Affiliation:
1. Kocaeli University , Kocaeli , Turkey
Abstract
Abstract
This paper deals with an inverse coefficient problem of simultaneously identifying the thermal conductivity
k
(
x
)
k(x)
and radiative coefficient
q
(
x
)
q(x)
in the 1D heat equation
u
t
=
(
k
(
x
)
u
x
)
x
-
q
(
x
)
u
u_{t}=(k(x)u_{x})_{x}-q(x)u
from the most available Dirichlet and Neumann boundary measured outputs.
The Neumann-to-Dirichlet and Neumann-to-Neumann operators
Φ
[
k
,
q
]
(
t
)
:=
u
(
ℓ
,
t
;
k
,
q
)
\Phi[k,q](t):=u(\ell,t;k,q)
,
Ψ
[
k
,
q
]
(
t
)
:=
-
k
(
0
)
u
x
(
0
,
t
;
k
,
q
)
\Psi[k,q](t):=-k(0)u_{x}(0,t;k,q)
are introduced, and main properties of these operators are derived.
Then the Tikhonov functional
J
(
k
,
q
)
=
1
2
∥
Φ
[
k
,
q
]
-
ν
∥
L
2
(
0
,
T
)
2
+
1
2
∥
Ψ
[
k
,
q
]
-
φ
∥
L
2
(
0
,
T
)
2
J(k,q)=\tfrac{1}{2}\lVert\Phi[k,q]-\nu\rVert^{2}_{L^{2}(0,T)}+\tfrac{1}{2}\lVert\Psi[k,q]-\varphi\rVert^{2}_{L^{2}(0,T)}
of two functions
k
(
x
)
k(x)
and
q
(
x
)
q(x)
is introduced, and an existence of a quasi-solution of the inverse coefficient problem is proved.
An explicit formula for the Fréchet gradient of the Tikhonov functional is derived through the weak solutions of two appropriate adjoint problems.
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Partial Differential Equations,
AMC, Rhode Island, 2002.
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Methods for Inverse Heat Conduction Problems,
Peter Lang, Frankfurt/Main, 1998.
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An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation,
J. Inverse Ill-Posed Probl. 14 (2006), 435–463.
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