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:10.1186/1687-2770-2013-43Published: 1 March 2013
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.