Research
Semigroup approach for identification of the unknown diffusion coefficient in a linear parabolic equation with mixed output data
1 Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No. 156, Balcova, Izmir, 35330, Turkey
2 Department of Mathematics, Kocaeli University, Umuttepe, Izmit, Kocaeli, 41380, Turkey
Boundary Value Problems 2013, 2013:43 doi:10.1186/1687-2770-2013-43
Published: 1 March 2013Abstract
This article presents a semigroup approach for the mathematical analysis of the inverse
coefficient problems of identifying the unknown coefficient 
in the linear parabolic equation 
with mixed boundary conditions 
, 
. The aim of this paper is to investigate the distinguishability of the input-output
mappings 
, 
via semigroup theory. In this paper, we show that if the null space of the semigroup
consists of only zero function, then the input-output mappings 
and 
have the distinguishability property. It is shown that the types of the boundary
conditions and the region on which the problem is defined have a significant impact
on the distinguishability property of these mappings. Moreover, in the light of measured output data (boundary observations) 
or/and 
, the values 
and 
of the unknown diffusion coefficient at 
and 
, respectively, can be determined explicitly. In addition to these, the values 
and 
of the unknown coefficient at and , respectively, are also determined via the input data. Furthermore, it is shown that
measured output data
and 
can be determined analytically by an integral representation. Hence the input-output
mappings 
, 
are given explicitly in terms of the semigroup.
