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Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm

Regularized solution of an inverse source problem for a time
fractional diffusion equation
Huy Tuan Nguyen a,b, Dinh Long Le b, Van Thinh Nguyen c,∗
a

Department of Mathematics and Computer Science, University of Science, Vietnam National University, Ho Chi Minh City, Vietnam
Institute of Computational Science and Technology, Ho Chi Minh City, Vietnam
c
Department of Civil and Environmental Engineering, Seoul National University, Republic of Korea
b

a r t i c l e

i n f o

a b s t r a c t

Article history:

Received 17 November 2014
Revised 20 February 2016
Accepted 13 April 2016
Available online xxx

In this paper, we study on an inverse problem to determine an unknown source term
in a time fractional diffusion equation, whereby the data are obtained at the later time.
In general, this problem is illposed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the
exact solution and its regularized solutions is obtained. We also propose two methods,
a priori and a posteriori parameter choice rules, to estimate the convergence rate of the
regularized methods. In addition, the proposed regularized methods have been verified by
numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the posteriori parameter
choice rule method converges to the exact solution faster than the priori parameter choice
rule method.

Keywords:
Cauchy problem
Ill-posed problem
Convergence estimates

© 2016 Elsevier Inc. All rights reserved.

1. Introduction
Diffusion equations with fractional order derivatives play an important role in modeling of contaminant diffusion processes. One of such problems was raising by Adam and Gelhar [1]; during analyzing the field data of dispersion in a heterogeneous aquifer, they could not explain a long-tailed profile of spatial density distribution by a classical diffusion–advection
equation with integer order derivatives. The long-tailed profile is of asymptotic behavior of fundamental solution near t = 0,
which can be described by a time-fractional diffusion equation as follows:

Dtα u(x, t ) −

∂

∂u
D (x )
= F (x, t ), (x, t ) ∈ (0, π ) × (0, T ),
∂x
∂x

(∗)

where u = u(x, t ) denotes a concentration of contaminant at a position x and time t, α ∈ (0, 1) is a fractional order and an
important parameter for anomaly of diffusion, F(x, t) is a source/sink function, D(x) is the diffusivity, and Dtα is the Caputo
fractional derivative of order α defined by:

Dtα u(t ) =

∗

t

1

(1 − α )

0

u (s )
ds,
(t − s )α

Corresponding author. Tel.: +82 28807355.
E-mail address: (V.T. Nguyen).

/>S0307-904X(16)30210-4/© 2016 Elsevier Inc. All rights reserved.

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and (.) denotes the standard Gamma function. Note that if the fractional order α tends to unity, the fractional derivative
Dtα u converges to the canonical first-order derivative du
[2], and thus the problem (∗) reproduces the canonical diffusion
dt
model. See, e.g., [2,3] for the definition and properties of Caputos derivative.
The fractional initial boundary value problem was firstly considered by Nigmatullin [4]. After that, several applications
of the fractional calculus and derivatives in applied sciences were developed; and the research on theoretical analysis and
numerical methods for solving direct problems, such as initial value and/or boundary value problems for the time fractional
diffusion equation were growing. For a well-posedness analysis, we refer to [5–9]; for numerical methods and simulations,
see [10–16] and references therein. Unfortunately, some input data and parameters of the diffusion equations in practical
problems may be unknown, such as initial and boundary data, diffusion coefficients, and source terms; therefore we have
to determine them by additional measurement data which can be yielded from a fractional diffusion inverse problem. The
fractional inverse problem provides an efficient tool for the modeling of the anomalous diffusion processes observed in
various fields of science and engineering, such as in biology [17,18], physics [19,20], chemistry [21], and hydrology [22].
Murio [23] considered an inverse problem of recovering boundary functions from transient data at an interior point

for a 1D semi-infinite half-order time-fractional diffusion equation. Liu and Yamamoto [24] applied a quasi-reversibility
regularization method to solve a backward problem for the time-fractional diffusion equation. Wei et al. [25–27] studied
an inverse source problem for a spatial fractional diffusion equation using quasi-boundary value and truncation methods.
Recently, Kirane et al. [2,28] studied conditional well-posedness to determine a space dependent source for one-dimensional
and two-dimensional time-fractional diffusion equations. Rundell et al. [29–31] considered an inverse problem for a onedimensional time-fractional diffusion problem. However, there are only a few studies on determination of a sources term
depending on both time and space for a time fractional diffusion equation.
In this work, we focus on an inverse problem for the following time-fractional diffusion equation:

Dtα u(x, t ) + Au(x, t ) = F (x, t ), (x, t ) ∈
u(x, t ) = 0, x ∈ ∂ ,
u(x, 0 ) = 0, x ∈ ,
where T > 0 and 0 < α < 1;
given by:
d

Au(x, t ) =
i=1

∂
∂ xi

× ( 0, T ),

is a bounded domain in Rd with the sufficient smooth boundary ∂ , and the operator A is

d

ai j (x )
j=1

∂
u(x, t ) + B(x )u(x, t ), x ∈ ·
∂xj

where ai j = a ji , 1 ≤ i, j ≤ d. Moreover, we assume that the operator A is uniformly elliptic on
smooth: there exists a constant μ > 0 such that:

μ

d

ξi2 ≤

i=1

d

(1.1)

a i j ( x ) ξi ξ j , x ∈

(1.2)
and that its coefficients are

, ξ ∈ Rd ·

(1.3)

i, j=1

and the coefficients satisfy:

ai j ∈ C 1 ( ), B ∈ C ( ), B ( x ) ≤ 0, x ∈

·

(1.4)

Problem (1.1) is the forward problem when the source function F = F (x, t ) is given appropriately. Whereas, an inverse
source problem based on Problem (1.1) is to determine the source term F at a previous time from its value at the final time
T as follows:

u(x, T ) = h(x ), x ∈

.

(1.5)

where h ∈ H 2 ( ) ∩ H01 ( ). Assuming that the source term F = F (x, t ) can be split into a product of R(t)f(x), where R(t)
is known in advance. We assume the time-dependent source term R(t) is obtained from observation data R (t) in such a
way that R (t ) − R(t ) L1 (0,T ) ≤ and is a noise level from a measurement. The space-dependent source term f(x) is also
determined from the observation of h(x) at the final data t = T by h ∈ L2 ( ) with the noise level of

h −h

L2 ( )

≤ .

and satisfied:

(1.6)

It is known that the inverse source problem mentioned above is ill-posed in general, i.e., a solution does not always exist,
and in the case of existence of a solution, which does not depend continuously on the given data. In fact, from a small
noise of a physical measurement, the corresponding solutions may have a large error. This makes a troublesomeness for the
numerical computation, hence a regularization is required.
If α = 1, the inverse source Problems (1.1) and (1.2) is a classical ill-posed problem and has been studied in [20,32].
However, for the fractional inverse source problem, up-to-date, there are only very few works; for example, Sakamoto and
Yamamoto [9] used the data u(x0 , t)(x0 ∈ ) to determine R(t) once f(x) was given, where the authors obtained a Lipschitz
stability for R(t). Zhang and Wei [27] used the Fourier truncation method to solve an inverse source problem with R(t ) = 1
in Problem (1.1) for one-dimensional problem with special coefficients. The inverse source problem for the time-fractional
diffusion equation with R(t) depended on time still has a limited achievement. Actually, this problem recently has been
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introduced by Jin and Rundell on Page 18 of [33], however a further regularized solution for this problem has not been
focused in their paper.
Motivated by above reasons, in this study, we apply the Tikhonov regularization method to solve the fractional inverse
source problem with variable coefficients in a general bounded domain. We estimate a convergence rate under a priori
bound assumption of the exact solution and a priori parameter choice rule. Because the priori bound is difficult to obtain in practical application, so we also estimate a convergence rate under the posteriori parameter choice rule which is

independent on the priori bound.
The paper is organized as follows. In Section 2, we introduce some preliminary results. The ill-posedness of the fractional
inverse source Problem (1.1) and a conditional stability are provided in Section 3. In Section 4, we propose a Tikhonov regularization method and give two convergence estimates under a priori assumption for the exact solution and two regularization parameter choice rules. Finally two numerical examples to verify our proposed regularized methods are shown in
Section 5. Eventually, a conclusion is given in Section 6.
2. Preliminary results
Throughout this paper, we use the following definition and lemmas.
Definition 2.1 (see [3]). The Mittag–Leffler function is:

Eα ,β (z ) =

∞
k=0

zk
, z∈C
(α k + β )

where α > 0 and β ∈ R are arbitrary constants.
Lemma 2.1 (see [3]). Let λ > 0, then we have:

d
Eα ,1 (−λt α ) = −λt α −1 Eα ,α (−λt α ), t > 0, 0 < α < 1.
dt
Lemma 2.2 (see [34]). For 0 < α < 1, ρ > 0, we have 0 < Eα ,1 (−ρ ) < 1. Moreover, Eα ,1 (−ρ ) is completely monotonic, that is

(−1 )m

dm
Eα ,1 (−ρ ) ≥ 0,
dρ m

∀m ∈ N ∩ {0}

Lemma 2.3. For 0 < α < 1, ρ > 0, we have 0 ≤ Eα ,α (−ρ ) ≤
ρ > 0.

1
(α ) . Moreover, Eα ,α (−ρ ) is a monotonic decreasing function with

Lemma 2.4 (see [3]). For α > 0 and β ∈ R, we have:

Eα ,β (z ) = zEα ,α +β (z ) +

1

(β )

, z∈C

Lemma 2.5 (see [35]). Assume R(t) ∈ L1 (0, T), for 0 < α < 1 and λ > 0, we have:

Dtα

t
0

(t − s )α−1 Eα,α (−λ(t − s )α )R(s )ds = R(t ) − λ

t
0

(t − s )α−1 Eα,α (−λ(t − s )α )R(s )ds, 0 ≤ t ≤ T .

Lemma 2.6. Let R : [0, T ] → R be a positive continuous function such that ∈ ft ∈ [0, T] |R(t)| > 0. Set
Then we have:

inft∈[0,T ] |R(t )|[1 − Eα ,1 (−λ1 T α )]

λp

T

≤

0

Gα (s, λ p )R(s )ds ≤

R

C[0,T ]

λp

.

R

C[0,T ]

= supt∈[0,T ] |R(t )|.
(2.7)

where, λp is defined in Theorem 2.1 and

Gα (s, λ p ) = (T − s )α −1 Eα ,α (−λ p (T − s )α )ds.
Proof. Using Lemmas 2.1 and 2.3, we have:
T
0

Gα (s, λ p )ds =
=
=

T
0
T
0

(T − s )α−1 Eα,α (−λ p (T − s )α )ds

|sα−1 Eα,α (−λ p sα )|ds = −

1 − Eα ,1 (−λ p T α )

λp

≤

1

λp

T

1

λp

0

d
Eα ,1 (−λ p sα )ds
ds

.

This implies that:
T
0

Gα (s, λ p )R(s )ds ≤ sup |R(t )|
t∈[0,T ]

T
0

Gα (s, λ p )ds ≤

R

C[0,T ]

λp

.

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and
T
0

Gα (s, λ p )R(s )ds ≥ inf

t∈[0,T ]

|R(t )|

T

0

Gα (s, λ p )ds ≥

inft∈[0,T ] |R(t )|[1 − Eα ,1 (−λ1 T α )]

λp

.

The proof is completed.
3. The inverse source problem
First, we introduce a few properties of the eigenvalues of the operator A on an open, connected and bounded domain
with Dirichlet boundary conditions (see also in Chapter 6 of [36]).
Theorem 2.1. (Eigenvalues of the Laplace operator)
1. Each eigenvalues of A is real. The family of eigenvalues {λ p }∞
p=1 satisfy 0 ≤ λ1 ≤ λ2 ≤ λ3 ≤ ..., and λp → ∞ as p → ∞.
2. There exists an orthonormal basis {φ p }∞
of L2 ( ), where φ p ∈ H01 ( ) is an eigenfunction corresponding to λ:
p=1

Aφ p (x ) = −λ p φ p (x ), x ∈
φ p ( x ) = 0, x ∈ ∂ ,

(3.8)

for p = 1, 2, . . .
Let 0 = γ < ∞. By Hγ ( ) we denote the space of all functions g ∈ L2 ( ) with the property:
∞

( 1 + λ ) 2γ |g p |2 < ∞,

(3.9)

p=1

∞
γ
2
2γ
2
where g p =
g(x )φ p (x )dx. Then we also define g H γ ( ) =
p=1 (1 + λ p ) |g p | . If γ = 0 then H ( ) is L ( ). This space
is introduced by Brezis [37] (see Chapter V) and Feng et al. [38] (see page 179).

3.1. The formula and uniqueness of the source term f
Now we use the separation of variables to yield the solution of (1.1). Suppose that the exact u is defined by Fourier
series:
∞

u(x, t ) =

u p (t )φ p (x ), with u p (t ) = u(., t ), φ p (x ) .

(3.10)

p=1

Then the eigenfunction expansions can be defined by the Fourier method. That is, we multiply both sides of (1.1) by φ p (x)

and integrate the equation with respect to x. Using the Green formula and φ p |∂ = 0, we obtain an uncouple system of the
initial value problem for the fractional differential equations for the unknown Fourier coefficient u p (t ) :

Dtα u p (t ) = λ p u p (t ) + Fp (t ), 0 < t < T
u p (0 ) = u(x, 0 ), ϕ p (x )

(3.11)

where Fp (t ) = F (x, t ), φ p (x ) . As Sakamoto and Yamamoto [9], the formula of the solution corresponding to the initial value
problem for (3.11) is obtained as follows:

u p (t ) = Eα ,1 (−λ pt α )u p (0 ) +

t
0

(t − s )α−1 Eα,α (−λ p (t − s )α )Fp (s )ds.

It follows from u(x, 0 ) = 0 and Fp (s ) = R(s ) f (x ), φ p (x ) that,

h ( x ), φ p ( x ) = u p ( T ) =

T
0

(T − s )α−1 Eα,α (−λ p (T − s )α )R(s )ds f (x ), φ p (x ) .

A simple transformation gives:

f (x ) =

∞
p=1

T
0

h ( x ), φ p ( x ) φ p ( x )
.
(T − s )α−1 Eα,α (−λ p (T − s )α )R(s )ds

(3.12)

and by denote:

Gα (s, λ p ) = (T − s )α −1 Eα ,α (−λ p (T − s )α ),

(3.13)

then we obtain the formula of the source function f :

f (x ) =

∞
p=1

h ( x ), φ p ( x )
T
0

φ p (x )
.
Gα (s, λ p )R(s )ds

(3.14)

In the following theorem, we provide the uniqueness property of the inverse source problem.
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Theorem 3.1. Let R : [0, T ] → R be as in Lemma 2.6, then the solution u(x, t), f(x) of Problems (1.1) and (1.2) is unique.
Proof. Let f1 and f2 be the source functions corresponding to the final values h1 and h2 respectively. Suppose that h1 = h2
then we prove that f1 = f2 . In fact, it is well-known that Eα ,α (−λ p (t − s )α ) ≥ 0 for s ≤ t. Since R(t) ≥ R0 > 0 for t ∈ [0, T],
we have:
T
0

T

Gα (s, λ p )R(s )ds ≥ R0

0

(T − s )α−1 Eα,α (−λ p (T − s )α )ds

= R0 T α Eα ,α +1 (−λ p T α ) > 0.

(3.15)

From (3.14) and (3.15), we get:
∞

f 1 (x ) − f 2 (x ) =

h1 ( x ) − h2 ( x ), φ p ( x )
T
0

p=1

φ p ( x ) = 0.

Gα (s, λ p )R(s )ds

(3.16)

The proof is completed.
3.2. The ill-posedness of the inverse source problem
Theorem 3.2. The inverse source problem is ill-posed.
Proof. Defining a linear operator K: L2 ( ) → L2 ( ) as follows:

∞

K f (x ) =

T

Gα (s, λ p )R(s )ds

0

p=1

f ( x ), φ p ( x )

φ p (x ) =

k(x, ξ ) f (ξ )dξ ,

(3.17)

where,
∞

k(x, ξ ) =

T

Gα (s, λ p )R(s )ds

0

p=1

φ p ( x )φ p ( ξ ).

Due to k(x, ξ ) = k(ξ , x ) we know K is self-adjoint operator. Next, we are going to prove its compactness. Defining the finite
rank operators KN as follows:
N

T

KN f (x ) =

0

p=1

Gα (s, λ p )R(s )ds

f ( x ), φ p ( x )

φ p ( x ).

(3.18)

Then, from (3.17) and (3.18), we have:
∞

KN f − K f

2

T

=
p=N+1
∞

0

R

≤
p=N+1

≤

R

Gα (s, λ p )R(s )ds

2
C ([0,T ] )
2
p

λ

| f ( x ), φ p ( x ) |2

| f ( x ), φ p ( x ) |2

2
∞
C ([0,T ] )
2
N
p=N+1

λ

2

| f ( x ), φ p ( x ) |2 .
(3.19)

This implies that:

KN f − K f

≤

R

2
C[0,T ]
2
N

λ

f

2
L2 ( )

=

R

C[0,T ]

λN

f

L2 ( ) .

Therefore, KN − K → 0 in the sense of operator norm in L(L2 ( ); L2 ( )) as N → ∞. Also, K is a compact operator. Next,
the singular values for the linear self-adjoint compact operator K are:

ψp =

T
0

Gα (s, λ p )R(s )ds,

(3.20)

and corresponding eigenvectors is φ p which is known as an orthonormal basis in L2 ( ). From (3.17), the inverse source
problem introduced above can be formulated as an operator equation.

K f ( x ) = h ( x ),

(3.21)

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and by Kirsch [39], we conclude that it is ill-posed. To illustrate an ill-posed problem, we introduce the following example.
Let us choose the input final data hm (x ) = φ√m (x ) . By (3.14), the source term corresponding to hm is:
λm

∞

f (x ) =

hm ( x ), φ p ( x )

m

T
0

p=1

Gα (s, λ p )R(s )ds

φ p (x ) =

φ√
m (x )
, φ p (x )
λm

∞
T
0

p=1

Gα (s, λ p )R(s )ds

φ p (x ) =

φm ( x )
λm

T
0

Gα (s, λm )R(s )ds

.

(3.22)

Let us choose other input final data g = 0. By (3.14), the source term corresponding to g is f = 0. An error in L2 norm
between two input final data is:

hm − g

=

L2 ( )

φm ( x )
λm

L2

( )

1

=

λm

.

Therefore,

lim

m→+∞

hm − g

1

= lim

L2 ( )

λm

m→+∞

= 0.

(3.23)

And an error in L2 norm between two corresponding source term is:

fm − f

=

L2 ( )

φm ( x )
Gα (s, λm )R(s )ds

λ

T
m 0

T
0

From (3.24) and using the inequality

fm − f

≥

L2 ( )

λm
R

C[0,T ]

1

=

L2 ( )

Gα (s, λm )R(s )ds ≤

λ

T
m 0

R C[0,T ]

λm

Gα (s, λm )R(s )ds

.

(3.24)

as in Lemma 2.6, we obtain:

,

This leads to:

lim

m→+∞

fm − f

L2 ( )

λm

> lim

m→+∞

R

C[0,T ]

= +∞.

(3.25)

Combining (3.23) and (3.25), we conclude that the inverse source problem is ill-posed.
3.3. Conditional stability of the source term f
In this section, we introduce a conditional stability by the following theorem.
Theorem 3.3. If

f

L2 ( )

f

Hγ ( )

≤ M for M > 0 then,

≤ C (γ , M ) h

γ
γ +1

L2 ( )

,

where,
1

C (γ , M ) =

M γ +1
γ

γ

inft∈[0,T ] |R(t )| γ +1 [1 − Eα ,1 (−λ1 T α )] γ +1

.

Proof. From (3.14) and Hölder inequality, we have:
∞
2
L2 ( )

f

h ( x ), φ p ( x )

=
p=1

T
0

Gα (s, λ p )R(s )ds

2

∞

=

1

∞

≤

| h ( x ), φ p ( x ) |2
T
2γ +2
p=1 | 0 Gα (s, λ p )R (s )ds|
∞

≤

| f ( x ), φ p ( x ) |2
T
2γ
p=1 | 0 Gα (s, λ p )R (s )ds|

γ

2

| h(x ), φ p (x ) | γ +1 | h(x ), φ p (x ) | γ +1
| 0T Gα (s, λ p )R(s )ds|2
p=1
γ +1

∞

| h ( x ), φ p ( x ) |

γ
γ +1

2

p=1
1

γ +1

h

2γ
γ +1

L2 ( )

.

(3.26)

Using Lemma 2.6, we have:
T
0

Gα (s, λ p )R(s )ds

2γ

≥

inft∈[0,T ] |R(t )|2γ [1 − Eα ,1 (−λ1 T α )]2γ

λ2pγ

,

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and this inequality leads to:

λ2pγ | f (x ), φ p (x ) |2
inft∈[0,T ] |R(t )|2γ [1 − Eα ,1 (−λ1 T α )]2γ
p=1

∞

∞

| f ( x ), φ p ( x ) |2
≤
T
2γ
p=1 | 0 Gα (s, λ p )R (s )ds|

f 2H γ
.
inft∈[0,T ] |R(t )|2γ [1 − Eα ,1 (−λ1 T α )]2γ

=

(3.27)

Combining (3.26) and (3.27), we obtain:
2

f

2
L2 ( )

f

≤
inft∈[0,T ] |R(t )|

2γ
γ +1

γ +1

Hγ

1 − Eα ,1 (−λ1

Tα)

2γ
γ +1

h

2γ
γ +1

L2 ( )

≤ C ( γ , M )2 h

2γ
γ +1

L2 ( )

.

4. Regularization of the inverse source problem using the Tikhonov method
As mentioned above, applying the Tikhonov regularization method we solve the inverse source problem, which minimizes
the function f in the following quantity in L2 ( ).
2

Kf −h

+ β2 f

2

,

and its minimized value fβ satisfies:

K ∗ K f β ( x ) + β 2 f β ( x ) = K ∗ h ( x ).

(4.28)

Due to the singular value decomposition for the compact self-adjoint operator K as in (3.20), we have:

f β (x ) =

∞

T
0

Gα (s, λ p )R(s )ds

2
p=1 β + |

Gα (s, λ p )R(s )ds|2

T
0

< h ( x ), φ p ( x ) > φ p ( x ).

(4.29)

If the measured data (R (t), h (x)) of (R(t), h(x)) with a noise level of

h−h

L2 ( )

≤ ,

R−R

C[0,T ]

and satisfied:

≤ ,

(4.30)

then we can present the regularized solution as follows:
∞

f β (x ) =

p=1

T
0

Gα (s, λ p )R (s )ds

β +|
2

T
0

Gα (s, λ p )R (s )ds|2

< h ( x ), φ p ( x ) > φ p ( x ).

(4.31)

4.1. A priori parameter choice
Afterwards, we will give an error estimation for
for the regularization parameter.

f (x ) − f β (x )

L2 ( )

and show convergence rate under a suitable choice

Theorem 4.1. Let f be as Theorem 3 and the noise assumption (4.30) hold. Then,
1

a. If 0 < γ ≤ 1, and choose β = ( M ) γ +1 , we have a convergence estimate:
1

f (x ) − f β (x )

L2 ( )

≤

3M γ +1
1
+ +
2 inft∈[0,T ] |R (t )|λ1
2

1

M1 (R )2 + 1 M γ +1

γ
γ +1

.

1

b. If γ > 1, choose β = ( M ) 2 , we have a convergence estimate:
1

f (x ) − f β (x )

L2 ( )

≤

3M 2
1

1
+ + M2 ( R ) M 2
2 inft∈[0,T ] |R (t )|λ1
2

1
2

,

where,

M1 ( R ) =

Tα
2 inft∈[0,T ] |R(t )| 1 − Eα ,1 (−λ1 T α )

, M2 ( R ) =

1 −γ

T α λ1
.
2 inft∈[0,T ] |R(t )|[1 − Eα ,1 (−T α )]

Proof. We first give two lemmas as follows:
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Lemma 4.1. Assume that (4.30) holds. Then we have the following estimate:

f β (x ) − f β (x )

3 f L2 ( )
+
.
2 inft∈[0,T ] |R (t )| 2β

≤

L2 ( )

(4.32)

Proof. From (4.29) and (4.31), we have:
∞

f β (x ) − f β (x ) =

T
0

Gα (s, λ p )R(s )ds

β2 + |

p=1

∞

+

T
0

Gα (s, λ p )R(s )ds|2
T
0

β2
β +|

T
0

p=1

∞

T
0

+

β2 + |

× h ( x ), φ p ( x )
∞

T
0

β2 + |

p=1

Gα (s, λ p )R (s )ds|2

β2 + |
T
0

T
0

Gα (s, λ p )R(s )ds||

β2 + |

Gα (s, λ p )R(s )ds|2

φ p (x )

< h ( x ), φ p ( x ) > φ p ( x )

Gα (s, λ p )R (s )ds|2

T
0

h ( x ), φ p ( x )

φ p (x )

Gα (s, λ p )(R(s ) − R (s ))ds

Gα (s, λ p )R(s )ds|2

T
0

T
0

h ( x ) − h ( x ), φ p ( x )

Gα (s, λ p )(R(s ) − R (s ))ds|

p=1

+

T
0

Gα (s, λ p )R (s )ds

β2 + |

Gα (s, λ p )R (s )ds|2

∞
2

T
0

Gα (s, λ p )R (s )ds

T
0

2
p=1 β + |

=

−

T
0

Gα (s, λ p )R (s )ds|

Gα (s, λ p )R (s )ds|2

φ p (x )

Gα (s, λ p )R (s )ds
T
0

Gα (s, λ p )R (s )ds|2

h ( x ) − h ( x ), φ p ( x )

φ p ( x ) = A1 + A2 + A3 ,

(4.33)

where,
∞

β2

A1 =

β2 + |

p=1

T
0

Gα (s, λ p )(R(s ) − R (s ))ds

Gα (s, λ p )R(s )ds|2

T
0

β2 + |

T
0

Gα (s, λ p )R (s )ds|2

h ( x ), φ p ( x )

φ p ( x ),
(4.34)

Gα (s, λ p )(R(s ) − R (s ))ds

T
0

∞

A2 =

β2 + |

p=1
∞

T
0

A3 =
p=1

β +|
2

T
0

Gα (s, λ p )R(s )ds

β2 + |

Gα (s, λ p )R(s )ds|2

Gα (s, λ p )R (s )ds
T
0

T
0

Gα (s, λ p )R (s )ds|2

T
0

T
0

Gα (s, λ p )R (s )ds

h ( x ), φ p ( x )

Gα (s, λ p )R (s )ds|2

h ( x ) − h ( x ), φ p ( x )

φ p ( x ),

φ p ( x ).

We continue to estimate the error by diving into three steps.
Step 1. Estimate A1
Using the

a2

+

b2

L2 ( ) .

≥ 2ab, ∀a, b ≥ 0,

∞

A1

2
L2 ( )

β2

≤
p=1

β +|
2

∞

|β

≤
p=1
∞
p=1

Gα (s, λ p )R(s )ds|

2 T
0

|

T
0

4 inft∈[0,T ] |R (t )|2 |
R−R

∞

2
C[0,T ]

4 inft∈[0,T ] |R (t )|

2

β +|
2

T
0

Gα (s, λ p )R (s )ds|

Gα (s, λ p )(R(s ) − R (s ))ds|2

2
C[0,T ]

2
p=1

2

Gα (s, λ p )(R(s ) − R (s ))ds

Gα (s, λ p )R(s )ds|2 |

R−R

≤
≤

4β 2 |

T
0

T
0

T
0

T

0

Gα (s, λ p )R (s )ds|2

2

| h ( x ), φ p ( x ) |2

| h ( x ), φ p ( x ) |2

Gα (s, λ p )ds|2
T
0

| h ( x ), φ p ( x ) |2
Gα (s, λ p )ds|2 | Gα (s, λ p )R(s )ds|2

| f ( x ), φ p ( x ) |2 =

T
0

R−R

2
C[0,T ]

4 inft∈[0,T ] |R (t )|2

f

2
L2 ( ) .

(4.35)

Hence,

A1

L2 ( )

≤

Step 2. Estimate A2

R − R C[0,T ] f L2 ( )
f L2 ( )
≤
.
2 inft∈[0,T ] |R (t )|
2 inft∈[0,T ] |R (t )|

(4.36)

L2 ( ) .

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We have:
∞

A2

2
L2 ( )

Gα (s, λ p )(R(s ) − R (s ))ds

T
0

≤

β +|

T
0

2

p=1

≤

Gα (s, λ p )R(s )ds|

|

p=1
T
0

∞

≤

p=1 |

≤

|

p=1
∞

≤
≤

R−R

|

T
0

Gα (s, λ p ) R(s ) − R (s ) ds

T
0

2
C[0,T ]
T
0

2

T
0

T
0

T
0

Gα (s, λ p )R (s )ds

Gα (s, λ p )R (s )ds|

Gα (s, λ p )R(s )ds|2 |

Gα (s, λ p )R(s )ds|4 |

Gα (s, λ p )R(s )ds|2 |

T
0

R−R

∞

T
0

β +|

2

2

Gα (s, λ p )(R(s ) − R (s ))ds

T
0

∞

Gα (s, λ p )R(s )ds

T
0

T
0

T
0

2

2

| h ( x ), φ p ( x ) |2

Gα (s, λ p )R (s )ds|2

Gα (s, λ p )R (s )ds|4

| h ( x ), φ p ( x ) |2

2

Gα (s, λ p )R (s )ds|2

| h ( x ), φ p ( x ) |2

2

Gα (s, λ p )ds

| h ( x ), φ p ( x ) |2
| Gα (s, λ p )R(s )ds|2

Gα (s, λ p )R (s )ds|2

T
0

2
C[0,T ]

| h ( x ), φ p ( x ) |2
inft∈[0,T ] |R (t )| | Gα (s, λ p )R(s )ds|2
p=1
2

R−R

∞

2
C[0,T ]

inft∈[0,T ] |R (t )|

T

0

| f ( x ), φ p ( x ) |2 =

2
p=1

R−R

2
C[0,T ]

inft∈[0,T ] |R (t )|2

2
L2 ( ) .

f

(4.37)

Hence,

A2

L2 ( )

≤

R − R C[0,T ]

f
inft∈[0,T ] |R (t )|

∞
2
L2 ( )

T
0

≤

β2 +

p=1

≤

≤

f L2 ( )
.
inft∈[0,T ] |R (t )|

(4.38)

L2 ( ) .

Step 3. Estimate A3
We have:

A3

L2 ( )

∞

1
4β 2

2

Gα (s, λ p )R (s )ds
T
0

Gα (s, λ p )R (s )ds

2

| h ( x ) − h ( x ), φ p ( x ) |2 =

p=1

h ( x ) − h ( x ), φ p ( x )

1
4β 2

h−h

2
L2 ( )

≤

2

4β 2

.

(4.39)

Hence,

A3

L2 ( )

≤

2β

.

(4.40)

Combining from (4.36) to (4.40), we obtain:

f β (x ) − f β (x )

≤ A1

L2 ( )

≤

L2 ( )

+ A2

L2 ( )

+ A3

L2 ( )

3 f L2 ( )
+
.
2 inft∈[0,T ] |R (t )| 2β

(4.41)

The proof of lemma is completed.
In order to obtain the boundedness of bias, we need the priori condition. By Tikhonov’s theorem, the L−1 restricted to
the continuous image of a compact set M. Thus, we assume f is in a compact subset of L2 ( ). From now on, we assume that
f H 2 ( ) ≤ M for γ > 0.
Lemma 4.2. Suppose that

f (x ) − f β (x )

L2 ( )

f

≤

Hγ ( )

≤ M. Then the following estimate holds:

M M1 (R )2 + 1β γ , 0 < γ ≤ 1
M2 ( R )M β , γ > 1

(4.42)

Proof. From (3.14) and (4.29) and using Parseval equality, we get:
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+∞

f (x ) − f β (x )

2
L2 ( )

β 4 | h ( x ), φ p ( x ) |2

=

|

p=1

T
0

Gα (s, λ p )R(s )ds|2

β2 + |

T
0

Gα (s, λ p )R(s )ds|2

2

γ 2γ

β 4 λ−2
λ p | h ( x ), φ p ( x ) |2
p

+∞

=

|

p=1

T
0

Gα (s, λ p )R(s )ds|2

β2 + |

T
0

Gα (s, λ p )R(s )ds|2

λ2pγ | h(x ), φ p (x ) |2
= sup |D( p)|2 f
T
2
p∈N
p=1 | 0 Gα (s, λ p )R (s )ds|

2

+∞

≤ sup |D( p)|2
p∈N

2
Hγ ( ) .

(4.43)

Hence, D(p) is given by:

D ( p) =

β 2 λ−p γ
.
β 2 + | 0T Gα (s, λ p )R(s )ds|2

(4.44)

Next, we continue to estimate D(p). Infact, we have:

D ( p) ≤

β 2 λ−p γ
β T α λ1p−γ
≤

.
T
2 inft∈[0,T ] |R(t )| 1 − Eα ,1 (−λ1 T α )
2β 0 Gα (s, λ p )R(s )ds

(4.45)

We divide into two following cases:
Case 1: γ ≥ 1; in this case, we note that:

λ1p−γ =

1

λγp −1

≤

1

λγ1 −1

1 −γ

= λ1

.

(4.46)

Combining (4.43), (4.45) and (4.46), we obtain:

f (x ) − f β (x )

L2 ( )

≤

β T α λ11−γ
f
2 inft∈[0,T ] |R(t )|[1 − Eα ,1 (−T α )]

Hγ ( ) .

(4.47)

Case 2: 0 < γ < 1. Choose any m such that m ∈ (0, 1). We rewrite N by N = P1 ∪ P2 where,
1 −γ

≤ β −m },

1 −γ
p

> β −m },

P1 = { p ∈ N, λ p
P2 = { p ∈ N, λ

(4.48)

From (4.43) and (4.47), we have:

f (x ) − f β (x )

2
L2 ( )

β T α λ1p−γ
2 inft∈[0,T ] |R(t )| 1 − Eα ,1 (−λ1 T α )

= sup
p∈P1

P2

+
p=1

≤

β 2 λ−p γ
β 2 + | 0T Gα (s, λ p )R(s )ds|2

p∈P2

≤
Choose m = 1 − γ and from

f (x ) − f β (x )

2

λ2pγ | f (x ), φ p (x ) |2

2 inft∈[0,T ] |R(t )| 1 − Eα ,1 (−λ1 T α )
−2γ

P2

β 2−2m f

2
Hγ ( )

β 2−2m f

2
Hγ ( )

λ2pγ | f (x ), φ p (x ) |2

p=1
2

Tα
2 inft∈[0,T ] |R(t )| 1 − Eα ,1 (−λ1
f (x )

λ2pγ | f (x ), φ p (x ) |2

p=1

2

Tα

+ sup λ p

2 P1

Hγ ( )

Tα)

2 mγ

+ β 1−γ

f

2
Hγ ( ) .

(4.49)

≤ M we have:

2
L2 ( )

≤ M1 ( R ) 2 β 2γ M 2 + β 2γ M 2 = β 2γ M 2 ( M1 ( R ) 2 + 1 ) .

L2 ( )

≤ βγ M

This implies that:

f (x ) − f β (x )

M1 (R )2 + 1.

(4.50)

Now, we continue to prove the theorem.
If 0 ≤ γ ≤ 1 then from Lemmas 4.1 and 4.2, we get:
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f (x ) − f β (x )

L2 ( )

f (x ) − f β (x )
f L2 ( )
3

≤

f β (x ) − f β (x )

L2 ( ) +

11

L2 ( )

+
+ M M1 (R )2 + 1β γ
2in f t∈[0,T ] |R (t )|
2β
3 M
≤
+ M M1 (R )2 + 1β γ
γ +
2β
2in f t∈[0,T ] |R (t )|λ1
≤

1

Since β = ( M ) γ +1 and

f (x ) − f β (x )

f

L2 ( )

1

≤

L2 ( )

f

λγ1

Hγ ( )

≤

M

λγ1

, we obtain:

1
3M

1
+ M γ +1
2 inft∈[0,T ] |R (t )|λ1
2

≤

γ
γ +1

1

M1 (R )2 + 1M γ +1

+

1

3M γ +1
1
+ +
2 inft∈[0,T ] |R (t )|λ1
2

≤

1

M1 (R )2 + 1 M γ +1

γ
γ +1

γ
γ +1

.

(4.51)

If γ > 1 then from Lemmas 4.1 and 4.2, we get:

f (x ) − f β (x )

L2 ( )

f (x ) − f β (x )

≤

L2 ( )

+ f β (x ) − f β (x )

L2 ( )

3 f L2 ( )
+
+ M2 ( R )M β .
2 inft∈[0,T ] |R (t )| 2β

≤

(4.52)
Since β = ( M )

1
2

and

f (x ) − f β (x )

f

L2 ( )

L2 ( )

1

≤

f

λγ1

Hγ ( )

≤

M

λγ1

, we obtain:

3M
γ
2 inft∈[0,T ] |R (t )|λ1

≤

+

1 1
M2
2

1
2

1

+ M2 ( R )M 2

1
2

1

3M 2
1
1
+ M2 ( R ) M 2
γ +
2
2 inft∈[0,T ] |R (t )|λ1

≤

1
2

.

(4.53)

4.2. A posteriori parameter choice
In this section, we consider a posteriori regularization parameter choice in Morozov’s discrepancy principle (see in [35]).
As usual, it follows the lemma below.
Lemma 4.3. Set

ρ (β ) =

∞
p=1

2

β2
β +

T
0

2

| h ( x ), φ p ( x ) |2 .

2

Gα (s, λ p )R (s )ds )

(4.54)

If 0 < < h L2 ( ) , then the following results hold:
(a) ρ (β ) is a continuous function.
(b) ρ (β ) → 0 as β → 0.
(c) ρ (β ) → h L2 ( ) as β → ∞.
(d) ρ (β ) is a strictly increasing function.
Theorem 4.2. Assume the priori condition and the noise assumption hold, and there exists k > 1 such that 0 < k < h
we choose a unique regularization parameter β > 0 such that:

. Then,

1

γ +1

If 0 ≤ γ ≤ 1, since β =

f (x ) − f β (x )
If γ > 1, since β =

f (x ) − f β (x )

L2 ( )

M

≤
1
2

M

L2 ( )

≤

we have:
γ
γ +1

γ

γ +1
√
M

+ α
+ 2
T Eα ,α +1 (−λ1 T α )N
λ1 N

1

γ +1

γ

γ
γ +1

γ

+

M γ +1 K
.
γ
Nλ1

(4.55)

we have:
1
2

√

2
M
+ α
+ 2
α
λ1 N T Eα,α+1 (−λ1 T )N
1
2

γ

1

1

1
2

+

M2K
γ ,
Nλ1

(4.56)

where inft∈[0,T ] |R (t )| = N, supt∈[0,T ] |R(t )| = K.
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Firstly, we can receive the following estimation:
∞

k2

2

β2
β 2 + | 0T Gα (s, λ p )R (s )ds|2

=
p=1
∞

2

β2
T
2
β + | 0 Gα (s, λ p )R (s )ds|2

≤2
p=1
∞

β2

+2

β2 + |

p=1

T
0
T
0

| h ( x ), φ p ( x ) |2
2

| h ( x ) − h ( x ), φ p ( x ) |2
2

Gα (s, λ p )R(s )ds

Gα (s, λ p )R (s )ds|2

λγp

λ2pγ | h(x ), φ p (x ) |2

.
| 0T Gα (s, λ p )R(s )ds|2

(4.57)

From (4.57), we get:
∞

k2

2

| h ( x ) − h ( x ), φ p ( x ) |2

≤2
p=1

∞

+2
p=1

β 2 0T Gα (s, λ p )R(s )ds
T
γ
[β 2 + | 0 Gα (s, λ p )R (s )ds|2 ]λ p
∞

≤2

2

∞
2

+2

λ2pγ | h(x ), φ p (x ) |2
| 0T Gα (s, λ p )R(s )ds|2

β 2 0T Gα (s, λ p )R(s )ds
T
γ
[β 2 + | 0 Gα (s, λ p )R (s )ds|2 ]λ p

+2
p=1

≤2

2

E ( p )2

p=1

2

λ2pγ | h(x ), φ p (x ) |2
| 0T Gα (s, λ p )R(s )ds|2

λ2pγ | h(x ), φ p (x ) |2
,
| 0T Gα (s, λ p )R(s )ds|2
(4.58)

where,

β 2 0T Gα (s, λ p )R(s )ds
.
β 2 + | 0T Gα (s, λ p )R (s )ds|2 λγp

E ( p) =

(4.59)

We estimate E(p) as follows:

E ( p) =
≤

β 2 0T Gα (s, λ p )R(s )ds
β 2 + | 0T Gα (s, λ p )R (s )ds|2 λγp
β 2 0T Gα (s, λ p )R(s )ds
T
γ
2λ p β 0 Gα (s, λ p )R (s )ds

β supt∈[0,T ] |R(t )| 0T Gα (s, λ p )ds
T

γ
2λ1 inft∈[0,T ] |R (t )| 0 Gα (s, λ p )ds
β supt∈[0,T ] |R(t )|
≤
.
γ
2λ1 inft∈[0,T ] |R (t )|
≤

(4.60)

Therefore, combining (4.58) and (4.60), we conclude that:

k2

2

( k2 − 2 )

2

2

β 2 supt∈[0,T ] |R(t )|2
f 2H γ ( )
2γ
2λ1 inft∈[0,T ] |R (t )|
β 2 supt∈[0,T ] |R(t )|2
≤
f 2H γ ( )

2γ
2λ1 inft∈[0,T ] |R (t )|
β 2 supt∈[0,T ] |R(t )|2
≤
f 2H γ ( ) ,
2γ
2λ1 inft∈[0,T ] |R (t )|(k2 − 2 )2
≤2

2

+

(4.61)

which gives the required results.
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Secondly, we have estimate:

∞

f (x ) − f β (x )

h ( x ), φ p ( x )

≤

L2 ( )

T
0

p=1
∞

h ( x ), φ p ( x )

+

−

φ p (x )

Gα (s, λ p )R (s )ds

T
0

p=1

∞

φ p (x )

Gα (s, λ p )R(s )ds

h ( x ), φ p ( x )
T
0

−

φ p (x )

Gα (s, λ p )R (s )ds
T
0

β +

Gα (s, λ p )R (s )ds
T
0

2

L2 ( )

Gα (s, λ p )R (s )ds

h ( x ), φ p ( x )

2

h ( x ), φ p ( x )

φ p (x )
L2 ( )

h ( x ), φ p ( x )

≤

T
0

p=1
∞

φ p (x )
h ( x ), φ p ( x ) φ p ( x )
− T
Gα (s, λ p )R(s )ds
0 Gα (s, λ p )R (s )ds

2

L2 ( )

h ( x ), φ p ( x )

+

T
0

p=1
∞

φ p (x )
h ( x ), φ p ( x ) φ p ( x )
− T
Gα (s, λ p )R (s )ds
0 Gα (s, λ p )R (s )ds

h ( x ), φ p ( x )

+

T
0

p=1

φ p (x )
−
Gα (s, λ p )R (s )ds
β2 +

T

0

L2 ( )

Gα (s, λ p )R (s )ds
T
0

Gα (s, λ p )R (s )ds

φ p (x )
L2 ( )

= B1 + B2 + B3 ,

(4.62)

where,
∞

B1 =
p=1
∞

B2 =
p=1
∞

B3 =
p=1

h ( x ), φ p ( x )
T
0

φ p (x )

h ( x ), φ p ( x )
T
0

−

Gα (s, λ p )R(s )ds

φ p (x )

Gα (s, λ p )R (s )ds

h ( x ), φ p ( x )
T
0

φ p (x )

Gα (s, λ p )R (s )ds

h ( x ), φ p ( x )

−

T
0

φ p (x )

Gα (s, λ p )R (s )ds

h ( x ), φ p ( x )

φ p (x )
−
T
G
(
s,
λ
)
R
(s )ds β 2 +
α
p
0

T
0

,
L2 ( )

,
L2 ( )

Gα (s, λ p )R (s )ds
Gα (s, λ p )R (s )ds

T
0

2

h ( x ), φ p ( x )

φ p (x )

.
L2 ( )

(4.63)
We continue to estimate the error by diving into three following steps.
Step 1 Estimate B21 , we have:
∞

B21 ≤

| h ( x ), φ p ( x ) |2 |
T
2
p=1 | 0 Gα (s, λ p )R (s )ds|
T

0

∞

≤

T
0

p=1

≤

Gα (s, λ p )ds

R−R

2

Gα (s, λ p )ds

2
C[0,T ]

inft∈[0,T ] |R (t )|2

f

2

T
0

Gα (s, λ p )(R(s ) − R (s ))ds|2

R−R

|

T
0

Gα (s, λ p )R (s )ds|2

2
C[0,T ]

| h ( x ), φ p ( x ) |2
| Gα (s, λ p )R(s )ds|2
inft∈[0,T ] |R (t )|
T
0

2
L2 ( ) .

(4.64)

Hence,

B1 ≤

R − R C[0,T ]
f
inft∈[0,T ] |R (t )|

(4.65)

L2 ( )

Setp 2 Estimate B22 , we have:
∞

B22 ≤

| h ( x ) − h ( x ), φ p ( x ) |2
T
2
p=1 | 0 Gα (s, λ p )R (s )ds|
∞

≤

| h ( x ) − h ( x ), φ p ( x ) |2
T
2
2
p=1 | 0 Gα (s, λ p )ds| inft∈[0,T ] |R (t )|
∞

≤
p=1

≤

T 2α [E

| h ( x ) − h ( x ), φ p ( x ) |2
α 2
2
α ,α +1 (−λ1 T )] inft∈[0,T ] |R (t )|
h−h

T 2α [E

α ,α +1 (−λ1

2

L2 ( )
T α )]2 inf

t∈[0,T ]

|R (t )|2

.

(4.66)

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Hence,

B2 ≤

h − h L2 ( )
.
T α [Eα ,α +1 (−λ1 T α )] inft∈[0,T ] |R (t )|

(4.67)

Step 3 Estimate B23 , we have:

B23 ≤

β2
| h ( x ), φ p ( x ) |
T
2

β + | 0 Gα (s, λ p )R (s )ds|2

1

|

T
0

Gα (s, λ p )R (s )ds|2

β2
| h ( x ), φ p ( x ) |
T
2
β + | 0 Gα (s, λ p )R (s )ds|2

≤
≤ k2

2

2

2

.

(4.68)

From (4.58) and (4.68), we get:

B23 ≤ 2

2

+

β 2 supt∈[0,T ] |R(t )|2
f
2γ
4 inft∈[0,T ] |R (t )|2 λ1

√
2 +

(4.69)

a2 + b2 ≤ a + b ∀a, b ≥ 0. Hence,

From (4.69) and using the inequality:

B3 ≤

Hγ ( ) .

β supt ∈[0,T ]|R(t )|
γ f
2 inft∈[0,T ] |R (t )|λ1

(4.70)

Hγ ( )

From (4.62), we receive:

f (x ) − f β (x )

≤

L2 ( )

R − R C[0,T ]
f
inft∈[0,T ] |R (t )|
√
+ 2 +

From (4.71) and

f (x ) − f β (x )

f

≤

L2 ( )

f

+

β supt ∈[0,T ]|R(t )|

T α [E
γ

2 inft∈[0,T ] |R (t )|λ1

Hγ ( )

f

h − h L2 ( )
α
α ,α +1 (−λ1 T )] inft∈[0,T ] |R (t )|
Hγ ( ) .

(4.71)

putting inft∈[0,T ] |R (t )| = N, supt∈[0,T ] |R(t )| = K, we get:

√
M
M K
+ α
+ 2 +β γ .
γ
α
λ1 N T [Eα,α+1 (−λ1 T )]N

λ1 N

≤

L2 ( )

1

λγ1

L2 ( )

Now, we continue to prove the theorem.
1

γ +1

If 0 ≤ γ ≤ 1, since β =

f (x ) − f β (x )

L2 ( )

If γ > 1, since β =

f (x ) − f β (x )

M

≤

√
M
+ α
+ 2 +
γ
α
T
[
E
(
−
λ
T
)
]
N
λ1 N
α ,α +1
1

≤

γ +1

1
2

M

γ
γ +1

1

γ +1

γ

γ +1
√
M
+ α
+ 2
γ
α
T Eα ,α +1 (−λ1 T )N
λ1 N

1

γ

M γ +1

K
γ
Nλ1
γ

γ
γ +1

+

M γ +1 K
.
γ
Nλ1

(4.72)

we have:

≤

L2 ( )

we have:

≤

√
M
+ α
+ 2 +
γ
α
λ1 N T [Eα,α+1 (−λ1 T )]N
1

2

√
2
M
+ α
+ 2
γ
α
λ1 N T Eα,α+1 (−λ1 T )N
1
2

1
2

1

1

M2

K
γ
Nλ1

1

1
2

+

M2K
γ .
Nλ1

(4.73)

5. Numerical experiments
To verify our proposed methods, we carry out numerically above regularization methods. Two different numerical examples corresponding to T = 1, and α = 0.2 are shown in this section. The couple of (h , R ) which are determined below, play
as measured data with a random noise as follows:

h (. ) = h (. ) 1 +

rand (. )
,
h L2 ( )

R (. ) = R(. ) + rand (. ),

(5.74)

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where rand() ∈ (−1, 1 ) is a random number. We can easily verify the validity of the inequality:

h−h

L2 ( )

≤ ,

R−R

C[0,T ]

≤ .

and

1

In addition, we can take the regularization parameter for the priori parameter choice rule, β pri = ( M ) 2 , where the value
of M plays a role as the priori condition computed by f H 2 (0,π ) . Similarly, based on the choice of k we can take the
regularization parameter for the posteriori parameter choice rule, β pos =
satisfied (4.61).
From (3.14), we can definite the exact solution, as follows:

f (x ) =

P

2

π

1

p=1

T
0

Gα (s, λ p )R(s )ds

1
2

M{k,R,R } ,

where M{k,R,R } is dependent on {k,

h(x ), sin( px ) sin( px ).

}

(5.75)

From (4.31), we can definite the regularized solution using Tikhonov method.

f β (x ) =

π

p=1

Gα (s, λ p )R (s )ds

T
0

P

2

[ β ( )] 2 +

T
0

Gα (s, λ p )R (s )ds

h (x ), sin( px ) sin( px ).

2

(5.76)

where P is a large enough number, and plays as a truncation number.
5.1. Example 1
In this example, we consider the function f is an exact data function. We particularly consider an one-dimensional case
of the problem (1.1) as follows with λ p = p2 and T = 1.

⎧ α
⎪
⎨ut (x, t ) − uxx (x, t ) = R(t ) f (x ), (x, t ) ∈ (0, π ) × (0, 1 ),
u(0, t ) = u(π , t ) = 0, t ∈ [0, 1],
⎪
⎩u(x, 0 ) = 0, x ∈ (0, π ),
u(x, 1 ) = h(x ), x ∈ (0, π ).

(5.77)

whereby,

t 1 −α
et − 1 ,
(2 − α )
1
h(x ) = 10−1
e − 1 sin(4x )(π − x ).
(2 − α )
R(t ) =

(5.78)

From (3.12) and (5.78), we can deduce the exact solution, as follows:

f (x ) = 10−1

1
0

P

e−1
(2 − α )

p=1

[β ( )

]2

Gα (s, p2 )R (s )ds

+

1
0

Gα (s,

p2

)R (s )ds

2

sin(4x )(π − x ).

(5.79)

and

s 1 −α
es − 1 + rand (. ),
(2 − α )

R (s ) =

Gα (s, p2 ) = (1 − s )α −1 Eα ,α (−p2 (1 − s )α ).

(5.80)

From (4.31) and (5.80), we can deduce the regularized solution, as follows:

fβ (x ) = 10−1

e−1

(2 − α )
1
0

P

×

p=1

where

1
0

[ β ( )] 2 +

Gα (s, p2 )R (s )ds =

1+

rand (. )
h L2 ( )

Gα (s, p2 )R (s )ds
1
0

Gα (s, p2 )R (s )ds

2

sin(4x )(π − x ).

1
α −1 E
2
α

α ,α (−p (1 − s ) )R
0 (1 − s )

ds =

1
[1
p2

− Eα ,1 (−p2 )]

(5.81)

1
0 ( R ( si )

+ .|rand (si )| )ds.

Next, we establish the regularized solution according to composite Simpson’s rule. In general, the whole numerical procedure is summarized in the following steps:
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Table 1
Error estimation between the exact solution and its regularized
solution in Example 1.
β

1E−01
1E−02
1E−03
1E−04
1E−05
1E−06
1E−07
1E−08
1E−09
1E−10

β

β

β

E1 pri

E2 pri

E1 pos

E2 pos

8.15E−01
8.02E−02
7.64E−03
6.38E−04
6.15E−05
5.94E−06
5.75E−07
4.53E−08
3.54E−09
4.64E−10

9.04E−01
8.86E−02
8.50E−03
7.34E−04
8.20E−05
7.95E−06
6.84E−07
6.73E−08
5.25E−09
5.24E−10

6.97E−02
8.16E−03
7.24E−04
7.19E−05
5.16E−06
6.09E−07
4.09E−08
4.09E−09

3.85E−10
4.53E−11

7.97E−02
9.16E−03
8.58E−04
7.34E−05
6.84E−06
7.33E−07
5.43E−08
7.55E−09
4.45E−10
5.95E−11

Step 1 Choose Q and L to generate the spatial and temporal discretization in such a manner as:

xi = i x,

x=

π

, i = 0, Q ,
Q
1
t = , j = 0, L.
L

t j = j t,

(5.82)

Of course, the higher value of Q and L will provide numerical results more accurate and stable, however in the following
numerical examples Q = L = 100 are satisfied.
Step 2 Setting fβ (xi ) = fβ ,i and f (xi ) = fi , constructing two vectors contained all discrete values of f β and f denoted by
and , respectively.

β

Q+1
,
β = [ f β ,0 , f β ,1 , . . . , f β ,Q ] ∈ R

= [ f0 , f1 , . . . , fQ ] ∈ RQ+1 .

(5.83)

Step 3 Error estimate between the exact and regularized solutions.
Relative error estimation:
Q
i=0

E1 =

| fβ (xi ) − f (xi )|2L2 (0,π )
Q
i=0

| f (xi )|2L2 (0,π )

.

(5.84)

Absolute error estimation:

E2 =

1
Q +1

Q

| fβ (xi ) − f (xi )|2L2 (0,π ).

i=0

(5.85)

In practice, it is very difficult to obtain the value of M for the priori parameter choice rule without having an exact solution. We thus try to take
rule, and β pos =
supt∈[0,T ] |R (t )|

1
2

M{k,R,R }

2γ
2λ1 (k2 −2 ) inft∈[0,T ] |R (t )|

f

f

1

H2 ( )

≤ M with M = 517 leading to β pri = ( M ) 2 for the priori parameter choice

with M{k,R,R } = 1858 for the posteriori parameter choice rule based on k = 1.5 and M{k,R,R } =
Hγ ( )

.

Figs. 1 and 2 below show a comparison between the exact solution and its regularized solutions for both parameter
choice rule methods, the priori and posteriori, in Example 1, respectively. Table 1 shows the absolute and relative error
estimates between the exact solution and its regularized solutions for both methods, the priori and posteriori parameter
choice rules, in Example 1.
5.2. Example 2
In this example, we consider the function f is an exact data function, and a two-dimensional case of the problem
(1.1) with λ p = p2 + q2 and T = 1 as follows:

⎧ α
ut (x, y, t ) − uxx (x, y, t ) − uyy (x, y, t ) = R(t ) f (x, y ), (x, y ) ∈ (0, π ) × (0, π ), t ∈ [0, 1],
⎪
⎪
⎨u(0, y, t ) = u(π , y, t ) = 0, (y, t ) ∈ (0, π ) × [0, 1],
u(x, 0, t ) = u(x, π , t ) = 0, (x, t ) ∈ (0, π ) × [0, 1],

⎪
⎪
⎩u(x, y, 0 ) = 0, (x, y ) ∈ (0, π ) × (0, π ),
u(x, y, 1 ) = h(x, y ), (x, y ) ∈ (0, π ) × (0, π ).

(5.86)

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Fig. 1. A comparison between the exact solution and its regularized solution for the priori parameter choice rule in Example 1.

whereby,

t 1 −α
et − 1 ,
(2 − α )
1
h(x, y ) = 2−1
e − 1 sin(4x ) sin(π − y ).

(2 − α )
R(t ) =

(5.87)

From (3.12) and (4.31), we can obtain the exact and regularized solutions, as follows:
Q

Q

f (x, y ) =

e−1

2−1
q=1 q=1

fβ (x, y ) = 2−1
Q

e−1

p=1 q=1

where

1
0

1

0

Q

1
0

Gα (s, p2 + q2 )R (s )ds

h

L2 ( )×L2 ( )

Gα (s, ( p2 + q2 ))R (s )ds

[ β ( )] 2 +

sin(4x ) sin(π − y ),

rand (. )

1+

(2 − α )

×

1

(2 − α )

1
0

Gα (s, p2 + q2 )R (s )ds =

Gα (s, p2 + q2 )R (s )ds
1
α −1 E
2
α ,α ( − ( p
0 (1 − s )

2

sin(4x ) sin(π − y ).

+ q2 )(1 − s )α )R ds =

(5.88)

1
[1 − Eα ,1 (−( p2
p2 +q2

+ q2 ))]

1
0 ( R ( si )

+

.|rand (si )| )ds.
Next, we establish the regularized solution according to composite Simpson’s rule. In this example, we choose P and
Q are large enough, unlike the first example, we can have f H 2 (0,π ) < M with M = 112230 from the analytical solution,
1

which implies that β pri = ( M1 ) 2 for the priori parameter choice, and β pos =
supt∈[0,T ] |R (t )|

1
2

M{k,R,R }

ori parameter choice rule based on k = 1.8 and M{k,R,R } = 2γ 2
f
2λ1 (k −2 ) inft∈[0,T ] |R (t )|
procedure is summarized in the following steps:

with M{k,R,R } = 239587 for the posteriHγ ( ) .

In general, the whole numerical

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Fig. 2. A comparison between the exact solution and its regularized solution for the posterior parameter choice rule in Example 1.

Step 1 Choose Q and K (in our computations, Q = K = 100 are chosen) to discretize spatial and temporal domain, as follows:

xi = i x,

x=

y j = j y,

y=

tk = k t,

π
Q

, i = 0, Q ,

π

, j = 0, K ,
K
1

t = , k = 0, L.
L

(5.89)

Step 2 Setting fβ )(i, j )= f (., ., 0 )β xi , y j and f(i, j)=f(., ., 0)(xi , yj ), constructing two following vectors contained all discrete
values of f(., ., 0)β , and f(., ., 0) denoted by β and , respectively.

⎡

f β ( 0, 0 )
⎢ f β ( 1, 0 )
⎢
⎢ f ( 2, 0 )
β = ⎢ β
⎢

⎣
⎡

...
fβ (Q, 0 )

f ( 0, 0 )
⎢ f ( 1, 0 )
⎢ f ( 2, 0 )
=⎢
⎢
⎣ ...
f (Q, 0 )

f β ( 0, 1 )
f β ( 1, 1 )
f β ( 2, 1 )

...
...
...
..
.
...

...
fβ (Q, 1 )
f ( 0, 1 )
f ( 1, 1 )
f ( 2, 1 )
...
f (Q, 1 )

...
...
...
..
.
...

f β ( 0, K − 1 )
f β ( 1, K − 1 )
f β ( 2, K − 1 )

⎤

f β ( 0, K )
f β ( 1, K ) ⎥
⎥
fβ (2, K ) ⎥ ∈ RQ+1 × RK+1

⎥
⎥
⎦
...
...
fβ (Q, K − 1 )
fβ (Q, K )
⎤
f ( 0, K − 1 )
f ( 0, K )
f ( 1, K − 1 )
f ( 1, K ) ⎥
⎥
f ( 2, K − 1 )
f (2, K ) ⎥ ∈ RQ+1 × RK+1 .
⎥
...
... ⎦
f (Q, K − 1 )
f (Q, K )

Step 3 Error estimate between the exact and regularized solutions;

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Fig. 3. A comparison between exact solution and its regularized solution for the priori parameter choice rule method in Example 2.

Relative error estimation:

E1 =

Q
i=0

K
j=0

| f β ( xi , y j ) − f ( xi , y j )|2

Q
i=0

K
j=0

| f ( xi , y j )|2

.

(5.90)

Absolute error estimation:

E2 =

1
1
Q +1K+1

Q

K

i=0 j=0

| f β ( xi , y j ) − f ( xi , y j )|2 .

(5.91)

Figs. 3 and 4 below show a comparison between the exact solution and its regularized solutions for both parameter
choice rule methods, the priori and posteriori, in Example 2, respectively. Table 2 shows the absolute and relative error
estimates between the exact solution and its regularized solutions for both parameter choice rule methods in Example 2.

From Figs. 1 and 2 combined with Table 1 of the first example, it shows that the posterior parameter choice rule method
converges to the exact solution faster than the prior parameter choice rule method. This property has been confirmed again
by Figs. 3 and 4 combined with Table 2 in the second example. Particularly, the error estimate shown in Tables 1 and2
show that the posterior parameter choice rule method converges to the exact solution with one order faster than the prior
parameter choice rule method. Nevertheless, from two examples, it also shows our proposed regularized methods have a
very good convergence rate to the exact solution once tends to 0.
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Fig. 4. A comparison between exact solution and its regularized solution for the posteriori parameter choice rule method in Example 2.

Table 2
Error estimation between the exact solution and its regularized
solution in Example 2.
β

1E−01
1E−02
1E−03
1E−04
1E−05

1E−06
1E−07
1E−08
1E−09
1E−10

β

β

β

E1 pri

E2 pri

E1 pos

E2 pos

7.12E−02
6.88E−03
6.58E−04
6.33E−05
7.10E−06
5.89E−07
5.07E−08
4.35E−09
4.21E−10
3.23E−11

9.70E−02
9.42E−03
8.66E−04
7.35E−05
8.81E−06
7.58E−07
6.36E−08
5.55E−09
3.12E−10
5.43E−11

8.14E−03
7.09E−04
7.92E−05
5.81E−06
6.16E−07
5.54E−08
4.90E−09
3.88E−10
3.36E−11
4.23E−12

7.64E−03
8.99E−04
8.39E−05
6.74E−06
5.97E−07
5.09E−08
4.41E−09

3.35E−10
2.16E−11
4.51E−12

6. Conclusion
In this study, we solved the inverse problem to recover the source term for the time fractional diffusion equation with
the time dependent coefficient by applying Tikhonov method. In the theoretical results, we obtained the error estimates of
two priori and posterior parameter choice rule methods based on the priori condition. In the numerical results, it shows
that the proposed regularized solutions are converged to the exact solutions. Furthermore, it also shows that the posteriori
parameter choice rule method is better than the priori parameter choice rule method in term of the convergence rate. In
the future work, we will continue to study some source terms for multiple diffusion equations.
Please cite this article as: H.T. Nguyen et al., Regularized solution of an inverse source problem for a time fractional
diffusion equation, Applied Mathematical Modelling (2016), />

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Acknowledgments
This work was supported by the Institute for Computational Science and Technology at Ho Chi Minh City, Vietnam under
the project named Fractional Diffusion–Wave Equations and Application to Soil Contaminant, and the NRF Research Grant
of Korea. The authors also would like to thank the editors and anonymous reviewers for their very valuable constructive
comments to improve this manuscript.
References
[1] E.E. Adams, W.L. Gelhar, Field study of dispersion in a heterogeneous aquifer 2: Spatial moments analysis, Water Resour. Res. 28 (1992) 3293307.
[2] M. Kirane, A.S. Malik, M.A. Al-Gwaiz, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary

conditions, Math. Methods Appl. Sci. 36 (9) (2013) 1056-1069.
[3] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering,, 198, Academic Press Inc., San Diego, CA, 1990.
[4] R. Nigmatulin, The realization of the generalized transfer equation in a medium with fractal geometry., Phys. Stat. Sol. B 133 (1986) 425–430.
[5] D.S. Eidelman, N.A. Kochubei, Cauchy problem for fractional diffusion equations, J. Differ. Equ. 199 (2) (2004) 211–255.
[6] A. Hanyga, Multidimensional solutions of time-fractional diffusion–wave equations., R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2020) (2002)
933–957.
[7] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl. 351 (1) (2009) 218–223.
[8] Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal. 15 (1) (2012) 141160.
[9] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion–wave equations and applications to some inverse problems,
J. Math. Anal. Appl. 382 (1) (2011) 426–447.
[10] F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model.
38 (15-16) (2014) 3871–3878.
[11] S. Shen, F. Liu, J. Chen, I. Turner, V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput. 218 (22)
(2012) 10861–10870.
[12] W. McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J. 52 (2) (2010) 123–138.
[13] K. Mustapha, W. McLean, Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J. Numer. Anal. 51
(1) (2013) 491–515.
[14] Y. Zhang, Z.Z. Sun, H.L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, J. Comput. Phys. 265 (2014)
195210.
[15] X. Zhao, Z.Z. Sun, Compact Crank–Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium, J. Sci. Comput. 62 (2015)
747–771.
[16] X. Zhao, Q. Xu, Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient, J. Appl. Math. Model. 38 (2014)
3848–3859.
[17] F. Liu, K. Burrage, Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl. 62
(2011) 822–833.
[18] B. Yu, X. Jiang, C. Wang, Numerical algorithms to estimate relaxation parameters and caputo fractional derivative for a fractional thermal wave model
in spherical composite medium, J. Appl. Math. Comput. 274 (2016) 106–118.
[19] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (20 0 0) 1–77.
[20] W. Fan, X. Jiang, H. Qi, Parameter estimation for the generalized fractional element network zener model based on the Bayesian method, Physica A
427 (2015) 40–49.

[21] S.B. Yuste, L. Acedo, K. Lindenberg, Reaction front in an a+b → c reaction–subdiffusion process, Phys. Rev. E 69 (2004) 036126.
[22] F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker–Planck equation, J Comput. Appl. Math. 166 (2004) 209–219.
[23] D.A. Murio, Stable numerical solution of fractional-diffusion inverse heat conduction problem, Comput. Math. Appl. 53 (2007) 1492–1501.
[24] J.J. Liu, M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal. 89 (2010) 1769–1788.
[25] T. Wei, J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math.
78 (2014) 95–111.
[26] J.G. Wang, Y.B. Zhou, T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer.
Math. 68 (2013) 3957.
[27] Z.Q. Zhang, T. Wei, Identifying an unknown source in time-fractional diffusion equation by a truncation method, Appl. Math. Comput. 219 (11) (2013)
5972–5983.
[28] M. Kirane, A.S. Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional
derivative in time, Appl. Math. Comput. 218 (1) (2011) 163–170.
[29] B. Jin, W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Probl. 28 (7) (2012) 075010.19 pp.
[30] W. Rundell, X. Xu, L. Zuo, The determination of an unknown boundary condition in a fractional diffusion equation, Appl. Anal. 92 (7) (2013) 1511–1526.
[31] Y. Luchko, W. Rundell, M. Yamamoto, L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction–diffusion
equation, Inverse Probl. 29 (2013) 065019.16pp.
[32] H.E. Roman, P.A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A 27 (1994) 3407–3410.
[33] B. Jin, W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Probl. 31 (2015) 035003.40pp.
[34] R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (20 0 0) 107–125.
[35] A.A. Kilbas, H.M. Srivastava, J. J.Trujillo, Theory and Application of Fractional differnetial equations, North-Holland Mathematics Studies, 204, Elsevier
Science B.V, Amsterdam, 2006.
[36] L.C. Evans, Partial Differential Equation, 19, American Mathematical Society, Providence, Rhode Island, 1997.
[37] H. Brezis, Analyse Fonctionelle, Masson, Paris, 1983.
[38] X.L. Feng, L. Elden, C.L. Fu, Numerical approximation of solution of nonhomogeneous backward heat conduction problem in bounded region, J. Math.
Comp. Simul. 79 (2) (2008) 177–188.
[39] A. Kirsch, An introduction to the mathematical theory of inverse problem, second ed., 2011, Springer.

Please cite this article as: H.T. Nguyen et al., Regularized solution of an inverse source problem for a time fractional
diffusion equation, Applied Mathematical Modelling (2016), />