Список литературы:
[1] Гусятников, П.Б. К вопросу об информированности игроков в дифференциальной
игре // Прикладная математика и механика. 1972. Т.36, № 5. С. 917924.
[2] Пшеничный Б.Н., Остапенко В.В. Дифференциальные игры. Киев, 1992. 260 с.
[3] Остапенко В.В., Амиргалиева С.Н., Остапенко Е.В. Выпуклый анализ и
дифференциальные игры. – Алматы, 2005. – 392 с.
УДК 51
Erdogan A.S.
1
1
Assoc. Prof., Suleyman Demirel University, Аlmaty, Каzakhstan
e-mail: abdullah.erdogan@sdu.edu.kz
ON A RIGHT HAND SIDE IDENTIFICATION PROBLEM OF A PARABOLIC
EQUATION
Abstract. In many physical phenomena, especially in temperature overspecification
partial differential equations with an unknown source function appears. The present paper is
devoted to the study of the wellposedness of the approximate solution of a righthand side
identification problem for a parabolic equation.
Key words: Identification problem, stability estimates
Introduction
Inverse parabolic problems is of significant importance in mathematical sciences,
applied sciences and engineering. In many physical phenomena, for instance in the process of
transportation, diffusion and conduction of natural materials, the parabolic partial differential
equation is induced (see [1] and the references therein). In inverse problems, the optimal
overdetermination conditions are analyzed in some classical boundary conditions or/and
similar conditions given at a point. The problem of determining the temperature at one end of a
rod from temperature measurements at an interior point is an example of an inverse heat
conduction problem (IHCP) which has been extensively studied [2].
120
Considerable efforts have been expanded in formulating numerical solution methods
that are both accurate and efficient. Methods of numerical solutions of parabolic problems with
parameters have been studied by many researchers (see, [36]). One usually focuses himself on
the uniqueness and the stability of the inverse problem. For the uniqueness, we refer [7]. The
discussion of the stability is preparatory to the numerical implementation for the inverse
problem in the theoretical respect. During the last decades, some numerical techniques have
been proposed to solve a onedimensional IHCP. Determination of a control function in three
dimensional parabolic equation and in polar coordinate system are also investigated. Among
them the finite difference method and the finite element method are so far the principal
numerical tool of choice for the modeling and simulation of the IHCP.
Problem formulation
One application of inverse heat conduction problem in engineering and science is to
predict the thermal conductivity from the measured temperature profiles. The inverse
estimation of thermal conductivity by the measured temperature profiles has been studied by
many researchers. This work is devoted to the study of the wellposedness of the approximate
solution of the righthand side identification problem
l
x
x
x
u
T
t
l
t
u
t
u
T
t
l
x
x
t
f
x
Q
t
P
x
t
u
x
t
u
x
a
x
t
u
xx
t
0
,
=
0,
,
0,0
=
,
=
,0
,
<
<
0
,
<
<
0
,
,
=
,
,
)
(
,
(1)
where
.
=
2
2
1
1
x
q
t
p
x
q
t
p
x
q
t
p
x
Q
t
P
n
n
Here
x
t
u ,
and
n
i
t
p
i
,
1,2,
=
are unknown functions,
,
, x
t
f
,
,
1,2,
=
)
(
n
i
x
q
i
),
(x
and
x
a
are given sufficiently
smooth functions,
0
>
x
a
and
0
>
is a sufficiently large number. For solving the
parabolic
inverse
problem
(1),
the
overdetermined
conditions
=
,
,
,
=
,
,
=
,
2
2
1
1
t
s
t
u
t
s
t
u
t
s
t
u
n
n
where
n
s
s
s
,
,
,
2
1
are inner points,
t
i
,
,
1,2,
=
n
i
T
t
0
are sufficiently smooth functions are determined. Let us assume
=
0
q
0
=
l
q
and
n
s
q
s
q
s
q
,
,
2
1
are different from zero. In this paper, for the clarity
2
=
n
is taken.
Stability analysis of solutions
t
p
x
t
u
1
,
,
and
t
p
2
are given by the help of an
auxiliary problem. To formulate the auxiliary problem, the transformation
,
,
=
,
2
2
1
1
x
q
t
x
q
t
x
t
w
x
t
u
(2)
where
0
=
0
,
=
1
1
0
1
ds
s
p
t
t
and
0
=
0
,
=
2
2
0
2
ds
s
p
t
t
is determined. Taking partial derivative of both sides of equation (2), we get
x
q
t
p
x
q
t
p
x
t
w
x
t
u
t
t
2
2
1
1
,
=
,
(3)
and
.
,
=
,
2,
2
1,
1
x
q
t
x
q
t
x
t
w
x
t
u
xx
xx
xx
xx
(4)
Using the overdetermined conditions, we can write
1
2
2
1
1
1
1
1
1
,
=
,
=
s
q
t
s
q
t
s
t
w
s
t
u
t
and
.
,
=
,
=
2
2
2
2
1
1
2
2
2
s
q
t
s
q
t
s
t
w
s
t
u
t
If
121
0,
=
=
,
2
2
2
1
1
2
1
1
2
1
s
q
s
q
s
q
s
q
s
s
J
one can easily show that
2
1
2
2
2
2
1
2
1
1
1
,
,
,
=
s
s
J
s
q
s
t
w
t
s
q
s
t
w
t
t
(5)
and
.
,
,
,
=
2
1
2
2
2
1
1
1
1
1
2
s
s
J
s
t
w
t
s
q
s
t
w
t
s
q
t
(6)
Replacing equations (3)(6) in (1), we reach to the following auxiliary problem
l
x
x
x
u
T
t
l
t
u
t
u
T
t
l
x
x
q
x
q
s
q
s
q
s
q
s
q
s
q
s
t
w
s
q
s
t
w
s
q
t
s
q
t
x
q
x
q
s
q
s
q
s
q
s
q
s
q
s
t
w
s
q
s
t
w
s
q
t
s
q
t
x
t
f
x
t
w
x
t
w
x
a
x
t
w
xx
xx
xx
t
0
,
=
0,
,
0,0
=
,
=
,0
,
<
<
0
,
<
<
0
,
,
,
,
,
=
,
,
)
(
,
2
2,
1
2
2
1
2
2
1
1
2
1
1
1
1
2
2
1
1
1
1
2
1
1,
1
2
2
1
2
2
1
1
1
2
2
2
2
1
1
2
2
2
2
1
(7)
under the same assumptions on
x
q
.
Theoretical considerations
In this section, coercive stability estimates of problem (1) are obtained where additional
condition is observed with and without noise. To formulate our results, we introduce the
Banach space
,
0, L
C
0,1
, of all continuous functions
x
defined on
L
0,
with
0
=
=
0
L
'
satisfying a Hölder condition for which the following norm is finite
.
sup
max
=
0,
<
0
0
h
x
h
x
x
L
L
h
x
x
L
x
C
In a Banach space ,
E with the help of a positive operator
A
we introduce the fractional
spaces
1,
<
<
,0
E
consisting of all
E
v
for which the following norm is finite:
.
exp
sup
=
1
0
>
E
E
E
v
A
A
v
v
Positive constants will be indicated by
M
which can be differ in time.
Theorem 1 1 Let
,
0,
2
2
L
C
x
L
C
T
C
x
t
f
0,
,
0,
,
2
and
T
C
t
'
0,
. Then
for the solution of problem (1) the following coercive stability estimates
,
0,
0,
,
,
,
,
,
,
,
]
[0,
]
[0,
0,
2
,
0,
2
2
0,
2
2
,
0,
2
,
0,
T
C
C
T
C
C
T
C
'
C
T
C
C
T
C
t
L
f
L
T
q
x
a
M
q
x
M
L
u
L
u
T
C
'
T
C
q
x
M
p
0,
0,
1
,
122
,
0,
0,
,
,
,
,
,
,
0,
2
,
0,
2
2
T
C
C
T
C
C
L
f
L
T
q
x
a
M
T
C
'
T
C
q
x
M
p
0,
0,
2
,
T
C
C
T
C
C
L
f
L
T
q
x
a
M
0,
2
,
0,
2
2
0,
0,
,
,
,
,
,
,
hold.
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