed for $0<t<\tau$ and the problem $(P_{0})$ is ill-posed.

What we are interested in is to find stable approximate solutions for the ill-
posed inverse problem $(P_{0})$.

##  3\. Operator Theoretic formulation of the inverse problems

For a few observations on operators that appear in this section, we shall make
use of the following proposition based on basic results from functional
analysis. For the sake of completeness of the exposition, we provide its proof
as well.

######  Proposition 3.1.

Let ${\mathcal{H}}$ be an infinite dimensional separable Hilbert space and let
$\\{v_{n}:n\in{\mathbb{N}}\\}$ be an orthonormal basis of ${\mathcal{H}}$. Let
$(\mu_{n})$ be a bounded sequence of real numbers and
$A:{\mathcal{H}}\to{\mathcal{H}}$ be defined by

$Av=\sum_{n=1}^{\infty}\mu_{n}\langle v,v_{n}\rangle_{\mathcal{H}}v_{n},\quad
v\in\mathcal{H},$

where $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ denotes the inner product on
${\mathcal{H}}$. Then $A$ is a self-adjoint, bounded linear operator. Further,
we have the following:

  1. (i)

If $\mu_{n}\to 0$, then $A$ is a compact operator.

  2. (ii)

If $\mu_{n}\not=0$ for all $n\in{\mathbb{N}}$, then $A$ is injective and range
of $A$ is dense.

  3. (iii)

If there exists $c_{0}>0$ such that $|\mu_{n}|\geq c_{0}$ for all
$n\in{\mathbb{N}}$, then $A$ is injective, range of $A$ is closed, and its
inverse from the range is continuous.

In particular, if assumptions in (ii) and (iii) are satisfied, then $A$ is
bijective and its inverse is continuous.

###### Proof.

Using the boundedness of $(\mu_{n})$ it follows from Riesz-Fischer theorem
(cf. [7]) that $A$ bounded linear operator on ${\mathcal{H}}$ with
$\|A\|\leq\sup\\{|\mu_{n}|:n\in{\mathbb{N}}\\}$. Also, since
$\mu_{n}\in{\mathbb{R}}$, we that $A$ is a self-adjoint operator.

(i) Suppose $\mu_{n}\to 0$. For $n\in{\mathbb{N}}$, let
$A_{n}:{\mathcal{H}}\to{\mathcal{H}}$ be defined by

$A_{n}v=\sum_{j=1}^{n}\mu_{j}\langle v,v_{j}\rangle_{\mathcal{H}}v_{j},\quad
v\in\mathcal{H}.$

Then we see that

$\|A-A_{n}\|\leq\sup\\{|\mu_{j}|:j>n\\}\to 0.$

Since each $A_{n}$ is a finite rank operator, it follows that (cf. Theorem 9.-
in [7]) $A$ is a compact operator .

(ii) Suppose $\mu_{n}\not=0$ for all $n\in{\mathbb{N}}$. Then for
$v\in\mathcal{H}$, we have

$Av=0\iff\mu_{n}\langle
v,v_{n}\rangle_{\mathcal{H}}=0\,\,\forall\,n\in{\mathbb{N}}\iff\langle
v,v_{n}\rangle_{\mathcal{H}}=0\,\,\forall\,n\in{\mathbb{N}}\iff v=0.$

Hence, $A$ is injective. Now, to see that $R(A)$, the range of $A$, is dense
in ${\mathcal{H}}$, let $w\in{\mathcal{H}}$ be such that $\langle
Av,w\rangle=0$ for all $v\in{\mathcal{H}}$. Then, in particular, we have

$\mu_{n}\langle\varphi_{n},w\rangle=\langle
A\varphi_{n},w\rangle=0\quad\forall n\in{\mathbb{N}}.$

From this, using again the fact that $\mu_{n}\not=0$ for all
$n\in{\mathbb{N}}$, we have $w=0$. Thus, we have proved that
$R(A)^{\perp}=\\{0\\}$, which implies, by projection theorem, that $R(A)$ is
dense.

(iii) Suppose there exists $c_{0}>0$ such that $|\mu_{n}|\geq c_{0}$ for all
$n\in{\mathbb{N}}$. Then we have

$\|Av\|_{\mathcal{H}}\geq
c_{0}\|v\|_{\mathcal{H}}\quad\forall\,v\in\mathcal{H}.$

From this, the conclusions follow.

The last part of the theorem is obvious. ∎

Now, consider the operator $A_{\alpha}:L^{2}[0,\pi]\to L^{2}[0,\pi]$ defined
by

$A_{\alpha}f=\sum_{n=1}^{\infty}E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\langle
f,\varphi_{n}\rangle\varphi_{n},\ f\in L^{2}[0,\pi].$

Using Lemma 1.1, we see that

$E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\to 0\quad\hbox{as}\quad
n\to\infty.$

Hence, by Proposition 3.1, $A_{\alpha}$ is a compact operator of infinite
rank. Hence, in view of equation (7), the fact that the problem $(P_{0})$ is
ill-posed also follows from the observation that (7) is same as solving the
compact operator equation

(18) $A_{\alpha}f=g,$

which is an ill-posed problem.

Again, by Proposition 3.1, $A_{\alpha}$ is one-one and its range is dense in
$L^{2}[0,\pi]$. Hence, if $g\in L^{2}[0,\pi]$ satisfies the the Picard
condition (9), then it is in the range of $A_{\alpha}$ and
$f_{0}:=u_{\alpha}(\cdot,0)$ is the the generalized solution of the operator
equation (18), that is,

$f_{0}=A_{\alpha}^{\dagger}g,$

where $A_{\alpha}^{\dagger}$ denotes the Moore-Penrose inverse of $A_{\alpha}$
(cf. [8]).

Next, we observe from (6) that

$\langle f_{0},\varphi_{n}\rangle=\frac{\langle
u_{\alpha}(\cdot,t),\varphi_{n}\rangle}{E_{\alpha}(\lambda_{n}^{2}t^{\alpha})}\quad\forall\,n\in{\mathbb{N}},$

so that equation (7) leads to

(19)
$g=\sum_{n=1}^{\infty}\frac{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}\langle
u_{\alpha}(\cdot,t),\varphi_{n}\rangle\varphi_{n}.$

Interchanging $t$ and $\tau$ in (14), we obtain

$\frac{C_{1}}{C_{2}}\frac{t^{\alpha}}{(1+\tau^{\alpha})}\leq\frac{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}\leq\frac{C_{2}}{C_{1}}\frac{(1+t^{\alpha})}{\tau^{\alpha}}.$

Hence, by Proposition 3.1, it follows that for $0<t<\tau$,
$B_{\alpha,t}:L^{2}[0,\pi]\to L^{2}[0,\pi]$ defined by

$B_{\alpha,t}f=\sum_{n=1}^{\infty}\frac{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}\langle
f,\varphi_{n}\rangle\varphi_{n},\quad f\in L^{2}[0,\pi],$

is a bijective bounded linear operator with continuous inverse. Thus, the
problem $(P_{t})$ of recovering $u_{\alpha}(\cdot,t)$ from $g$, which
corresponds to the equation (19) is same as the problem of solving the
operator equation

$B_{\alpha,t}f=g,$

which is a well-posed problem.

##  4\. The Regularization

In view of the expression (11), for each $\alpha\in(0,1)$ and $t\in(0,\tau)$,
we define the map $R_{t,\alpha}:L^{2}[0,\pi]\to L^{2}[0,\pi]$ as

(20)
$R_{t,\alpha}\psi=\sum_{n=1}^{\infty}\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\langle\psi,\varphi_{n}\rangle\varphi_{n},\quad\psi\in
L^{2}[0,\pi].$

In view of (14), we see that $R_{t,\alpha}:L^{2}[0,\pi]\to L^{2}[0,\pi]$ is a
well-defined bounded linear operator with

$\|R_{t,\alpha}\|\leq\frac{C_{2}}{C_{1}}\Big{(}\frac{1+\tau\alpha}{t^{\alpha}}\Big{)}.$

Note that if $g=u_{\alpha}(\cdot,\tau)$, then by (11),

$R_{t,\alpha}g=u_{\alpha}(\cdot,t),\quad 0<t<\tau.$

The following theorem shows that, for each $\alpha\in(0,1)$, the family
$\\{R_{t,\alpha}:0<t<\tau\\}$ of operators defined above is a regularization
family for the ill-posed inverse problem $P_{0}$.

######  Theorem 4.1.

If $g=u_{\alpha}(\cdot,\tau)$, then

$\|u_{\alpha}(\cdot,0)-u_{\alpha}(\cdot,t)\|\to 0\quad\hbox{as}\quad t\to 0.$

###### Proof.

By the representation of $u_{\alpha}(\cdot,t)$ in (6), we have

$u_{\alpha}(\cdot,t)-u_{\alpha}(\cdot,0)=\sum_{n=1}^{\infty}[E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1]\langle
f_{0},\varphi_{n}\rangle\varphi_{n}.$

so that

(21)
$\|u_{\alpha}(\cdot,t)-u_{\alpha}(\cdot,0)\|^{2}=\sum_{n=1}^{\infty}|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|^{2}|\langle
f_{0},\varphi_{n}\rangle|^{2}.$

From the definition of $E_{\alpha}(\cdot)$, we have

$\displaystyle E(z)-1$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{z^{k}}{\Gamma(\alpha
k+1)}=z\sum_{k=1}^{\infty}\frac{z^{k-1}}{\Gamma(\alpha k+1)}$ $\displaystyle=$
$\displaystyle z\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha
k+\alpha+1)}=zE_{\alpha,\alpha+1}(z),$

where for $\alpha,\beta>0$, $E_{\alpha,\beta}(\cdot)$ is the generalized
Mittag-Leffler function defined by

$E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\beta)}.$

Hence, we have

(22)
$E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1=(-\lambda_{n}^{2}t^{\alpha})E_{\alpha,\alpha+1}(-\lambda_{n}^{2}t^{\alpha}).$

It is known that (cf. [10]) that if there exists $C>0$ such that for
$0<\alpha<2$ and $\beta\in{\mathbb{R}}$,

$|E_{\alpha,\beta}(z)|\leq\frac{C}{1+|z|}$

for all $z\in{\mathbb{C}}$ and for $\mu<|\arg(z)|\leq\pi$, where

$\frac{\pi\alpha}{2}<\mu<\min\\{\pi,\pi\alpha\\}.$

Now, taking $z=-\lambda_{n}^{2}t^{\alpha}$ and $0<\alpha<1$, we have
$\arg(z)=\pi$ and $\min\\{\pi,\pi\alpha\\}=\pi\alpha$. Hence, in this case the
required conditions on $\arg(z)$ is automatically satisfied. Thus, we have

$|E_{\alpha,\alpha+1}(-\lambda_{n}^{2}t^{\alpha})|\leq\frac{C}{1+\lambda_{n}^{2}t^{\alpha}}.$

Hence,

(23)
$|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|=|-\lambda_{n}^{2}t^{\alpha}E_{\alpha,\alpha+1}(-\lambda_{n}^{2}t^{\alpha})|\leq\frac{C\lambda_{n}^{2}t^{\alpha}}{1+\lambda_{n}^{2}t^{\alpha}}\leq
C.$

In particular, $|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|\to 0$ as $t\to 0$
for each $n\in{\mathbb{N}}$. Thus,

$|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|^{2}|\langle
f_{0},\varphi_{n}\rangle|^{2}\to 0\hbox{ as }\,t\to 0\hbox{ for each
}\,n\in{\mathbb{N}}$

and

$|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|^{2}|\langle
f_{0},\varphi_{n}\rangle|^{2}\leq C^{2}|\langle f_{0},\varphi_{n}\rangle|^{2}$

with $\sum_{n=1}^{\infty}|\langle
f_{0},\varphi_{n}\rangle|^{2}\leq\|f_{0}\|^{2}$. Hence, by the dominated
convergence theorem, the relation (21) implies that

$\|u_{\alpha}(\cdot,t)-u_{\alpha}(\cdot,0)\|\to 0\hbox{ as }\quad t\to 0.$

This completes the proof. ∎

##  5\. Error Estimate under Noisy Data

If the data is noisy, say we have $\tilde{g}\in L^{2}[0,\pi]$ in place of $g$
such that

$\|g-\tilde{g}\|\leq\delta$

for some known noise level $\delta>0$, then using the expression in (11), the
corresponding solution at $t$ can be taken as

$\tilde{u}_{\alpha}(\cdot,t)=R_{\alpha,t}\tilde{g},$

where $R_{\alpha,t}$ is defined as in (20). Thus,

(24)
$\tilde{u}_{\alpha}(\cdot,t)=\sum_{n=1}^{\infty}\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\langle\tilde{g},\varphi_{n}\rangle\varphi_{n}.$

Hence, we obtain

$\|u_{\alpha}(\cdot,t)-\tilde{u}_{\alpha}(\cdot,t)\|^{2}=\sum_{n=1}^{\infty}\Big{|}\frac{E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})}{E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})}\Big{|}^{2}|\langle
g-\tilde{g},\varphi_{n}\rangle|^{2}$

so that using (14),

$\|u_{\alpha}(\cdot,t)-\tilde{u}_{\alpha}(\cdot,t)\|\leq\frac{C_{2}}{C_{1}}\Big{(}\frac{1+\tau^{\alpha}}{t^{\alpha}}\Big{)}\delta.$

Thus, we have proved:

######  Theorem 5.1.

For $0<t<\tau$,

$\|u_{\alpha}(\cdot,t)-\tilde{u}_{\alpha}(\cdot,t)\|\leq\frac{C_{2}}{C_{1}}\Big{(}\frac{1+\tau^{\alpha}}{t^{\alpha}}\Big{)}\delta.$

Thus we have proved the following theorem.

######  Theorem 5.2.

For $0<t<\tau$,

$\|u_{\alpha}(\cdot,0)-\tilde{u}_{\alpha}(\cdot,t)\|\leq\|u_{\alpha}(\cdot,0)-u_{\alpha}(\cdot,t)\|+\frac{C_{2}(1+\tau^{\alpha})}{C_{1}}\frac{\delta}{t^{\alpha}},$

where (by Theorem 4.1) $\|u_{\alpha}(\cdot,0)-u_{\alpha}(\cdot,t)\|\to 0$ as
$t\to 0$.

##  6\. Error Estimates Under Source Conditions

Assumption (A): There exists an index function
$\varphi:(0,\infty)\to[0,\infty)$ such that

(25) $\|u_{\alpha}(\cdot,t)-f_{0}\|\leq c_{0}\varphi(t)$

for some $c_{0}>0$.

######  Theorem 6.1.

Under the Assumption (A), let $\psi(t):=t\varphi(t)$ for $t>0$, and for
$\delta>0$, let $t_{\delta}:=\psi^{-1}(\delta)$. Then

$\|f_{0}-\tilde{u}_{\alpha}(\cdot,t_{\delta})\|=O(\varphi(\psi^{-1}(\delta)).$

###### Proof.

By (25) and Theorem 5.1,

$\displaystyle\|f_{0}-\tilde{u}_{\alpha}(\cdot,t)\|$ $\displaystyle\leq$
$\displaystyle\|u_{\alpha}(\cdot,t)-u_{\alpha}(\cdot,0)\|+\|u_{\alpha}(\cdot,t)-\tilde{u}_{\alpha}(\cdot,t)\|$
$\displaystyle\leq$ $\displaystyle
c_{0}\varphi(t)+\frac{}{}\frac{C_{2}(1+\tau^{\alpha})}{C_{1}}\frac{\delta}{t^{\alpha}}$

Note that

$\varphi(t)=\frac{\delta}{t^{\alpha}}\iff\psi(t):=t^{\alpha}\varphi(t)=\delta.$

Hence, by choosing $t=t_{\delta}:=\psi^{-1}(\delta)$, we obtain

$\|f_{0}-\tilde{u}_{\alpha}(\cdot,t)\|=O(\varphi(t_{\delta}))=O(\varphi(\psi^{-1}(\delta)).$

∎

Now, we specify an index function $\varphi$ and a source set $M_{\varphi}$
such that (25) is satisfied whenever $f_{0}\in M_{\varphi}$.

Let $K_{\alpha}:L^{2}[0,\pi]\to L^{2}[0,\pi]$ be defined by

$K_{\alpha}\varphi=\sum_{n=1}^{\infty}E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\langle\varphi,\varphi_{n}\rangle\varphi_{n},\quad\varphi\in
L^{2}[0,\pi].$

Since $E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\in{\mathbb{R}}$ and
$E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\to 0$ as $n\to\infty$, it follows
that (cf. [7]) $K_{\alpha}$ is a compact, self-adjoint operator and

(26)
$K_{\alpha}\varphi_{n}=E(-\lambda_{n}^{2}\tau^{\alpha})\varphi_{n}\quad\forall\,n\in{\mathbb{N}}.$

In view of (7), we see that $f_{0}:=u_{\alpha}(\cdot,0)$ is the solution of
the compact operator equation

$K_{\alpha}f=g.$

Let

(27) ${\mathcal{M}}_{\alpha,\rho}:=\\{K_{\alpha}u:\|u\|\leq\rho\\}.$

and assume that

$f_{0}\in{\mathcal{M}}_{\alpha,\rho}.$

Then $f_{0}=K_{\alpha}u$ for some $u\in L^{2}[0,\pi]$ with $\|u\|\leq\rho$,
and hence by (26), we have

$\langle f_{0},\varphi_{n}\rangle=\langle
K_{\alpha}u,\varphi_{n}\rangle=\langle
u,K_{\alpha}\varphi_{n}\rangle=E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})\langle
u,\varphi_{n}\rangle\quad\forall\,n\in{\mathbb{N}}.$

Hence, from (21), we have

$\displaystyle\|u_{\alpha}(\cdot,t)-f_{0}\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|^{2}|\langle
f_{0},\varphi_{n}\rangle|^{2}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|^{2}||E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})|^{2}\langle
u,\varphi_{n}\rangle|^{2}$

Now, (23) and Lemma 1.1 imply that

$|E_{\alpha}(-\lambda_{n}^{2}t^{\alpha})-1|\,|E_{\alpha}(-\lambda_{n}^{2}\tau^{\alpha})|\leq
C_{\alpha}\frac{\lambda_{n}^{2}t^{\alpha}}{(1+\lambda_{n}^{2}t^{\alpha})(1+\lambda_{n}^{2}\tau^{\alpha})},$

where

(28) $C_{\alpha}:=\frac{CC_{2}}{\Gamma(1-\alpha)},$

with $C$ and $C_{2}$ as in (22) and Lemma 1.1, respectively. Note that

$\frac{\lambda_{n}^{2}t^{\alpha}}{(1+\lambda_{n}^{2}t^{\alpha})(1+\lambda_{n}^{2}\tau^{\alpha})}=\frac{(\lambda_{n}^{2}t^{\alpha})/(\lambda_{n}^{2}\tau^{\alpha})}{(1+\lambda_{n}^{2}t^{\alpha})}\frac{\lambda_{n}^{2}\tau^{\alpha}}{(1+\lambda_{n}^{2}\tau^{\alpha})}\leq\frac{t^{\alpha}}{\tau^{\alpha}}.$

Thus, we arrive at the estimate

$\|u_{\alpha}(\cdot,t)-f_{0}\|^{2}\leq
C_{\alpha}^{2}\Big{(}\frac{t^{\alpha}}{\tau^{\alpha}}\Big{)}^{2}\sum_{n=1}^{\infty}|\langle
u,\varphi_{n}\rangle|^{2}.$

Thus, we have proved the following theorem.

######  Theorem 6.2.

If $f_{0}\in{\mathcal{M}}_{\alpha,\rho}$, then

$\|u_{\alpha}(\cdot,t)-f_{0}\|\leq\rho
C_{\alpha}\frac{t^{\alpha}}{\tau^{\alpha}},$

where ${\mathcal{M}}_{\alpha,\rho}$ and $C_{\alpha}$ are as in (27) and (28),
respectively.

######  Remark 6.3.

Theorem 6.2 shows that the function $\varphi$ defined by

$\varphi(t)=t^{\alpha},\quad t>0,$

satisfies the Assumption (A) with $c_{0}=\rho C_{\alpha}/\tau^{\alpha}$.

In view of Theorem 6.2 and Theorem 5.2, if
$f_{0}\in{\mathcal{M}}_{\alpha,\rho}$, then we have

$\displaystyle\|f_{0}-\tilde{u}_{\alpha}(\cdot,t)\|$ $\displaystyle\leq$
$\displaystyle\|u_{\alpha}(\cdot,t)-u_{\alpha}(\cdot,0)\|+\|u_{\alpha}(\cdot,t)-\tilde{u}_{\alpha}(\cdot,t)\|$
$\displaystyle\leq$ $\displaystyle\rho
C_{\alpha}\frac{t^{\alpha}}{\tau^{\alpha}}+\frac{C_{2}(1+\tau^{\alpha})}{C_{1}}\frac{\delta}{t^{\alpha}}.$

Now,

$\frac{t^{\alpha}}{\tau^{\alpha}}=\frac{\delta}{t^{\alpha}}\iff
t^{2\alpha}=\tau^{\alpha}\delta\iff
t^{\alpha}=\sqrt{\tau^{\alpha}\delta}\iff\frac{t^{\alpha}}{\tau^{\alpha}}=\sqrt{\frac{\delta}{\tau^{\alpha}}}.$

Thus, we have proved the following theorem.

######  Theorem 6.4.

If $f_{0}\in{\mathcal{M}}_{\alpha,\rho}$ and
$t_{\delta}:=\sqrt{\tau}\delta^{1/\alpha}$, then

$\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|\leq\Big{(}\rho
C_{\alpha}+\frac{C_{2}(1+\tau)}{C_{1}}\Big{)}\sqrt{\frac{\delta}{\tau^{\alpha}}}.$

where ${\mathcal{M}}_{\alpha,\rho}$ and $C_{\alpha}$ are as in (27) and (28),
respectively. In particular,

$\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|=O(\sqrt{\delta}).$

######  Remark 6.5.

By the definition of ${\mathcal{M}}_{\alpha,\rho}$ and from the standard
regularization theory (cf. [2, 8]), it follows that the estimate obtained in
Theorem 6.4 is optimal for the source set ${\mathcal{M}}_{\alpha,\rho}$.

##  7\. Numerical Illustrations

In this section, we shall consider some numerical examples to illustrate the
level of approximation of the regularized solutions.

For numerical computations, we divide the given space domain $[0,\pi]$ into a
finite number of equal subintervals with step size $h$ where
$h=x_{i+1}-x_{i},\;i=0,1,2,\ldots,N-1$ with $x_{0}=0$, and $x_{N}=\pi$. All
the simulations are carried out using MATLAB R2022a with step size
$h=\pi/100$.

We have observed that the problem of finding
$u_{\alpha}(\cdot,t),\,0<t\leq\tau$ from the knowledge of
$g:=u_{\alpha}(\cdot,\tau)$ is a well-posed problem, and
$\|u_{\alpha}(\cdot,t)-f_{0}\|\rightarrow 0$ as $t\rightarrow 0$ (see Theorem
3.1).

For computational purpose, we take $\tau=1$ and consider
$g=u_{\alpha}(\cdot,\tau)$ obtained from (6) by taking
$f_{0}(x)=x(\pi-x)e^{-x},\;0\leq x\leq\pi$, and then compute
$u_{\alpha}(\cdot,t)$ according to the formula (11).

To approximate the integrals involved in the computation of $g$ and
$u_{\alpha}(\cdot,t)$, we make use of the composite trapezoidal rule. For the
illustration of the convergence
$\|u_{\alpha}(\cdot,t)-f_{0}\|_{L^{2}}\rightarrow 0$ as $t\rightarrow 0$, we
take $t_{i}=10^{-(i+2)}$ for $i=1,2,\ldots,7$ and compute the exact expression
$f_{0}(x)$ as well as $u_{\alpha}(x,t_{i})$ and show in Figures $1$ to $4$ for
various $\alpha\in\left\\{0.2,\;0.4,\;0.6,\;0.8\right\\}$.

Figure 1. Solution profiles of $f_{0}(x)$ and $u_{\alpha}(x,t_{i})$ for
$\alpha=0.2$ Figure 2. Solution profiles of $f_{0}(x)$ and
$u_{\alpha}(x,t_{i})$ for $\alpha=0.4$ Figure 3. Solution profiles of
$f_{0}(x)$ and $u_{\alpha}(x,t_{i})$ for $\alpha=0.6$ Figure 4. Solution
profiles of $f_{0}(x)$ and $u_{\alpha}(x,t_{i})$ for $\alpha=0.8$

In Table $1$, we show the error $\|u_{\alpha}(\cdot,t)-f_{0}\|_{L^{2}}$ for
$t=t_{i},\,i=1,2,\ldots,7$ for different values of $\alpha$. We observe that
when $t$ decreases, the error $\|u_{\alpha}(\cdot,t)-f_{0}\|_{L^{2}}$
decreases. This validates our theoretical result in Theorem 4.1.

Table 1. $L^{2}$-errors between $f_{0}$ and $u_{\alpha}(\cdot,t_{i})$ for $\alpha=0.2,\;0.4,\;0.6$ and $0.8$ | $\alpha=0.2$ | $\alpha=0.4$ | $\alpha=0.6$ | $\alpha=0.8$
---|---|---|---|---
$t_{i}$ value | $\|u_{\alpha}(\cdot,t)-f_{0}\|_{L^{2}}$ | $\|f_{0}-u_{\alpha}(\cdot,t)\|_{L^{2}}$ | $\|u_{\alpha}(\cdot,t)-f_{0}\|_{L^{2}}$ | $\|u_{\alpha}(\cdot,t)-f_{0}\|_{L^{2}}$
$10^{-3}$ | 3.2829(-1) | 1.4958(-1) | 5.2366(-2) | 1.5080(-2)
$10^{-4}$ | 2.5593(-1) | 7.5081(-2) | 1.5476(-2) | 2.6205(-3)
$10^{-5}$ | 1.9370(-1) | 3.4924(-2) | 4.2280(-3) | 4.2730(-4)
$10^{-6}$ | 1.4253(-1) | 1.5401(-2) | 1.1058(-3) | 6.8572(-5)
$10^{-7}$ | 1.0225(-1) | 6.5509(-3) | 2.8063(-4) | 2.6230(-5)
$10^{-8}$ | 7.1751(-2) | 2.7197(-3) | 7.1304(-5) | 2.5992(-5)
$10^{-9}$ | 4.9403(-2) | 1.1102(-3) | 2.8563(-5) | 2.6255(-5)

For the illustration of the case with the noisy data, we take
${\tilde{g}}(x)=g(x)+\frac{\delta}{2}$ with some noise level $\delta>0$. Note
that $\|{\tilde{g}}-g\|_{L^{2}}\leq\delta$. Now we take
$t_{\delta}:=\sqrt{\tau}\,\delta^{1/\alpha}$ as per Theorem 6.3 and compute
$\tilde{u}_{\alpha}(x,t_{\delta})$ using the formula (24) for several values
of $\delta$ and for some values of $\alpha$.

Taking $\delta_{i}=10^{-(i+2)},i=1,2,\ldots,7$ we compute the exact expression
$f_{0}$ as well as $\tilde{u}_{\alpha}(x,t_{\delta_{i}})$ and show them in
Figures $5$ to $8$ for various
$\alpha\in\left\\{0.2,\;0.4,\;0.6,\;0.8\right\\}$.

Figure 5. Solution profiles of $f_{0}(x)$ and
$\tilde{u}_{\alpha}(x,t_{\delta_{i}})$ for $\alpha=0.2$ Figure 6. Solution
profiles of $f_{0}(x)$ and $\tilde{u}_{\alpha}(x,t_{\delta_{i}})$ for
$\alpha=0.4$ Figure 7. Solution profiles of $f_{0}(x)$ and
$\tilde{u}_{\alpha}(x,t_{\delta_{i}})$ for $\alpha=0.6$ Figure 8. Solution
profiles of $f_{0}(x)$ and $\tilde{u}_{\alpha}(x,t_{\delta_{i}})$ for
$\alpha=0.8$

In Table $2$, we show the error
$\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|_{L^{2}}$ for different values
of $\alpha$ and $\delta_{i},i=1,2,\ldots,7$. Note that when $\delta$ value
decreases, the error $\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|_{L^{2}}$
decreases. This validates our theoretical result in Theorem 6.3.

Table 2. $L^{2}$-errors between $f_{0}$ and $\tilde{u}_{\alpha}(\cdot,t_{\delta_{i}})$ for $\alpha=0.2,\;0.4,\;0.6$ and $0.8$ | $\alpha=0.2$ | $\alpha=0.4$ | $\alpha=0.6$ | $\alpha=0.8$
---|---|---|---|---
$\delta_{i}$ value | $\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|_{L^{2}}$ | $\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|_{L^{2}}$ | $\|f_{0}-\tilde{u}_{\alpha}(\cdot,t_{\delta})\|_{L^{2}}$ | $\|\tilde{u}_{\alpha}(\cdot,t_{\delta})-f_{0}\|_{L^{2}}$
$10^{-3}$ | 8.2528(-2) | 8.6054(-2) | 8.6816(-2) | 8.3284(-2)
$10^{-4}$ | 3.2857(-2) | 3.4183(-2) | 3.4288(-2) | 3.2853(-2)
$10^{-5}$ | 1.1881(-2) | 1.2337(-2) | 1.2327(-2) | 1.1816(-2)
$10^{-6}$ | 4.0572(-3) | 4.2074(-3) | 4.1935(-3) | 4.0215(-3)
$10^{-7}$ | 1.3385(-3) | 1.3870(-3) | 1.3802(-3) | 1.3240(-3)
$10^{-8}$ | 4.3110(-4) | 4.4651(-4) | 4.4386(-4) | 4.2573(-4)
$10^{-9}$ | 1.3645(-4) | 1.4123(-4) | 1.4027(-4) | 1.3453(-4)

Acknowledgement: The first author M. Thamban Nair gratefully acknowledges the
support received from BITS Pilani, K.K. Birla Goa Campus, where he is a
Visiting Professor from August 1, 2023 after superannuation from I.I.T.
Madras, Chennai.

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