Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

Abstract

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k ( x ) in the linear parabolic equation u t ( x , t ) = ( k ( x ) u x ( x , t ) ) x , with Dirichlet boundary conditions u ( 0 , t ) = ψ 0 , u ( 1 , t ) = ψ 1 . Main goal of this study is to investigate the distinguishability of the input–output mappings Φ [ ⋅ ] : K → C 1 [ 0 , T ] , Ψ [ ⋅ ] : K → C 1 [ 0 , T ] via semigroup theory. In this paper, we show that if the null space of the semigroup T ( t ) consists of only zero function, then the input–output mappings Φ [ ⋅ ] and Ψ [ ⋅ ] have the distinguishability property. Moreover, the values k ( 0 ) and k ( 1 ) of the unknown diffusion coefficient k ( x ) at x = 0 and x = 1 , respectively, can be determined explicitly by making use of measured output data (boundary observations) f ( t ) : = k ( 0 ) u x ( 0 , t ) or/and h ( t ) : = k ( 1 ) u x ( 1 , t ) . In addition to these, the values k ′ ( 0 ) and k ′ ( 1 ) of the unknown coefficient k ( x ) at x = 0 and x = 1 , respectively, are also determined via the input data. Furthermore, it is shown that measured output data f ( t ) and h ( t ) can be determined analytically, by an integral representation. Hence the input–output mappings Φ [ ⋅ ] : K → C 1 [ 0 , T ] , Ψ [ ⋅ ] : K → C 1 [ 0 , T ] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k ( x ) at the end points x = 0 and x = 1 .