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slides
Determine the spacial term of a two-dimensional heat source
Authors
Alain Pham Ngoc Dinh
Phan Thanh Nam
Dang Duc Trong
Publication date
11 July 2008
Publisher
View
on
arXiv
Abstract
We consider the problem of determining a pair of functions
(
u
,
f
)
(u,f)
(
u
,
f
)
satisfying the heat equation
u
t
−
Δ
u
=
φ
(
t
)
f
(
x
,
y
)
u_t -\Delta u =\varphi(t)f (x,y)
u
t
​
−
Δ
u
=
φ
(
t
)
f
(
x
,
y
)
, where
(
x
,
y
)
∈
Ω
=
(
0
,
1
)
×
(
0
,
1
)
(x,y)\in \Omega=(0,1)\times (0,1)
(
x
,
y
)
∈
Ω
=
(
0
,
1
)
×
(
0
,
1
)
and the function
φ
\varphi
φ
is given. The problem is ill-posed. Under a slight condition on
φ
\varphi
φ
, we show that the solution is determined uniquely from some boundary data and the initial temperature. Using the interpolation method and the truncated Fourier series, we construct a regularized solution of the source term
f
f
f
from non-smooth data. The error estimate and numerical experiments are given.Comment: 18 page
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Last time updated on 14/04/2021