Advances in Systems Science and Applications (2013) Vol.13 No.4 392-400

Stability Results for an Inverse Parabolic Problem
J. Damirchi1 and A. shidfar2

1Faculty of Mathematics, Statistics and Computer Science,
Department of Mathematics, Semnan University, Semnan, Iran.
2Department of Mathematics, Iran University of Science and

Technology, Narmak, Tehran 16844, Iran.

Abstract

In this paper, a one dimensional inverse parabolic problem in a quarter plane
will be considered. The unknown function in a boundary is estimated from an
over specified condition at a fixed location inside the region by solving an ill-
posed integral equation. The Tikhonov Regularization method of the 1st order
is applied in order to stabilize the solution of the ill-posed problem. The solution
of the inverse problem is defined by minimization of the Tikhonov functional.
Some analytical results for regularization parameter determination and stability
of solution of inverse problem are derived.
Keywords Tikhonov Regularization Method, Inverse Parabolic Problem, Regu-
larization Parameter, Stable Solution, Ill-Posed Problem

1 Introduction

Inverse parabolic problems play a crucial role in applied Mathematics, Physics
and engineering science. They arise for example, in the study of heat conduc-
tion processes, diffusion, control theory [1-10]. In recent years, a lot of attention
has been devoted to the study of inverse parabolic problems. Hence, the last
20 years have seen growing attention paid in the literature to the developmen-
t, analysis, and implementation of accurate methods for the solution of inverse
parabolic problems, i.e., the determination of unknown boundary condition g(t)
in the parabolic partial differential equation.

In this paper, we investigate an inverse parabolic problem in quarter plane to
obtain an unknown function g(t) from over specified data p(t) at a fixed location
inside the body. This inverse problem can be reduced to the operator integral
equation Ag = G. It is well known that the operator equation Ag = G is an
ill-posed problem, when a solution is unstable with respect to small variations
in input data. Since in the usual case where only a measured or computation
approximation Gδ is available, some kind of regularization methods is required
in order to obtain a reasonable stable approximation Gδ to g [6, 11-15]. In order
to obtain stable solution for this ill-posed problem the Tikhonov regularization
method is applied for operator equation Ag = G for retrieving solution in a sta-
ble manner. Applying general result in the theory of Tikhonov regularization
method for ill-posed inverse problem which consists in solving the unconstrained
minimization problem, we can find stable solution. Since the regularization pa-



Advances in Systems Science and Applications (2013) Vol.13 No.4 393

rameter play an important role in applying the Tikhonov regularization method
to the operator equation, we obtain this parameter based on the error in input
data directly, which can be one of the advantage of our method for selecting reg-
ularization parameter with respect to other methods [16-19].

The organization of this paper is as follows, In forcecoming section, mathemati-
cal formulation for this inverse parabolic problem in a quarter plane is introduced.
In section 3, Thikhonov regularization method is stated and we use this method
to construct a stable solution for this ill-posed problem. Some theoretical result-
s will be proved about the solution of this ill-posed problem and the choice of
regularization parameter. It will be shown that the solution of Thikhonov reg-
ularization method is stable under small errors in input data and the existence
of this stable solution is shown. We conclude this article with a brief conclusive
discussion in section 4. Finally, some references are given at the end of this paper.

2 Mathematical Formulation

In this section, we consider the following inverse parabolic problem in a quarter
plane:

ut = uxx, 0 < x, 0 < t < T (1)

u(x, 0) = f(x), 0 < x (2)

u(0, t) = g(t), 0 < t < T, (3)

|u(x, t)| ≤ M, 0 < x, 0 < t < T (4)

where f(x) and g(t) are piecewise known continuous functions and M is a
positive number.The problem consist of using an overspecified data p(t), which
is given by

p(t) = u(1, t), 0 < t < T (5)

to determine the unknown function g(t).
For the forward problem (1)-(4), the unique bounded solution u(x, t) is given

by [20],

u(x, t) =
x√
4π

∫ t

0

g(τ)√
(t− τ)3

e
− x2

4(t−τ)dτ

+
1√
4πt

∫ ∞

0
(e−

(x−ξ)2

4t − e−
(x+ξ)2

4t )f(ξ)dξ

(6)

Due to the setting x = 1, and using (5) we obtain

∫ t

0

e
− 1

4(t−τ)√
(t− τ)3

g(τ)dτ = 2
√
πp(t)− 1√

t

∫ ∞

0
(e−

(x−ξ)2

4t − e−
(x+ξ)2

4t )f(ξ)dξ (7)



394 J. Damirchi: Stability Results for an Inverse Parabolic Problem

which is written in the form∫ t

0
h(t− τ)g(τ)dτ = 2

√
πp(t)− 1√

t

∫ ∞

0
(e−

(x−ξ)2

4t − e−
(x+ξ)2

4t )f(ξ)dξ

where the kernel h(t, τ) = 1√
(t−τ)3

e
− 1

4(t−τ) is a continuous function on [0, T ] ×
[0, T ].

Now, we formulate (7) in term of an operator integral equation :

(Ag)(t) = G(t), 0 < t < T (8)

where the operator A is defined by

(Ag)(t) =

∫ t

0

e
− 1

4(t−τ)√
(t− τ)3

g(τ)dτ, 0 < t < T

and

G(t) = 2
√
πp(t)− 1√

t

∫ ∞

0
(e−

(x−ξ)2

4t − e−
(x+ξ)2

4t )f(ξ)dξ, 0 < t < T

The above integral equation of the first kind (7) cannot be reduce into an inte-
gral equation of the second kind by differentiation and the problem is inherently
ill-posed. For 1 ≤ p < ∞, A is a compact linear operator in Lp[0, T ]. Zero is not
an eigenvalue of A and is the only point in the spectrum of A, thus A−1 exist
and is unbounded, so if G(t) on 0 ≤ t ≤ T is in the range of A, g(t) is uniquely
determined from g = A−1G. In practice, with G(t) obtained from measurement,
small error in G(t) lead to enormous errors in g(t) because A−1 is unbounded.
In order to regularize the problem, we use Tikhonov regularization method for
finding approximate stable solution for ill-posed integral equation (8). In the
next section we describe Tikhonov regularization method.

3 Tikhonov Regularization Method for Ill-Posed Integral Equation

It is well known that the Tikhonov regularization method is one of the useful tools
for solving an ill-posed problem of the form (7)[12-13, 19]. Since in practical pur-
pose, the input data are non smooth, we apply Tikhonov regularization method
to construct stable solution for solving ill-posed equation (8). Based on the main
concept of Tikhonov regularization method, we introduce smoothing function-
al Mα[g,G] which is called also Thikhonov functional and stabilizing functional
Ω(g) as follows:

Mα[g,G] = ∥Ag −G∥2L2[0,T ] + αΩ(g)



Advances in Systems Science and Applications (2013) Vol.13 No.4 395

α is regularization parameter and Ω(g) is a stabilizer of the 1th order constant
coefficient which is defined as:

Ω(g) =

∫ T

0
(g2(τ) + g′2(τ))dτ

We construct a regularize solution for the integral equation (8) by using the
following minimization problem which is state by the following Theorem:
Theorem 1. For every function G ∈ L2[0, T ], and any positive α, there exists
an element gα ∈ W 1

2 , such that smoothing functional Mα[g,G] attain it greatest
lower bound.

infMα[g,G] = Mα[gα, G]

Proof. We have

Mα[g,G] =

∫ T

0
(

∫ t

0

e
− 1

4(t−τ)√
(t− τ)3

g(τ)dτ −G(t))2dt+ α

∫ T

0
(g2(τ) + g′2(τ))dτ

A condition for a minimum of this functional is vanishing of its first variation,
this is written in the form

1

2

d

dε
Mα[g + εη,G]|ε=0 =

∫ T

0
[

∫ T

τ

∫ t

0
h(t− s)h(t− τ)g(s)dsdt−

∫ T

τ
h(t− τ)dt

− α(g′′(τ)− g(τ))]η(τ)dτ + g′(τ)η(τ)|T0
=0

(9)

Here, η(τ) is an arbitrary variation of the function g(τ) such that both g(τ) and
g(τ) + εη(τ) belong to the class of admissible function.

Condition (9) will be satisfied if∫ T

τ

∫ t

0
h(t− s)h(t− τ)g(s)dsdt−

∫ T

τ
h(t− τ)dt = α(g′′(τ)− g(τ)) (10)

and

g′(0) = g′(T ) = 0 (11)

The equation (10) is called Euler-Lagrange equation, therefore the minimizer
gα(t) for Tikhonov functional is determined by the solution of Euler-Lagrange
equation corresponding to functional Mα[g,G]. The solution of (10) with bound-
ary condition (11) is unique by classical theorems in ODE.

Now, we can assume that the regularized solution of the above minimization
problem, gα as an regularizing operator R(G,α) such that gα = R(G,α) where



396 J. Damirchi: Stability Results for an Inverse Parabolic Problem

α = α(δ,Gα) in accordance with the error in the initial data G and δ, measuring
the error in data, we select α in a suitable way such that R(G,α) is a regularizing
operator for the equation (8), and gα = R(Gα, α(δ)) can be taken as an approxi-
mate stable solution for ill-posed problem (8).
Theorem 2. If gT (t) ∈ C[0, T ] be the exact solution of the original problem (8)
associated with the exact right hand member G = GT ; that, (AgT )(t) = GT (t).
Then, for every positive number ε, there exists a positive number δ(ε) such that
for every Gδ ∈ L2[0, T ] the inequality

∥GT (t)−Gδ(t)∥L2[0,T ] ≤ δ < δ(ε)

implies the inequality
∥gα(δ)(t)− gT (t)∥C[0,T ] ≤ ε

where gα = R(Gα(δ), α(δ)) be the solution of (8) associated with perturbed data
Gδ for all α satisfying α(δ) = δγ , 0 ≤ γ < 2.
Proof. Since gα is a minimizer of functional Mα, we have

Mα(δ)[gα, Gδ] ≤ Mα(δ)[gT , Gδ]

therefore,

∥Agα −Gδ∥2 ≤ Mα(δ)[gα, Gδ]

≤
∫ T

0
(AgT (t)−Gδ(t))

2dt+ α(δ)

∫ T

0
(g2T (τ) + g′2T (τ))dτ

=

∫ T

0
(GT (t)−Gδ(t))

2dt+ δγ
∫ T

0
(g2T (τ) + g′2T (τ))dτ

≤ δ2 + δγ
∫ T

0
(g2T (τ) + g′2T (τ))dτ

≤ δγ(1 +

∫ T

0
(g2T (τ) + g′2T (τ))dτ)

= δγN

with N = 1 +
∫ T
0 (g2T (τ) + g′2T (τ))dτ .

Consequently, the elements gT (t) and gα(δ)(t) belong to the compact subset E
of element g of C[0, T ] such that

E = {g(t)| ∥g∥2W 1
2
≤ N}

Since E is compact in C[0, T ], and the operator A is continuous, the mapping
A : E → AE is continuous and one to one, therefore the inverse mapping A−1 :
AE → E is also continuous. This means that,

∀ε > 0, ∃η(ε), ∥GT −Gα∥ ≤ η(ε), AgT = GT , Agα = Gα



Advances in Systems Science and Applications (2013) Vol.13 No.4 397

then

∥gT − gα∥C[0,T ] ≤ ε

on the other hand we have

∥GT −Gα∥2L2 =

∫ T

0
(Agα −GT (t))

2dt < δ2

an so,

∥gT (t)− gα(δ)(t)∥C[0,T ] = ∥A−1Agt −A−1Agα(δ)∥ ≤ ∥A−1∥∥Agt −Agα(δ)∥

on the other hand

∥Agt −Agα(δ)∥L2 ≤ ∥AgT −Gδ∥+ ∥Agα(δ) −Gδ∥
≤ ∥GT −Gδ∥+ ∥Agα(δ) −Gδ∥

≤ δ + δ
γ
2

√
N

≤ δ
γ
2 (1 +

√
N)

Therefore,

∥gT − gα(δ)∥ ≤ ∥A−1∥δ
γ
2 (1 +

√
N)

The above results show that δ(ε) should be chosen in the form

δ(ε) ≤ [
ε

∥A−1∥(1 +
√
N)

]
2
γ

such that the Theorem is satisfied.
The above theorem shows that when we construct regularizing operator by

minimizing the smoothing functional Mα[g,G], the regularization parameter α
can be obtain according to the error in the right hand member. By the next
theorem we will show that G depends continuously on the initial data f, p.
Theorem 3. If the exact data fT (x) and pT (t) satisfy (8) and the approximate
data fδ(x) and pδ(t) also satisfy (8), then inequalities ∥fT (x)− fδ(x)∥L2[0,∞] < δ
and ∥pT (t)− pδ(t)∥L2[0,T ] < δ imply that

∥GT (t)−Gδ(t)∥L2[0,T ] < δD, D = (9
√
2πT + 12π)

1
2



398 J. Damirchi: Stability Results for an Inverse Parabolic Problem

Proof. By using of Cauchy-Schwartz inequality, we have

∥GT (t)−Gδ(t)∥2

=∥2
√
π(pT (t)− pδ(t))−

1√
t

∫ ∞

0
(e−

(1−x)2

4t − e−
(1+x)2

4t )(fT (x)− fδ(x))dx∥2

≤3[

∫ T

0
4π(pT (t)− pδ(t))

2dt+

∫ T

0

1

t
(

∫ ∞

0
e−

(1−x)2

4t (fT (x)− fδ(x))dx)
2dt

+

∫ T

0

1

t
(

∫ ∞

0
e−

(1+x)2

4t (fT (x)− fδ(x))dx)
2dt]

≤3(4δ2π +

∫ T

0

1

t
(

∫ ∞

0
(e−

(1−x)2

4t )2dx)(

∫ ∞

0
(fT (x)− fδ(x))

2dx)dt

+

∫ T

0

1

t
(

∫ ∞

0
(e−

(1+x)2

4t )2dx)(

∫ ∞

0
(fT (x)− fδ(x))

2dx)dt)

≤3(4δ2π + δ2
∫ T

0

1

t

∫ ∞

0
e−

(1−x)2

2t dxdt+ δ2
∫ T

0

1

t

∫ ∞

0
e−

x2

2t dxdt)

≤3(4πδ2 + 3
√
2πTδ2)

=δ2(12π + 9
√
2πT )

Theorem (3) and (4) show that by choosing α in such a way that the choice
for α is consistent with the accuracy δ of the initial data, then the element
gα = R(Gα, α) obtained with the aid of the regularizing operator R(G,α), can
be taken as approximate stable solution of equation (8).

We summarize the above results by the following Theorem:
Theorem 4. If gT (t) is the exact solution of equation (8) with exact data fT (x)
and pT (t), then for every ε > 0 and the approximate data fδ(x) and pδ(t) also
satisfy (8), there exist δ(ε) and α(δ) such that inequalities ∥fT (x)−fδ(x)∥L2[0,∞] <
δ and ∥pT (t)− pδ(t)∥L2[0,∞] < δ imply that inequality

∥gT (t)−Gα(t)∥ < ε

where gα(δ) = R(Gδ, α(δ)).

4 Conclusion

In this paper, we have introduced Tikhonov regularization method and we have
shown why it is important to use it in order to solve ill-posed problems. We
have shown that the Tikhonov regularization technique is introduced to treat the
instability of obtaining a stable solution. The choice of regularization parameter
and stability of solution are proved. It is very interesting to extend these results
for higher dimensional problem and for nonstandard heat equation.



Advances in Systems Science and Applications (2013) Vol.13 No.4 399

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Corresponding author
J. Damirchi can be contacted at: damirchi.javad@gmail.com