ficient problem for a An undetermined coef fractional diffusion equation Zhidong Zhang

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Inverse Problems
Inverse Problems 32 (2016) 015011 (21pp)
doi:10.1088/0266-5611/32/1/015011
An undetermined coefficient problem for a
fractional diffusion equation
Zhidong Zhang
Department of Mathematics, Texas A&M University, College Station, TX, 77843,
USA
E-mail: [email protected]
Received 24 July 2015, revised 9 November 2015
Accepted for publication 30 November 2015
Published 18 December 2015
Abstract
We consider a fractional diffusion equation (FDE) CDta u = a (t ) uxx with an
undetermined time-dependent diffusion coefficient a(t). Firstly, for the direct
problem part, we establish the existence, uniqueness and some regularity
properties of the weak solution for this FDE with a fixed a (t ). Secondly, for
the inverse problem part, in order to recover a (t ), we introduce an operator
and show its monotonicity. With this property, we establish the uniqueness of
a(t) and create an efficient reconstruction algorithm to recover this coefficient.
Keywords: fractional diffusion, fractional inverse problem, reconstruct
diffusion coefficient, uniqueness, monotonicity, iteration algorithm
(Some figures may appear in colour only in the online journal)
1. Introduction
In this paper, we consider the following fractional diffusion equation (FDE) model:
⎧ CD a u (x , t ) - a (t ) u xx (x , t ) = 0, 0 < t  T , 0 < x < 1;
t
⎪
⎪ u (x , 0) = 0,
0 < x < 1;
⎨
0 < t  T;
⎪ u (1, t ) = 0,
⎪
0 < t  T , f (0) = 0;
⎩ u (0, t ) = f (t ) ,
(1.1)
where 0 < a < 1 and T is fixed. Model (1.1) is a diffusion model with time-dependent
diffusion coefficient a (t ). This model represents the direct problem—given f (t ), a (t ), solve
for the ‘temperature’ in the rod. We are interested in the inverse problem of recovering a(t)
from additional boundary data. Indeed, we impose the condition
0266-5611/16/015011+21$33.00 © 2016 IOP Publishing Ltd Printed in the UK
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Inverse Problems 32 (2016) 015011
Z Zhang
- a (t ) u x (0, t ) = g (t ) , 0 < t  T ,
which corresponds to imposing the value of heat flux at the end x = 0, and this inverse
problem becomes given f (t ), g (t ), seek the coefficient a (t ).
It should be noted that our model is one of ‘anomalous’ diffusion, which is modelled by a
time-fractional derivative CDta . Here, CDta (0 < a < 1) is the left-sided Djrbashian–Caputo
αth order derivative with respect to time t. The definition of CDta is
CD a u (x ,
t
t) =
1
G (n - a )
t
ò0
(t - t ) n - a - 1
dn
u (x , t ) dt
dt n
with Gamma function G( · ) and the nearest integer n which is bigger than a. In our case, we
are modeling subdiffusion process so that take n=1. The coefficient function a(t) is positive
and continuous. We denote the solution of model (1.1) by u (x, t; a ) to indicate its
dependence on the coefficient a(t).
A considerable literature has been built over the last 20 years showing that under certain
circumstance model (1.1) captures anomalous diffusion processes well, especially the subdiffusion process, in which the expected mean square path x 2 grows slower than that in the
Gaussian process for large time. At the microscopic level, the subdiffusion process can be
described by the continuous time random walk [23], of which the macroscopic counterpart is
a time-fractional differential equation. There are some important applications of the anomalous diffusion processes and fractional derivative. For instance, the thermal diffusion in
media with fractal geometry (see [24]), highly heterogeneous aquifer (see [1]), non-Fickian
diffusion in geological formations (see [2]) and underground environmental problem (see
[6]). Moreover, CDta can be applied to describe the viscoelasticity of some materials (see
[21, 30, 31]).
We note that in recent years, the direct problem of time-fractional diffusion has drawn
much attention. Some classical papers by several authors have shown various aspects of this
problem. For instance, Sakamoto and Yamamoto [27] studied some initial/boundary value
problems for fractional diffusion-wave equations; meanwhile the same problems for multiterm time-fractional diffusion equations are discussed in [15]. Luchko established the maximum principle for time-fractional diffusion equation in case of 0 < a < 1 in [20]. In the
numerical analysis aspect, Jin et al applied finite element method to analyze fractional
subdiffusion equations and gave the corresponding error estimates in [7, 8]. Reference [18]
offers some numerical estimates for the fractional advection–dispersion models which have
fractional derivatives both on time and space. However, the research on inverse problem of
time-fractional diffusion is still relatively scarce. Nonetheless, some interesting works have
been exhibited. Cheng et al [4] established the uniqueness for an inverse problem of fractional diffusion equation, which was one of the first mathematical works about fractional
inverse problems. Li et al [14] discussed the reconstructions of the space-dependent coefficient and the fractional order for a fractional diffusion equation. For the fractional diffusion
equation with multi-term time-fractional orders has been proved in [16]. In [33], the authors
used a Carleman estimate to deduce the stability in determining a zeroth-order coefficient with
one half order Caputo derivative. We also refer [3, 17, 32] for who are interested in Carleman
estimate in time-fractional diffusion equations. Rundell et al recovered an unknown
boundary condition for a time-fractional diffusion equation in [26]. Jin and Rundell [10] gave
a tutorial on inverse problems for anomalous diffusion equations, summarizing some interesting results and pointing out several open problems in fractional inverse and direct
problems.
For model (1.1), we note that if α tends to 1, then it reproduces the classical diffusion
equation with an undetermined coefficient a (t ). The classical parabolic problem was
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Inverse Problems 32 (2016) 015011
Z Zhang
completed by Jones in [11, 12], which was one of the first rigorous solutions of an undetermined coefficient inverse problem. Jones used a variational substitution to create an integral equation, somewhat curiously, this was of fractional order. Also he showed the unique
fixed point of the integral equation is the unknown coefficient a(t). Then he used monotonicity to prove existence and provided a complete stability and uniqueness analysis.
Moreover, for the same model with a fractional a Î (0, 2), Lopushanskyi and Lopushanska
[19] have used Green function to give the representation of solution u (x, t ) and an operator
equation about a(t), which ensure the existence and uniqueness of (u (x, t ), a (t )). This
method is really valuable since it can be extended to other time-fractional differential
equations.
This paper has two goals for the equation (1.1), which are analyzing the weak solution
with a fixed coefficient a(t) and the unique recovery of a(t) from the data
-a (t ) ux (0, t ) = g (t )(We prefer -a (t ) ux (0, t ) = g (t ) instead of the left end flux data
ux (0, t ) because physically, -a (t ) ux (0, t ) = g (t ) is usually measured rather than ux (0, t )).
For the first goal, which can be regarded as the direct problem, we establish the existence and
uniqueness for the weak solution and deduce some regularity results. To the best of my
knowledge, this is the first rigorous theoretical work on the direct problem of model (1.1). For
the second goal, i.e. the inverse problem, we create an operator whose fixed point is a(t) and
prove its monotonicity, uniqueness of its fixed points and deduce a reconstruction algorithm.
The rest of this paper has the following structure. In section 2, we deduce a modified
model from model (1.1) by a simple variational transformation and collect some preliminary
results of fractional calculus. In section 3, we consider the direct problem and give the
spectral representation of the weak solution for the modified model, showing the existence,
uniqueness and some regularity results of this weak solution. The inverse problem is discussed in section 4 by establishing an operator K of which the fixed point is a(t). Then we
prove some main properties of K, such as the monotonicity and the uniqueness of fixed points
which is followed from a reconstruction algorithm. In section 5, some numerical results are
presented to illustrate the theories we build.
2. Preliminary
2.1. Modified model
In order to simplify the boundary condition of model (1.1), we use the transformation
v (x, t ) = u (x, t ) + (x - 1) f (t ) to deduce the following model:
⎧ CD a v (x , t ) - a (t ) vxx (x , t ) = (x - 1)CD a f (t ) ,
t
⎪ t
⎨ v (0, t ) = v (1, t ) = 0, 0 < t  T ;
⎪ (
⎩ v x , 0) = 0, 0 < x < 1,
0 < t  T , 0 < x < 1;
(2.1)
with data
- a (t ) ( vx (0, t ) - f (t ) ) = g (t ) , 0 < t  T .
It is easily shown that model (2.1) has Dirichlet boundary condition and the coefficient a(t) in
model (2.1) is equivalent to that in model (1.1). Following [27] we can analyze the direct
problem of model (2.1) by spectral decomposition. In the next two subsections, we state some
necessary results for the analysis.
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Inverse Problems 32 (2016) 015011
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2.2. Mittag-Leffler function
In this part, we describe the Mittag-Leffler function which plays an important role in fractional diffusion equations. This is a two-parameter function defined as
Ea, b (z ) =
¥
å
k=0
zk
, z Î ,
G (ka + b )
which generalizes the natural exponential function in the sense that E1,1 (z ) = e z . We list some
important properties of the Mittag-Leffler function for future use.
Lemma 2.1. For l > 0, a > 0, t > 0 and n Î N, we have
dn
Ea,1 ( - lt a ) = - lt a - nEa, a - n + 1 ( - lt a ).
dt n
In particular, if we set n = 1, then there holds
d
Ea,1 ( - lt a ) = - lt a - 1Ea, a ( - lt a ).
dt
Proof.
See [27, lemma 3.2].
,
Lemma 2.2. If 0 < a < 1 and z > 0, then Ea, a ( - z )  0.
Proof.
This proof can be found in [22, 25, 29].
,
Lemma 2.3. For 0 < a < 1, Ea,1 ( - t a ) is completely monotonic, that is,
( - 1)n
Proof.
dn
Ea,1 ( - t a)  0, t > 0, n Î  .
dt n
See [5].
,
2.3. Some results for CD αt
The following results concern some properties of CDta , which are crucial for the future
development. The proofs of the lemmas below can be found in [28] and [20, theorem 1],
respectively.
Lemma 2.4. Define Riemann–Liouville αth order integral Ita as
Ita u (t ) =
1
G (a )
t
ò0
(t - t )a- 1u (t ) dt .
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Inverse Problems 32 (2016) 015011
Z Zhang
For 0 < a < 1, u (t ), CDta u (t ) Î C [0, T ], we have
CD a
t
◦ Ita u (t ) = Ita ◦ CDta u (t ) = u (t ) - u (0) , t Î [0, T ] .
Lemma 2.5. Fix 0 < a < 1 and given u (t ) Î C [0, T ] with CDta u Î C [0, T ] . If u(t) attains
its maximum over the interval [0, T ] at the point t = t0 Î (0, T ], then CDta0 u  0.
3. Direct problem
We call v (x, t; a ) is a weak solution to model (2.1) in L2 [0, 1]
corresponding with coefficient a(t) if v ( · , t; a ) Î H01 [0, 1] for t Î (0, T ] and for any
y Î H 2 [0, 1] Ç H01 [0, 1],
Definition 3.1.
⎧ CD a v (x , t ; a) , y (x ) - ( a (t ) v (x , t ; a) , y (x ) ) = ((x - 1) , y (x )) CD a f (t ) , t Î (0, T ] ;
)
xx
t
⎨( t
(
(
)
y
(
))
=
v
x
,
0;
a
,
x
0,
⎩
⎪
⎪
where ( · , · ) is the inner product in L2 (W).
In this section, we consider the weak solution v (x, t; a ) of model (2.1) under definition 3.1.
Specifically, we will establish the spectral representation, existence, uniqueness and some
regularity results for v (x, t; a ). To this end, we state some assumptions on f(t) and the
coefficient a(t).
Assumption 3.2.
(Direct Problem). Suppose a(t) and f(t) satisfy the following assumptions:
(a) f(t) and CDta f (t ) are continuous and non-negative on [0, T ] with f (0) = 0;
(b) a (t ) Î C +[0, T ] ≔ {y Î C [0, T ] : y (t ) > 0, t Î [0, T ]}.
We suppose assumption 3.2 be valid in this section. In order to illustrate assumption 3.2,
the following remark is set up.
Remark 3.1. Assumption 3.2(a ) and (b ) are crucial to deduce the regularity results of the
weak solution v (x, t; a), which can be seen in the future work. Also, without assumption
3.2(a ), [13, theorem 3.25] can not be used, which leads to the fails of the existence and
uniqueness of v (x, t; a ).
We prefer the condition CDta f (t ) is continuous and non-negative on [0, T ] rather that
f ¢ (t ) Î C +[0, T ] since the latter one is stronger that the former. It is not hard to show the
former condition can be derived from f ¢ (t ) Î C +[0, T ].
3.1. Spectral representation of the weak solution
For the operator -Lv = -vxx with homogeneous Dirichlet boundary condition,
{(n2p 2, sin npx ) : n Î +} is the eigensystem of -L and {sin npx : n Î +} forms an
orthogonal basis of L2 [0, 1].
Multiplying sin npx on the both sides of model (2.1) and integrating it on x over [0, 1]
give
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Inverse Problems 32 (2016) 015011
CD a (v (x ,
t
Z Zhang
t ; a) , sin npx ) - a (t ) ( vxx (x , t ; a) , sin npx ) = ((x - 1) , sin npx ) CDta f (t ) .
The zero boundary condition for v (x, t; a ) yields that
- ( vxx (x , t ; a) , sin npx ) = n2p 2 (v (x , t ; a) , sin npx ) ,
which leads to
CD a (v (x ,
t
t ; a) , sin npx ) + a (t ) n2p 2 (v (x , t ; a) , sin npx )
= ((x - 1) , sin npx ) CDta f (t ) , n Î +.
1
Set vn (t; a ) = 2 (v (x, t; a ), sin npx ) because of (sin npx, sin npx ) = 2 . The completeness of
{sin npx : n Î +} in L2 [0, 1] allows us to write v (x, t; a ) as
v (x , t ; a ) =
¥
å vn (t; a) sin npx.
n=1
1
Also, recall that (x - 1, sin npx ) = - np , then the following fractional ODE can be deduced
2 C a
Dt f (t ) , n Î +.
np
The initial condition v (x, 0; a ) = 0 and the linear independence of {sin npx : n Î +}
yield that vn (0; a ) = 0, n Î +.
To sum up, the spectral representation of v (x, t; a ) is
CD a v (t ;
t n
a) = - n2p 2a (t ) vn (t ; a) -
v (x , t ; a ) =
¥
å vn (t; a) sin npx,
(3.1)
n=1
where vn (t; a ) satisfies
CD a v (t ;
t n
a) + n2p 2a (t ) vn (t ; a) = -
2 C a
Dt f (t ) , vn (0; a) = 0, n Î +.
np
(3.2)
3.2. Existence, uniqueness and some properties of the weak solution
In this subsection, we show the existence and uniqueness of the weak solution (3.1). At first,
we state the following lemma, which is [13, theorem 3.25].
Lemma 3.1. For the Cauchy-type problem
CD a y (t )
t
= F (y , t ) , y (0) = c0 ,
if for any continuous y (t ), F (y, t ) Î C [0, T ], and $ A > 0 which is independent on
y Î C [0, T ] and t Î [0, T ] s.t. F (t , y1 ) - F (t , y2 )  A y1 - y2 , then there exists a unique
solution y (t ) Î C [0, T ] for the Cauchy-type problem which also satisfies CDta y (t ) Î C [0, T ].
The following lemma follows from lemma 3.1.
Lemma 3.2. Suppose assumption 3.2 holds. For model (2.1), there exists a unique weak
solution v (x, t; a )
each n Î +.
represented
by
(3.1)
6
with
vn (t; a ), CDta vn (t; a) Î C [0, T ] for
Inverse Problems 32 (2016) 015011
Z Zhang
Fix n Î +. Due to assumption 3.2, the right side of (3.2) satisfies the conditions of
lemma 3.1. Hence, for any n Î +, there exists a unique continuous solution vn (t; a ) of (3.2)
with CDta vn (t; a ) Î C [0, T ]. With the spectral representation (3.1), the existence and
uniqueness of each vn (t; a ) lead to the existence and uniqueness of the weak solution
v (x, t; a ). This completes the proof.
,
Proof.
Furthermore, there are two lemmas about vn (t; a ) which play important roles for deriving
the regularity of v (x, t; a ).
Lemma 3.3. For any a (t ) Î C +[0, T ] , there holds vn (t ; a )  0 on [0, T ] , n Î +.
For each a (t ) Î C +[0, T ] and n Î +, lemma 3.2 yields that vn (t; a ) Î C [0, T ],
which leads to vn (t; a ) attains its maximum over [0, T ] at t = t0 Î [0, T ].
If t0 = 0, with the fact that vn (0, a ) = 0 which comes from lemma 3.2, then we have 0 is
the maximum value of vn (t; a ) on [0, T ], which is the desired result. If t0 Î (0, T ], lemma 3.2
gives that vn (t; a ), CDta vn (t; a) Î C [0, T ], which ensure we can apply lemma 2.5 to vn (t; a ).
With lemma 2.5 and the definition of t0, we have CDta0 vn  0. Inserting CDta0 vn  0 into (3.2)
2
and recalling assumption 3.2 lead to n2p 2a (t0 ) vn (t0; a ) = - CDta0 vn - np CDta0 f  0, i.e.
vn (t0; a )  0 and complete the proof.
,
Proof.
Lemma 3.4. Given a1, a 2 Î C +[0, T ] with a1  a 2 on [0, T ] , we have
vn ( t ; a1 )  vn ( t ; a 2 )  0, t Î [0, T ] , n Î +.
Proof.
Equation (3.2) yields for each n Î +,
⎧C a
2 C a
2 2
⎪
⎪ Dt vn ( t ; a1 ) = - n p a1 (t ) vn ( t ; a1 ) - np Dt f (t ) ,
⎨
⎪ CD a v ( t ; a ) = - n2p 2a (t ) v ( t ; a ) - 2 CD a f (t ) ,
n
n
2
2
2
⎪
t
⎩ t
np
i.e.
CD a v ( t ;
t n
a1 ) + n2p 2a1 (t ) vn ( t ; a1 ) = CDta vn ( t ; a 2 ) + n2p 2a 2 (t ) vn ( t ; a 2 ) .
Let w (t ) = vn (t; a1 ) - vn (t; a 2 ). Lemma 3.2 ensures w (t ),
w (0) = 0. From (3.3), we deduce that
CD a w (t )
t
CD a w (t )
t
(3.3)
Î C [0, T ] and
+ n2p 2a1 (t ) w (t ) = n2p 2 ( a 2 (t ) - a1 (t ) ) vn ( t ; a 2 )  0, t Î [0, T ] ,
(3.4)
where the last equality follows from lemma 3.3 and the condition a1  a 2 .
w (t ) Î C [0, T ] yields that w(t) attains its maximum over [0, T ] at some t = t0 Î [0, T ].
If t0 = 0, we have w (t )  w (0) = 0. For the case t0 Î (0, T ], applying lemma 2.5 to w(t)
yields that CDta0 w  0, which together with (3.4) implies n2p 2a1 (t0 ) w (t0 )  - CDta0 w  0.
With the fact a1 Î C +[0, T ], we deduce w (t0 )  0 and w (t )  w (t0 )  0. The desired result
follows from w  0 and lemma 3.3.
,
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Inverse Problems 32 (2016) 015011
Z Zhang
3.3. Regularity results
In this subsection, we deal with the regularity properties of v (x, t; a ). Through the next two
lemmas, v (x, t; a )C ( [0, T ]; H 2 [0,1] ) is bounded by f C [0, T ] and CDta f C [0, T ].
Lemma 3.5. Suppose assumption 3.2 holds, then v (x, t ; a )C ( [0, T ]; L2 [0,1] )  C f C [0, T ] .
Pick t Î [0, T ], take the fractional integral Ita on both sides of (3.2) and use
lemma 2.4, then
Proof.
2
f (t ) , t Î [0, T ] , n Î +.
np
Lemma 3.3 together with assumption 3.2 and the definition of Ita gives that vn (t; a ) and
Ita [a (t ) vn (t; a )] are both nonpositive on [0, T ], which implies
vn (t ; a) + n2p 2Ita [ a (t ) vn (t ; a) ] = -
vn (t ; a) 
2
f (t ) on [0, T ] , n Î +.
np
(3.5)
Now, fix t Î [0, T ], calculate the L2 norm of v (x, t; a ) with respect to x, then
2
=
v (x , t ; a )
2
¥
å vn (t; a) sin npx
n=1
L2 [0,1]
=
L2 [0,1]
1
2
¥
å ( vn (t; a) )2 ,
(3.6)
n=1
where the last equality is due to the orthogonality of {sin npx : n Î +}. Equation (3.5) and
(3.6) yield
2

v (x , t ; a )
L2 [0,1]
i.e. v (x, t; a )
result.
L2 [0,1]
¥
1
4
(f (t ))2 å 2 2 = C (f (t ))2  C f C2 [0, T ] ,
2
n=1 n p
 C f C [0, T ] for each
t Î [0, T ], which leads to the expected
,
Lemma 3.6. With assumption 3.2, we have
v (x , t ; a)C ( [0, T ]; H 2 [0,1] )  Ca ( f C [0, T ] + CDta f C [0, T ] ),
where Ca > 0 only depends on the coefficient a (t ).
Assumption 3.2 states a Î C +[0, T ], which implies there exists constants
qa , Qa > 0 such that
Proof.
0 < qa < a (t ) < Qa on [0, T ] .
(3.7)
We denote these constants by qa , Qa due to their dependence on a(t). Then by lemma 3.4,
vn (t; qa )  vn (t; a)  0, which leads to
vn ( t ; qa )  vn (t ; a) , t Î [0, T ] , n Î +,
8
(3.8)
Inverse Problems 32 (2016) 015011
Z Zhang
and vn (t; qa ) is the solution of
CD a v ( t ;
t n
qa ) = - n2p 2qa vn ( t ; qa ) -
2 C a
Dt f (t ) .
np
For the above fractional ODE, [27] gives
t
2
a
CD a f (t )(t - t )a - 1E
2 2
a, a ( - n p qa (t - t ) ) dt ,
t
np 0
which allows us to deduce the regularity of vxx (x, t; a ) immediately.
Fix t Î [0, T ]. The representation (3.1) yields that
ò
vn ( t ; qa ) = -
¥
∣ v (x , t ; a)∣2H 2 [0,1] =  å n2p 2vn (t ; a) sin npx2L2 [0,1] =
n=1
1
2
(3.9)
¥
å n4p 4 ( vn (t; a) )2 ,
(3.10)
n=1
which implies the following estimate together with (3.8) and (3.9)
∣ v (x , t ; a)∣2H 2 [0,1] 
1
2
1

2
¥
⎡
t
å ⎣⎢ ò0
n=1
¥ ⎡
t
å ⎢⎣ ò0
n=1
⎤2
2 C a
Dt f (t ) n2p 2 (t - t )a - 1Ea, a ( - n2p 2qa (t - t )a ) dt ⎥
⎦
np
2 C a
Dt f (t )
np
C [0, T ]
2
´ n2p 2 (t - t )a - 1Ea, a ( - n2p 2qa (t - t )a ) dt ⎤⎦
=
t
ò0
The term
0
t
ò0
=-
1
2
¥
⎡
2
n=1
t
ò0
⎤2
n2p 2t a - 1Ea, a ( - n2p 2qa t a ) dt ⎥ .
⎦
(3.11)
n2p 2t a - 1Ea, a (-n2p 2qa t a ) dt is bounded by lemmas 2.1 and 2.2
n2p 2t a - 1Ea, a ( - n2p 2qa t a ) dt =
1
qa
t
å  np CDta f (t )C2 [0, T ] ⎢⎣ ò0
t
ò0
n2p 2t a - 1Ea, a ( - n2p 2qa t a ) dt
d
1
Ea,1 ( - n2p 2qa t a ) dt =
1 - Ea,1 ( - n2p 2qa t a )  qa-1,
dt
qa
(
)
(3.12)
where the last inequality follows from lemma 2.3. Inserting (3.12) into (3.11) gives
∣ v (x , t ; a)∣2H 2 [0,1] 
1
2
¥
4
å n 2p 2
CDta f (t )C2 [0, T ] qa-2
n=1
⎛ ¥ 4 ⎞
1
= qa-2 ⎜ å 2 2 ⎟ CDta f C2 [0, T ] ,
2
⎝ n=1 n p ⎠
i.e.
∣ v (x , t ; a)∣H 2 [0,1]  Ca CDta f (t )C [0, T ] ,
(3.13)
where Ca > 0 depends on a(t) only.
For v (x, t; a ) H1 [0,1], the spectral representation (3.1) gives
¥
∣ v (x , t ; a)∣2H1 [0,1] =  å npvn (t ; a) cos npx2L2 [0,1]
n=1
¥
1
=
2
å n2p 2 ( vn (t; a) )2  ∣v (x, t; a)∣2H
n=1
9
2 [0,1]
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Inverse Problems 32 (2016) 015011
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where the last inequality is ensured by (3.10). Then (3.13) yields
∣ v (x , t ; a)∣H1 [0,1]  Ca CDta f C [0, T ] .
(3.14)
With lemma 3.5, (3.13) and (3.14), we have
v (x , t ; a)H 2 [0,1]  Ca ( f C [0, T ] + CDta f C [0, T ] ), t Î [0, T ] ,
,
which gives the expected conclusion.
The regularity of CDta v (x, t; a ) is derived from lemma 3.6 directly.
Corollary 3.1. Suppose assumption 3.2 be valid, then we have
CDta v (x , t ; a)C ( [0, T ]; L2 [0,1] )  Ca CDta f C [0, T ] ,
where Ca is a positive constant which only depends on the coefficient a(t).
Proof.
Calculate CDta v (x, t; a ) directly from (3.1) and (3.2)
CD a v (x ,
t
t ; a) =
¥
⎡
å ⎣⎢ -n2p 2a (t ) vn (t; a) -
n=1
⎤
2 C a
Dt f (t ) ⎥ sin npx .
⎦
np
Fix an arbitrary t Î [0, T ], the above equality yields that
CDta v (x , t ; a)2L2 [0,1] =


1
2
¥
⎡
⎤2
2 C a
Dt f (t ) ⎥
⎦
np
¥
4
+ å 2 2 ∣CDta f (t )∣2
n
n=1 p
å ⎢⎣ -n2p 2a (t ) vn (t; a) -
n=1
¥
n 4p 4 (a (t ))2
å
n=1
¥
( vn (t; a) )2
å n4p 4 (a (t ))2 ( vn (t; a) )2 + C CDta f C2 [0, T ] .
n=1
The bound of
¥
å n = 1 n4p 4 (a (t ))2 (vn (t; a))2 follows from (3.7), (3.10) and (3.13)
¥
¥
n=1
n=1
å n4p 4 (a (t ))2 ( vn (t; a) )2  å n4p 4Qa2 ( vn (t; a) )2 = 2Qa2 ∣v (x, t; a)∣2H
2 [0,1]
 2Qa2 Ca2 CDta f C2 [0, T ] .
Consequently, we have CDta v (x, t; a)2L2 [0,1]  (2Qa2 Ca2 + C ) CDta f C2 [0, T ] , i.e.
CDta v (x , t ; a)L2 [0,1]  Ca CDta f C [0, T ] , t Î [0, T ] ,
,
which yields the desired result.
Remark 3.2. Lemma 3.6 and corollary 3.1 guarantee that model (2.1) is well defined in the
space L2 [0, 1].
3.4. Main theorem for the direct problem
In this subsection, we state the main theorem for the direct problem of model (2.1), which
follows from lemmas 3.2 and 3.6 and corollary 3.1.
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Inverse Problems 32 (2016) 015011
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Theorem 3.3 (Main theorem for the direct problem). Suppose assumption 3.2 holds, then
there exists a unique weak solution v (x, t; a ) for model (2.1) with the spectral representation
(3.1) and the following regularity estimates:
⎧
C a
⎪ v (x , t ; a )C ( [0, T ]; H 2 [0,1] )  Ca ( f C [0, T ] +  Dt f C [0, T ] ) ;
⎨
C a
⎪ C a
⎩  Dt v (x , t ; a)C ( [0, T ]; L2 [0,1] )  Ca  Dt f C [0, T ] ,
where Ca > 0 only depends on the coefficient a(t).
4. Inverse problem
In this section, we consider the inverse problem for model (2.1): using the data
-a (t )(vx (0, t; a) - f (t )) = g (t ) to reconstruct the coefficient a (t ). We only consider the
case a (t ) Î C +(0, T ], which is regarded as the admissible set of a (t ).
4.1. Operator K and its well-definedness
In order to reconstruct a (t ), we introduces an operator K : W  C +(0, T ] as
Ky (t ) ≔
g (t )
=
f (t ) - vx (0, t ; y)
f (t ) -
g (t )
,
¥
å n= 1npvn (t; y)
where the domain Ω is defined as
W ≔ { y Î C +(0, T ] : 0 < y (t )  g f , t Î (0, T ] }.
By the boundary condition -a (t )(vx (0, t; a) - f (t )) = g (t ), we can see the coefficient a(t)
is one of the fixed points of operator K. In order to analyze this K, we state some restrictions
for the data f and g.
Assumption 4.1.
Suppose f and g satisfy the following restrictions:
(a) f (t ) Î C [0, T ] Ç C +(0, T ] with f (0) = 0 and CDta f (t ) Î C [0, T ] is non-negative
on [0, T ];
(b) g (t ) Î C [0, T ] Ç C +(0, T ] and g (0) = 0.
The following two remarks explain the reasonableness and reason of assumption 4.1
respectively.
Remark 4.1. The function f(t) is input data so that we can control it, which means
assumption 4.1(a ) can be achieved.
For assumption 4.1(b ), given a (t ) Î C +(0, T ], lemma 3.2 ensures the continuity of
vn (t; a ) for each n Î +. This result and the continuity of f(t) and a(t) imply the continuity of
¥
g. Also, lemma 3.3 yields that -vx (0, t; a) = - å n = 1 npvn (t; a)  0 on (0, T ], which
together with a > 0 and assumption 4.1(a ) leads to g > 0 on (0, T ]. At last, the zero initial
condition and f (0) = 0 give vx (0, 0) - f (0) = 0, which implies g (0) = 0. The above
arguments guarantees the reasonableness of assumption 4.1(b ).
Assumption 4.1(a ) and (b ) guarantee the well-definedness of the space Ω in the
sense that the upper bound g/f is strictly larger than the lower bound 0 on the interval (0, T ].
Remark 4.2.
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Inverse Problems 32 (2016) 015011
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Also, assumption 4.1(a ) will be used to show the monotonicity of K and the uniqueness of its
fixed points.
Furthermore, it is clear that vx (0, 0) - f (0) = g (0) = 0 due to the zero initial condition
and assumption 4.1. That means we can not reconstruct a(t) at t = 0, but only over the
interval (0, T ], which can be seen from the definition of W.
For operator K, theorem 3.3 indicates that given a (t ) Î W, there exists a unique Ka.
¥
Also, assumption 4.1 and the result -å n = 1 npvn (t; a )  0 derived from lemma 3.3 yield the
¥
dominator f (t ) - å n = 1 npvn (t; a ) of Ka is strictly positive on (0, T ], which indicates there
is no singularities for Ka on (0, T ]. In addition, the positivity and continuity of the dominator
and numerator of Ka ensure Ka Î C +(0, T ]. Therefore, the operator K is well-defined and
maps Ω to C +(0, T ].
4.2. Monotonicity
We show the monotonicity of K in this part.
Theorem 4.2 (Monotonicity). With assumption 4.1, the operator K is monotone, i.e. given
a1, a 2 Î W with a1  a 2 on (0, T ], we have Ka1 (t )  Ka 2 (t ), t Î (0, T ].
Lemma 3.4 gives vn (t; a1 )  vn (t; a 2 )  0 on (0, T ], n Î +, which together with
¥
¥
assumption 4.1 yields f (t ) - å n = 1 npvn (t; a1 )  f (t ) - å n = 1 npvn (t; a 2 ) > 0. Consequently, with assumption 4.1(b ),
Proof.
g (t )

¥
f (t ) - å n = 1npvn ( t ; a1 )
f (t ) -
g (t )
å
¥
npvn ( t ;
n=1
a2 )
on (0, T ] ,
,
which implies the monotonicity.
The following lemma concerns the upper bound g/f of Ω.
Lemma 4.1. K (g f ) Î W.
¥
Lemmas 3.2 and 3.3 give that å n = 1 npvn (t; g f ) is nonpositive and continuous;
while assumption 4.1 yields f , g Î C +(0, T ]. These results ensure the continuity of K (g f )
g (t )
and 0 < K (g f ) = f (t ) - ¥ npv (t; g f )  g f on (0, T ], which lead to the expected
Proof.
result.
ån=1
n
,
4.3. Uniqueness
In this subsection, we deal with the uniqueness of fixed points of K. To this end, we introduce
a successive iteration as a 0 = g f , a n + 1 = Ka n , n Î . The reasonableness of the initial
guess g/f is guaranteed by lemma 4.1 and this iteration generates a series {a n : n Î }. The
next two lemmas state the above series will converge to a fixed point of K if it exists.
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Lemma 4.2. Suppose assumption 4.1 holds. If a1, a 2 Î W are two fixed points of K with
a1  a 2 , then a1 º a 2 on (0, T ].
Proof.
If a Î W is a fixed point of operator K, then
¥
å npa (t ) vn (t; a) = -g (t ) + a (t ) f (t ), t Î (0, T ].
n=1
Taking
Ita
on both sides of the above equality and recalling its linearity yield
¥
å npIta [ a (t ) vn (t; a) ] = -Ita g (t ) + Ita [a (t ) f (t )], t Î (0, T ].
(4.1)
n=1
Furthermore, the following equality follows from (3.2) and lemma 2.4
Ita [ a (t ) vn (t ; a) ] = -
2
1
f (t ) - 2 2 vn (t ; a) ,
n3p 3
np
which together with (4.1) leads to
Ita [a (t ) f (t )] +
¥
¥
1
2
å np vn (t; a) = -f (t ) å n2p 2
n=1
+ Ita g (t ) , t Î (0, T ] .
(4.2)
n=1
Now pick two fixed points a1, a 2 of K with a1  a 2 . For fixed points a1, a 2 , (4.2) holds
respectively. Combining them together yields
Ita [ a1 (t ) f (t ) ] +
¥
¥
1
1
å np vn ( t; a1) = Ita [ a 2 (t ) f (t ) ] + å np vn ( t; a 2 ).
n=1
(4.3)
n=1
Lemma 3.4 and a1  a 2 give vn (t; a1 )  vn (t; a 2 ), n Î +, i.e.
¥
1
¥
1
å np vn ( t; a1)  å np vn ( t; a 2 ), t Î (0, T ],
n=1
n=1
to Ita [(a1 (t )
which together with (4.3) leads
- a 2 (t )) f (t )]  0, t Î (0, T ]. Moreover, recall
that
and
the
definition
of
we
have
f > 0, a1  a 2
Ita ,
a
Therefore,
it
holds
It [(a1 (t ) - a 2 (t )) f (t )]  0, t Î (0, T ].
0  Ita [(a1 (t ) - a 2 (t )) f (t )]  0, i.e. Ita [(a1 (t ) - a 2 (t )) f (t )] = 0 on (0, T ]. The definition
of Ita and the non-positivity of (a1 - a 2 ) f imply (a1 (t ) - a 2 (t )) f (t ) = 0 on (0, t ] for each
t Î (0, T ], which leads to a1 - a 2 º 0 on (0, T ] due to f > 0.
,
Suppose assumption 4.1 be valid. If there exists a fixed point a (t ) Î W of
operator K, then the series {a n : n Î } is contained by Ω and converges to a (t ).
Lemma 4.3.
Proof. a 0 is the upper bound of Ω gives a  a 0 , which together with the monotonicity of K
yields Ka  Ka 0 , i. e. a  a1. Applying the monotonicity again to this inequality, it holds
Ka  Ka1, i. e. a  a 2 . Continuing this procedure, we deduce a  a n , n Î , which
implies a(t) is a lower bound of {a n : n Î }. On the other hand, lemma 4.1 gives a1  a 0 .
Apply the monotonicity of K to it, there holds Ka1  Ka 0 , i.e. a 2  a1. Repeat this argument
again and again, we obtain a n + 1  a n , n Î , which yields {a n : n Î } is decreasing.
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Inverse Problems 32 (2016) 015011
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Algorithm 1. Numerical algorithm.
Successive iteration algorithm to recover the coefficient a(t)
1: Set up f(t) and measure the data g (t ). f and g should satisfy assumption 4.1;
2: Set the initial guess a 0 = g f ;
3: for k=1, K, N do
4: Using the L1 time-stepping [8] to compute v (x, t; ak - 1),
which is the solution of model (2.1) with coefficient ak - 1;
5: Update the coefficient ak - 1 by ak = Kak - 1;
6: Check stopping criterion ak - ak - 1L2 [0, T ]   0 for some  0 > 0;
7: end for
8: output the approximate coefficient aN .
With the facts that {a n : n Î } is decreasing, a 0 is the upper bound of Ω and a (t ) Î W
is a lower bound of {a n : n Î }, the following results can be deduced: {a n : n Î } is
included by Ω and convergent in it. Denote the limit of {a n : n Î } by a. Clearly a is one
of the fixed points of K and a (t )  a (t ) due to a(t) is a lower bound of {a n : n Î }. Then
lemma 4.2 gives a (t ) = a (t ), which completes this proof.
,
The uniqueness of fixed points of K follows from lemma 4.3 immediately.
Theorem 4.3
(Uniqueness). With assumption 4.1, there is at most one fixed point of K in W.
Given two fixed points a1, a 2 Î W, lemma 4.3 yields that a n  a1 and a n  a 2 ,
which leads to a1 = a 2 and completes this proof.
,
Proof.
4.4. Main theorem for the inverse problem and the reconstruction algorithm
In this part, we state the main theorem for the inverse problem which follows from lemma 4.3
and theorem 4.3 immediately.
(Main theorem for the inverse problem). Let assumption 4.1 be valid. If
there exists a fixed point of K in W, then it is unique and coincides with the limit
of {a n : n Î }.
Theorem 4.4
Based on theorem 4.4, the reconstruction algorithm 1 is deduced. This algorithm is wellposed, meaning that the solution (a (t ), v (x, t; a)) depends continuously on the measured data
g, which can be seen from the definition of operator K. However, if we turn things around and
assume that g is given as part of the model and the ‘temperature’ f = u (0, t ) is measured,
then this problem is no longer such continuously dependence. Indeed, it can be easily shown
that the C0-norm of a(t) depends on  CDta f C [0, T ] and CDta contains the term f ¢ . This illposedness problem can be done by either Tikhonov regularization—adding a penalty term, or
mollification—replacing the data f by a mollifier f .
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Inverse Problems 32 (2016) 015011
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Figure 1. a 0 (t ) , a1 (t ) , a2 (t ) and a3 (t ) with a = 0.9 and  0 = 10-6 .
Figure 2. a (t ) and a(t) with a = 0.9 and  0 = 10-6 .
Moreover, the above situation also occurs in this inverse problem for the parabolic case
[11], i.e. a = 1, in the sense that the integral equation of a(t) created by the author contains
the terms f ¢ and g.
5. Numerical results for the inverse problem
In this section, we will present some numerical results of the reconstruction.
5.1. Numerical results for smooth coefficient
In this case, we let T=1 and a = 0.9, then pick the smooth data f (t ) = t 2 + t and
a (t ) = sin 4pt + 1.1. We regard this a(t) as the exact result. L1 time-stepping [8] yields the
output data g (t ). As expected, the g(t) we get is continuous and positive on (0, T ], which
satisfies assumption 4.1. Then we insert the data f(t) and g(t) into algorithm 1 and
pick  0 = 10-6.
Theorem 4.4 ensures that {a n : n Î } is decreasing and a (t ) = a (t ) = limn ¥ a n ,
which are presented by figures 1 and 2, respectively. From figure 2, we can see a (t ) is a good
approximation for a(t). More precisely, we can provide the L2 error as
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Inverse Problems 32 (2016) 015011
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Table 2. a - a L2 [0, T ] for different  0 with a = 0.9.
0
a - a L2 [0, T ] »
10−3
10−4
10−5
10−6
10−7
8 ´ 10-4
9 ´ 10-5
5 ´ 10-6
5 ´ 10-7
6 ´ 10-8
Figure 3. a 0 (t ) , a1 (t ) , a2 (t ) and a3 (t ) for nonsmooth case with a = 0.9 and  0 = 10-6 .
Table 3. The amounts of iterations for different α with  0 = 10-6 .
α
0.1
0.3
0.5
0.7
0.9
Amount of iterations from a 0 (t )
83
62
45
32
19
a - a L2 [0, T ] » 5 ´ 10-7 »  0, which implies the L2 error a - a L2 [0, T ] maybe bounded
by  0. In order to verity this relation, we try some more  0 and present the L2 errors
corresponding different  0 in table 2. This table gives a rough estimate as
a - a L2 [0, T ] »  0.
If we fix  0 = 10-6 and choose a different a Î (0, 1) for model (2.1), the numerical
results are still good, which means algorithm 1 does not strongly depend on α. However, the
α can affect the amount of iterations with the relation that the less α is, the more iterations we
need, which can be seen in table 3. For table 3, [9, figure 1] provides a reasonable interpretation. For time-fractional diffusion equation, the fundamental solution is in terms of
Mittag-Leffler function, which has the property that restricted α in (0, 1), the less α is, the
slower the decay rate of Ea,1 (-z ) is as z  +¥. This property explains the phenomenon in
table 3.
5.2. Numerical results for nonsmooth coefficient
Even though assumption 4.1 requires continuous data, algorithm 1 still works for the coefficient in L¥ (0, T ]. This means it is likely to weaken the restrictions on data to a certain
extent in numerical part.
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Figure 4. a (t ) and a(t) for nonsmooth case with a = 0.9 and  0 = 10-6 .
Figure 5. g(t) and gd (t ) with d = 5%.
For this numerical experiment, we pick a = 0.9, T = 1,  0 = 10-6 and set
1
f (t ) = t 2 + t , a (t ) = c( 0, T ) + c⎡⎣ T , 2 T ) + c⎡⎣ 2 T , T ⎤⎦ ,
3
3
2 3 3
where the exact coefficient a(t) has two discontinuities. The figures 3 and 4 illustrate theorem
4.4 holds for this experiment in the sense that figure 3 demonstrates {a n : n Î } is
decreasing, meanwhile figure 4 indicates the existence of a (t ) and a (t ) = a (t ). Similar to the
smooth case, a (t ) is still a good approximation for nonsmooth a(t) with the L2 error as
a - a L2 [0, T ] » 8 ´ 10-7 »  0.
5.3. Numerical results for noisy data
In this subsection, we consider the experiment with noisy data since every measurement in
practice should contain noise. Let g be the exact data, denote the noisy data by gδ with
relatively noise level d, i.e. g - gd C [0,1] gC [0,1]  d. Define the noisy operator Kδ as
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Inverse Problems 32 (2016) 015011
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Figure 6. a(t) and ad (t ) with d = 5% under the smooth case.
Figure 7. a(t) and ad (t ) with d = 5% under the nonsmooth case.
Table 4. a - ad L2 [0, T ] aL2 [0, T ] for different δ with  0 = 10-2 .
δ
1%
2%
3%
4%
5%
0.012
0.014
0.027
0.030
0.038
gd (t )
=
f (t ) - vx (0, t ; y)
f (t ) -
gd (t )
a - ad L2 [0, T ] aL2 [0, T ] »
Kd y (t ) ≔
¥
å n= 1npvn (t; y)
,
with domain
Wd ≔ { y Î C +(0, T ] : 0 < y (t )  gd (t ) f (t ) , t Î (0, T ] }.
Since δ is a slightly positive number, we can assume gδ satisfy assumption 4.1(b ), which
means theorem 4.4 still holds for Kd , and the uniqueness of the fixed point aδ of Kδ can be
deduced. If aδ exists, define ad,0 = gd f and ad, n + 1 = Kad, n , n Î . The series
{ad, n : n Î } converges to aδ decreasingly because of theorem 4.4 and we denote the
limit by ad . Algorithm 1 can be used to recover ad after a slightly modification—replacing g
and K by gδ and Kd , respectively.
Similar to the numerical experiment with smooth data, we fix
T = 1, a = 0.9,  0 = 10-2 and let f (t ) = t 2 + t, a (t ) = sin 4pt + 1.1. Then we add 5%
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Inverse Problems 32 (2016) 015011
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noise into the data g generated by L1 time-stepping and denote the data with noise by gd . By
the way, the reason we do not need to pick small  0 is that with noise d = 5%,  0 = 10-2 and
 0 = 10-6 will reproduce similar approximations and absolutely, the cost for  0 = 10-2 is
much less. Figure 5 shows g and gδ with d = 5%. With modified algorithm 1, we can obtain
the approximate coefficient ad , which comes from one experiment and is presented by
figure 6. In addition, the relative L2 error is a - ad L2 [0, T ] aL2 [0, T ] » 0.035 » d , which
illustrates the well-possedness of algorithm 1, and the relative L2 errors corresponding different δ are presented in table 4, which yields the rough error estimate:
a - adL2 [0, T ] aL2 [0, T ] » d .
We also attempt a numerical experiment with noise for the nonsmooth data. Similar to
the previous experiment, we pick a = 0.9, T = 1,  0 = 10-2 , d = 5% and
1
f (t ) = t 2 + t , a (t ) = c( 0, T ) + c⎡⎣ T , 2 T ) + c⎡⎣ 2 T , T ⎤⎦ .
3
3
2 3 3
Figure 7 shows ad (t ) and a(t) with the relative L2 error as
a - adL2 [0, T ] aL2 [0, T ] » 0.038 » d .
6. Concluding remarks
We have studied an undetermined coefficient problem for a fractional diffusion equation. For
the direct problem part, fixing the coefficient a(t), we provide the spectral representation of
the weak solution and prove its existence, uniqueness and some regularity properties. For the
inverse problem part, in order to recover the coefficient a (t ), we define an operator K from the
FDE model of which one of the fixed points coincides with a(t). Then we establish the
monotonicity of K, uniqueness of its fixed points and a numerical algorithm to recover the
coefficient a(t).
For this work, there are several questions deserving further investigation. For the direct
problem part, we can extend this time-fractional diffusion equation to the one with inhomogeneous right-hand side and nontrivial initial condition. In inverse problem part, at first,
we do not show the existence of a(t) since the domain Ω is not a complete subspace of
C +(0, T ]. However, it is possible to refine the domain and use monotonicity to show existence as Jones did in [11]. Secondly, our numerical experiments indicate that assumption 4.1
can be relaxed but this observation still awaits theoretical proofs. At last, the error estimate for
algorithm 1 and how the noise scale δ affects the error a - a L2 [0, T ] are still open and worth
further analysis.
Acknowledgments
The author is indebted to William Rundell, Bangti Jin and Zhi Zhou for their assistance in
this work.
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Inverse Problems 32 (2016) 015011
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