Open Access

Semigroup approach for identification of the unknown diffusion coefficient in a linear parabolic equation with mixed output data

Boundary Value Problems20132013:43

DOI: 10.1186/1687-2770-2013-43

Received: 3 January 2013

Accepted: 14 February 2013

Published: 1 March 2013

Abstract

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif in the linear parabolic equation u t ( x , t ) = ( k ( x ) u x ( x , t ) ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq2_HTML.gif with mixed boundary conditions k ( 0 ) u x ( 0 , t ) = ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq3_HTML.gif, u ( 1 , t ) = ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq4_HTML.gif. The aim of this paper is to investigate the distinguishability of the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq5_HTML.gif, Ψ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq6_HTML.gif via semigroup theory. In this paper, we show that if the null space of the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif consists of only zero function, then the input-output mappings Φ [ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq8_HTML.gif and Ψ [ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq9_HTML.gif 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) f ( t ) : = u ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq10_HTML.gif or/and h ( t ) : = k ( 1 ) u x ( 1 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq11_HTML.gif, the values k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq12_HTML.gif and k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq13_HTML.gif of the unknown diffusion coefficient k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif and x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif, respectively, can be determined explicitly. In addition to these, the values k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq16_HTML.gif and k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq17_HTML.gif of the unknown coefficient k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif and x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif, respectively, are also determined via the input data. Furthermore, it is shown that measured output data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif and h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif can be determined analytically by an integral representation. Hence the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq20_HTML.gif, Ψ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq21_HTML.gif are given explicitly in terms of the semigroup.

1 Introduction

Consider the following initial boundary value problem:
{ u t ( x , t ) = ( k ( x ) u x ( x , t ) ) x , ( x , t ) Ω T , u ( x , 0 ) = g ( x ) , 0 < x < 1 , k ( 0 ) u x ( 0 , t ) = ψ 0 , u ( 1 , t ) = ψ 1 , 0 < t < T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ1_HTML.gif
(1)

where Ω T = { ( x , t ) R 2 : 0 < x < 1 , 0 < t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq22_HTML.gif. The left flux ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq23_HTML.gif and the right boundary condition ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq24_HTML.gif are assumed to be constants. The functions c 1 > k ( x ) c 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq25_HTML.gif and g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq26_HTML.gif satisfy the following conditions:

(C1) k ( x ) H 1 , 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq27_HTML.gif;

(C2) g ( x ) H 2 , 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq28_HTML.gif, g ( 0 ) = ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq29_HTML.gif, g ( 1 ) = ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq30_HTML.gif.

Under these conditions, the initial boundary value problem (1) has the unique solution u ( x , t ) H 2 , 2 [ 0 , 1 ] H 1 , 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq31_HTML.gif [14].

Consider the inverse problem of determining the unknown coefficient k = k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq32_HTML.gif [59] from the following observations at the boundaries x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif and x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif:
u ( 0 , t ) = f ( t ) , k ( 1 ) u x ( 1 , t ) = h ( t ) , t ( 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ2_HTML.gif
(2)

Here u = u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq33_HTML.gif is the solution of the parabolic problem (1). The functions f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif, h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif are assumed to be noisy free measured output data. In this context, the parabolic problem (1) will be referred to as a direct (forward) problem with the inputs g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq26_HTML.gif and k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif. It is assumed that the functions f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif and h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif belong to H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq34_HTML.gif and satisfy the consistency conditions f ( 0 ) = g ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq35_HTML.gif, h ( 0 ) = k ( 1 ) g ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq36_HTML.gif.

We denote by K : = { k ( x ) H 1 , 2 [ 0 , 1 ] : c 1 > k ( x ) c 0 > 0 , x [ 0 , 1 ] } H 1 , 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq37_HTML.gif, the set of admissible coefficients k = k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq32_HTML.gif and introduce the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq38_HTML.gif, Ψ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq21_HTML.gif, where
Φ [ k ] = u ( x , t ; k ) | x = 0 , Ψ [ k ] = k ( x ) u x ( x , t ; k ) | x = 1 , k K , f ( t ) , h ( t ) H 1 , 2 [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ3_HTML.gif
(3)
Then the inverse problem [10] with the measured data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif and h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif can be formulated as the following operator equations:
Φ [ k ] = f , Ψ [ k ] = h , k K , f , h H 1 , 2 [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ4_HTML.gif
(4)

We denote by K : = { k ( x ) H 1 , 2 [ 0 , 1 ] : c 1 > k ( x ) c 0 > 0 , x [ 0 , 1 ] } H 1 , 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq39_HTML.gif, the set of admissible coefficients k = k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq32_HTML.gif. The monotonicity, continuity and hence invertibility of the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq5_HTML.gif and Ψ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq40_HTML.gif are given in [3, 4].

The aim of this paper is to study a distinguishability of the unknown coefficient via the above input-output mappings. We say that the mapping Φ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq5_HTML.gif (or Ψ [ ] : K H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq41_HTML.gif) has the distinguishability property if Φ [ k 1 ] Φ [ k 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq42_HTML.gif ( Ψ [ k 1 ] Ψ [ k 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq43_HTML.gif) implies k 1 ( x ) k 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq44_HTML.gif. This, in particular, means injectivity of the inverse mappings Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq45_HTML.gif and Ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq46_HTML.gif.

The purpose of this paper is to study the distinguishability of the unknown coefficient via the above input-output mappings. The results presented here are the first ones, to the knowledge of authors, from the point of view of semigroup approach [11] to inverse problems. This approach sheds more light on the identifiability of the unknown coefficient [12] and shows how much information can be extracted from the measured output data, in particular in the case of constant flux and boundary data [1215].

The paper is organized as follows. In Section 2, the analysis of the semigroup approach is given for the inverse problem with the measured data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif. A similar analysis is applied to the inverse problem with the single measured output data h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif given at the point x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif in Section 3. The inverse problem with two Neumann measured data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif and h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif is discussed in Section 4. Finally, some concluding remarks are given in Section 5.

2 Analysis of the inverse problem with measured output data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif

Consider now the inverse problem with one measured output data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif. In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, let us first arrange the parabolic equation as follows:
u t ( x , t ) ( k ( 0 ) u x ( x , t ) ) x = ( ( k ( x ) k ( 0 ) ) u x ( x , t ) ) x , ( x , t ) Ω T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equa_HTML.gif
Then the initial boundary value problem (1) can be rewritten in the following form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ5_HTML.gif
(5)
Here we assume that k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq12_HTML.gif was known. Later we will determine the value k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq12_HTML.gif. In order to formulate the solution of the parabolic problem (5) in terms of a semigroup, we need to define the following function:
v ( x , t ) = u ( x , t ) ψ 0 k ( 0 ) x + ψ 0 ψ 1 , x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ6_HTML.gif
(6)
which satisfies the following parabolic problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ7_HTML.gif
(7)

Here A [ v ( x , t ) ] : = k ( 0 ) d 2 v ( x , t ) / d x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq47_HTML.gif is a second-order differential operator, its domain is D A = { u H 2 , 2 ( 0 , 1 ) H 1 , 2 [ 0 , 1 ] : u x ( 0 ) = u ( 1 ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq48_HTML.gif. Since the initial value function g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq26_HTML.gif belongs to C 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq49_HTML.gif, it is obvious that g ( x ) D A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq50_HTML.gif.

Denote by T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif the semigroup of linear operators generated by the operator −A [5, 6]. Note that we can easily find the eigenvalues and eigenfunctions of the differential operator A. Furthermore, the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif can be easily constructed by using the eigenvalues and eigenfunctions of a differential operator A. For this reason, we first consider the following eigenvalue problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equb_HTML.gif
This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are λ n = k ( 0 ) ( 2 n 1 ) 2 π 2 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq51_HTML.gif for all n = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq52_HTML.gif and the corresponding eigenfunctions are ϕ n ( x ) = 2 cos ( ( 2 n 1 ) x π / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq53_HTML.gif. In this case, the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif can be represented in the following way:
T ( t ) U ( x , s ) = n = 0 ϕ n ( x ) , U ( x , s ) e λ n t ϕ n ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equc_HTML.gif
where ϕ n ( x ) , U ( x , s ) = 0 1 ϕ n ( x ) U ( x , s ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq54_HTML.gif. The null space of the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif of the linear operators can be defined as follows:
N ( T ) = { U ( x , s ) : ϕ n ( x ) , U ( x , s ) = 0 ,  for all  n = 1 , 2 , 3 , } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equd_HTML.gif

From the definition of the semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif, we can say that the null space of it is an empty set, i.e., N ( T ) = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq55_HTML.gif. This result is very important for the uniqueness of the unknown coefficient k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif.

The unique solution of the initial value problem (7) in terms of a semigroup T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif can be represented in the following form:
v ( x , t ) = T ( t ) v ( x , 0 ) + 0 t T ( t s ) ( ( k ( x ) k ( 0 ) ) ( v x ( x , t ) + ψ 0 k ( 0 ) ) ) x d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Eque_HTML.gif
Hence, by using identity (6), the solution u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq56_HTML.gif of the parabolic problem (5) in terms of a semigroup can be written in the following form:
u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 + T ( t ) ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 ψ 1 ) + 0 t T ( t s ) ( ( k ( x ) k ( 0 ) ) u x ( x , s ) ) x d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ8_HTML.gif
(8)
In order to arrange the above solution representation, let us define the following:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ10_HTML.gif
(10)
Then we can rewrite the solution representation in terms of ζ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq57_HTML.gif and ξ ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq58_HTML.gif in the following form:
u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 + T ( t ) ζ ( x ) + 0 t T ( t s ) ξ ( x , s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equf_HTML.gif
Substituting x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif into this solution representation yields
u ( 0 , t ) = ψ 1 ψ 0 + T ( t ) ζ ( 0 ) + 0 t T ( t s ) ξ ( 0 , s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equg_HTML.gif
Taking into account the overmeasured data u ( 0 , t ) = f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq59_HTML.gif, we get
f ( t ) = ( ψ 1 ψ 0 + T ( t ) ζ ( 0 ) + 0 t T ( t s ) ξ ( 0 , s ) d s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ11_HTML.gif
(11)

which implies that f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif can be determined analytically.

Differentiating both sides of the above identity with respect to x and using semigroup properties at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif yield
u x ( 0 , t ) = ψ 0 k ( 0 ) + z ( 0 , t ) + 0 t w ( 0 , t s , s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equh_HTML.gif
Using the boundary condition k ( 0 ) u x ( 0 , t ) = ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq3_HTML.gif, we can write k ( 0 ) = ψ 0 / u x ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq60_HTML.gif for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq61_HTML.gif which can be rewritten in terms of a semigroup in the following form:
k ( 0 ) = ψ 0 / ( ψ 0 + ψ 1 + z ( 0 , t ) + 0 t w ( 0 , t s , s ) d s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equi_HTML.gif
Taking limit as t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq62_HTML.gif in the above identity, we obtain the following explicit formula for the value k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq12_HTML.gif of the unknown coefficient k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif:
k ( 0 ) = ψ 0 / ( ψ 0 + ψ 1 + z ( 0 , 0 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equj_HTML.gif
The right-hand side of identity (11) defines explicitly the semigroup representation of the input-output mapping Φ [ k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq63_HTML.gif on the set of admissible unknown diffusion coefficients K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq64_HTML.gif:
Φ [ k ] ( x ) : = ψ 1 ψ 0 + T ( t ) ζ ( 0 ) + 0 t T ( t s ) ξ ( 0 , s ) d s , t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ12_HTML.gif
(12)
Let us differentiate now both sides of identity (8) with respect to t:
u t ( x , t ) = T ( t ) A ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 ψ 1 ) + ( ( k ( x ) k ( 0 ) ) u x ( x , t ) ) x + 0 t A T ( t s ) ( ( k ( x ) k ( 0 ) ) u x ( x , s ) ) x d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equk_HTML.gif
Using the semigroup property 0 t A T ( s ) u ( x , s ) d s = T ( t ) u ( x , t ) T ( 0 ) u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq65_HTML.gif, we obtain
u t ( x , t ) = k ( 0 ) T ( t ) g ( x ) 2 T ( 0 ) ( ( k ( x ) k ( 0 ) ) u x ( x , t ) ) x + T ( t ) ( ( k ( x ) k ( 0 ) ) u x ( x , 0 ) ) x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equl_HTML.gif
Taking x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif in the above identity, we get
u t ( 0 , t ) = k ( 0 ) T ( t ) g ( 0 ) T ( 0 ) k ( 0 ) u x ( 0 , 0 ) + T ( t ) ( k ( 0 ) u x ( 0 , 0 ) ) T ( 0 ) ( k ( 0 ) u x ( 0 , t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equm_HTML.gif
Since u ( 0 , t ) = f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq66_HTML.gif, we have u t ( 0 , t ) = f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq67_HTML.gif. Taking into account this and substituting t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq68_HTML.gif yield
f ( 0 ) = k ( 0 ) g ( 0 ) k ( 0 ) g ( 0 ) k ( 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equn_HTML.gif
Solving this equation for k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq16_HTML.gif and substituting u x ( 0 , 0 ) = g ( 0 ) / k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq69_HTML.gif, we obtain the following explicit formula for the value k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq16_HTML.gif of the first derivative k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq70_HTML.gif of the unknown coefficient at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif:
k ( 0 ) = k 2 ( 0 ) g ( 0 ) k ( 0 ) f ( 0 ) g ( 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ13_HTML.gif
(13)
Under the determined values k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq12_HTML.gif and k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq16_HTML.gif, the set of admissible coefficients can be defined as follows:
K 0 : = { k K : k ( 0 ) = ψ 0 ψ 0 + ψ 1 + z ( 0 , 0 ) , k ( 0 ) = k 2 ( 0 ) g ( 0 ) k ( 0 ) f ( 0 ) g ( 0 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equo_HTML.gif

The following lemma implies the relationship between the diffusion coefficients k 1 ( x ) , k 2 ( x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq71_HTML.gif at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif and the corresponding outputs f j ( t ) : = u ( 0 , t ; k j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq72_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq73_HTML.gif.

Lemma 2.1 Let u 1 ( x , t ) = u ( x , t ; k 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq74_HTML.gif and u 2 ( x , t ) = u ( x , t ; k 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq75_HTML.gif be solutions of the direct problem (5) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq76_HTML.gif. Suppose that f j ( t ) = u ( 0 , t ; k j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq77_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq73_HTML.gif, are the corresponding outputs and denote by Δ f ( t ) = f 1 ( t ) f 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq78_HTML.gif, Δ ξ ( x , t ) = ξ 1 ( x , t ) ξ 2 ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq79_HTML.gif. If the condition
k 1 ( 0 ) = k 2 ( 0 ) : = k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equp_HTML.gif
holds, then the outputs f j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq80_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq73_HTML.gif, satisfy the following integral identity:
Δ f ( τ ) = 0 τ T ( τ s ) Δ ξ ( 0 , s ) d s d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ14_HTML.gif
(14)

for each τ ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq81_HTML.gif.

Proof The solutions of the direct problem (5) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq82_HTML.gif can be written at x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq14_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equq_HTML.gif

respectively, by using representation (11). From identity (9) it is obvious that ζ 1 ( 0 , τ ) = ζ 2 ( 0 , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq83_HTML.gif for each τ ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq84_HTML.gif. Hence the difference of these formulas implies the desired result. □

This lemma with identity (14) implies the following.

Corollary 2.1 Let conditions of Lemma 2.1 hold. Then f 1 ( t ) = f 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq85_HTML.gif, t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq86_HTML.gif, if and only if
ϕ n ( x ) , Δ ξ ( 0 , s ) = ϕ n ( x ) , ξ 1 ( x , t ) ξ 2 ( x , t ) = 0 , t ( 0 , T ] , n = 0 , 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equr_HTML.gif

Since the Strum-Liouville problem generates a complete orthogonal family of eigenfunctions, the null space of a semigroup contains only zero function, i.e., N ( T ) = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq87_HTML.gif. Thus Corollary 2.1 states that f 1 f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq88_HTML.gif if and only if ξ 1 ( x , t ) ξ 2 ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq89_HTML.gif for all ( x , t ) Ω T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq90_HTML.gif. The definition of ξ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq91_HTML.gif implies that k 1 ( x ) = k 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq92_HTML.gif for all x [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq93_HTML.gif.

The combination of the conclusions of Lemma 2.1 and Corollary 2.1 can be given by the following theorem which states the distinguishability of the input-output mapping Φ [ ] : K 0 H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq94_HTML.gif.

Theorem 2.1 Let conditions (C1) and (C2) hold. Assume that Φ [ ] : K 0 H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq94_HTML.gif is the input-output mapping defined by (3) and corresponding to the measured output f ( t ) : = u ( 0 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq10_HTML.gif. Then the mapping Φ [ k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq63_HTML.gif has the distinguishability property in the class of admissible coefficients K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq95_HTML.gif, i.e.,
Φ [ k 1 ] Φ [ k 2 ] k 1 , k 2 K 0 , k 1 ( x ) k 2 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equs_HTML.gif

3 Analysis of the inverse problem with measured output data h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif

Consider now the inverse problem with one measured output data h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif at x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif. As in the previous section, let us arrange the parabolic equation as follows:
u t ( x , t ) ( k ( 1 ) u x ( x , t ) ) x = ( ( k ( x ) k ( 1 ) ) u x ( x , t ) ) x , ( x , t ) Ω T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equt_HTML.gif
Then the initial boundary value problem (1) can be rewritten in the following form:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ15_HTML.gif
(15)
In order to formulate the solution of the above parabolic problem in terms of a semigroup, let us use the same variable v ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq96_HTML.gif in identity (6), which satisfies the following parabolic problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ16_HTML.gif
(16)

Here B [ v ( x , t ) ] : = k ( 1 ) d 2 v ( x , t ) / d x 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq97_HTML.gif is a second-order differential operator, its domain is D B = { u H 2 , 2 ( 0 , 1 ) H 1 , 2 [ 0 , 1 ] : u x ( 0 ) = u ( 1 ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq98_HTML.gif. Since the initial value function g ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq26_HTML.gif belongs to H 2 , 2 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq99_HTML.gif, it is obvious that g ( x ) D B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq100_HTML.gif.

Denote by S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif the semigroup of linear operators generated by the operator −A [5, 6]. As in the previous section, we can easily find the eigenvalues and eigenfunctions of the differential operator B. Furthermore, the semigroup S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif can be easily constructed by using the eigenvalues and eigenfunctions of the differential operator B. For this reason, we first consider the following eigenvalue problem:
B ϕ ( x ) = λ ϕ ( x ) , ϕ x ( 0 ) = 0 ; ϕ ( 1 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equu_HTML.gif
This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are λ n = k ( 1 ) ( 2 n 1 ) 2 π 2 / 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq102_HTML.gif for all n = 0 , 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq103_HTML.gif and the corresponding eigenfunctions become ϕ n ( x ) = 2 cos ( ( 2 n 1 ) π / 2 x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq104_HTML.gif. Hence the semigroup S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif can be represented in the following form:
S ( t ) U ( x , s ) = n = 0 ϕ n ( x ) , U ( x , s ) e λ n t ϕ n ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equv_HTML.gif
The null space of the semigroup S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif of the linear operators can be defined as follows:
N ( S ) = { U ( x , s ) : ϕ n ( x ) , U ( x , s ) = 0 ,  for all  n = 1 , 2 , 3 , } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equw_HTML.gif

Since the Sturm-Liouville problem generates a complete orthogonal family of eigenfunctions, we can say that the null space of the semigroup S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif is an empty set, i.e., N ( S ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq105_HTML.gif. This result is very important for the uniqueness of the unknown coefficient k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq1_HTML.gif.

The unique solution of the initial value problem (16) in terms of a semigroup S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif can be represented in the following form:
v ( x , t ) = S ( t ) v ( x , 0 ) + 0 t S ( t s ) ( ( k ( x ) k ( 1 ) ) ( v x ( x , s ) + ψ 0 k ( 0 ) ) ) x d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equx_HTML.gif
Hence, by using identity (6), the solution u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq56_HTML.gif of the parabolic problem (15) in terms of a semigroup can be written in the following form:
u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 k ( 0 ) + S ( t ) ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 k ( 0 ) ψ 1 ) + 0 t S ( t s ) ( ( k ( x ) k ( 1 ) ) u x ( x , s ) ) x d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ17_HTML.gif
(17)
Defining the following:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ19_HTML.gif
(19)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ20_HTML.gif
(20)
The solution representation of the parabolic problem (17) can be rewritten in the following form:
u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 k ( 0 ) + S ( t ) ζ ( x ) + 0 t S ( t s ) χ ( x , s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equy_HTML.gif
Differentiating both sides of the above identity with respect to x and substituting x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif yield
u x ( 1 , t ) = ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equz_HTML.gif
Taking into account the overmeasured data k ( 1 ) u x ( 1 , t ) = h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq106_HTML.gif, we get
h ( t ) = k ( 1 ) ( ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ21_HTML.gif
(21)
Now we can determine the value k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq13_HTML.gif. From the overmeasured data k ( 1 ) u x ( 1 , t ) = h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq106_HTML.gif, the identity k ( 1 ) = h ( t ) / u x ( 1 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq107_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq108_HTML.gif can be rewritten in terms of a semigroup in the following form:
k ( 1 ) = h ( t ) ( ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equaa_HTML.gif
Taking limit as t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq62_HTML.gif in the above identity yields
k ( 1 ) = h ( 0 ) / ( ψ 0 k ( 0 ) + z 1 ( 1 , 0 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equab_HTML.gif
The right-hand side of the above identity defines the semigroup representation of the input-output mapping Ψ [ k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq109_HTML.gif on the set of admissible unknown diffusion coefficient K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq64_HTML.gif:
Ψ [ k ] ( x ) : = k ( 1 ) ( ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s ) , t [ 0 , T ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ22_HTML.gif
(22)
Differentiating both sides of identity (17) with respect to t, we get
u t ( x , t ) = S ( t ) B ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 k ( 0 ) ψ 1 ) S ( 0 ) ( ( k ( x ) k ( 1 ) ) u x ( x , t ) ) x + 0 t B S ( t s ) ( ( k ( x ) k ( 1 ) ) u x ( x , s ) ) x d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equac_HTML.gif
Using semigroup properties, we obtain
u t ( x , t ) = S ( t ) k ( 1 ) g ( x ) S ( 0 ) ( k ( x ) u x ( x , t ) + ( k ( x ) k ( 1 ) ) u x x ( x , t ) ) + S ( t ) ( ( k ( x ) k ( 1 ) ) u x ( x , 0 ) ) x S ( 0 ) ( ( k ( x ) k ( 1 ) ) u x ( x , t ) ) x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equad_HTML.gif
Taking x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif in the above identity, we get
u t ( 1 , t ) = S ( t ) k ( 1 ) g ( 1 ) 2 S ( 0 ) k ( 1 ) u x ( 1 , t ) + S ( t ) k ( 1 ) u x ( 1 , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equae_HTML.gif
Since u ( 1 , t ) = ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq110_HTML.gif, we have u t ( 1 , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq111_HTML.gif. Taking into account this and substituting t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq68_HTML.gif, we get
0 = k ( 1 ) g ( 1 ) k ( 1 ) u x ( 1 , 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equaf_HTML.gif
Solving this equation for k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq17_HTML.gif and substituting u x ( 1 , 0 ) = h ( 0 ) / k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq112_HTML.gif, we reach the following result:
k ( 1 ) = k 2 ( 1 ) g ( 1 ) h ( 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ23_HTML.gif
(23)
Then we can define the admissible set of diffusion coefficients as follows:
K 1 : = { k K : k ( 1 ) = h ( 0 ) ( ψ 0 k ( 0 ) + z 1 ( 1 , 0 ) ) , k ( 1 ) = k 2 ( 1 ) g ( 1 ) h ( 0 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equag_HTML.gif

The following lemma implies the relation between the coefficients k 1 ( x ) , k 2 ( x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq82_HTML.gif at x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif and the corresponding outputs h j ( t ) : = k j ( 1 ) u x ( 1 , t ; k j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq113_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq73_HTML.gif.

Lemma 3.1 Let u 1 ( x , t ) = u ( x , t ; k 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq74_HTML.gif and u 2 ( x , t ) = u ( x , t ; k 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq75_HTML.gif be solutions of the direct problem (16) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq76_HTML.gif. Suppose that h j ( t ) = u ( 1 , t ; k j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq114_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq73_HTML.gif, are the corresponding outputs and denote by Δ h ( t ) = h 1 ( t ) h 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq115_HTML.gif, Δ w 1 ( x , t , s ) = w 1 1 ( x , t , s ) w 1 2 ( x , t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq116_HTML.gif. If the condition
k 1 ( 1 ) = k 2 ( 1 ) : = k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equah_HTML.gif
holds, then the outputs h j ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq117_HTML.gif, j = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq73_HTML.gif, satisfy the following integral identity:
Δ h ( τ ) = k ( 1 ) 0 τ Δ w 1 ( 1 , τ s , s ) d s d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equ24_HTML.gif
(24)

for each τ ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq81_HTML.gif.

Proof The solutions of the direct problem (15) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq82_HTML.gif can be written at x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq15_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equai_HTML.gif

respectively, by using formula (20). From definition (18), it is obvious that z 1 1 ( 1 , τ ) = z 1 2 ( 1 , τ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq118_HTML.gif for each τ ( 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq81_HTML.gif. Hence the difference of these formulas implies the desired result. □

This lemma with identity (23) implies the following conclusion.

Corollary 3.1 Let the conditions of Lemma 3.1 hold. Then h 1 ( t ) = h 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq119_HTML.gif, t [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq86_HTML.gif, if and only if
ϕ n ( x ) , χ 1 ( x , t ) χ 2 ( x , t ) = 0 , t ( 0 , T ] , n = 0 , 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equaj_HTML.gif

hold.

Since the null space of it consists of only zero function, i.e., N ( S ) = { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq120_HTML.gif, Corollary 3.1 states that h 1 h 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq121_HTML.gif if and only if χ 1 ( x , t ) χ 2 ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq122_HTML.gif for all ( x , t ) Ω T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq90_HTML.gif. The definition of χ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq123_HTML.gif implies that k 1 ( x ) = k 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq92_HTML.gif for all x ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq124_HTML.gif.

Theorem 3.1 Let conditions (C1) and (C2) hold. Assume that Ψ [ ] : K 1 C 1 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq125_HTML.gif is the input-output mapping defined by (3) and corresponding to the measured output h ( t ) : = k ( 1 ) u x ( 1 , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq11_HTML.gif. Then the mapping Ψ [ k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq109_HTML.gif has the distinguishability property in the class of admissible coefficients K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq126_HTML.gif, i.e.,
Ψ [ k 1 ] Ψ [ k 2 ] k 1 , k 2 K 1 , k 1 ( x ) k 2 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equak_HTML.gif

4 The inverse problem with mixed output data

Consider now the inverse problem (1)-(2) with two measured output data f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq18_HTML.gif and h ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq19_HTML.gif. As shown before, having these two data, the values k ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq12_HTML.gif as well as k ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq13_HTML.gif can be defined by the above explicit formulas. Based on this result, let us define now the set of admissible coefficients K 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq127_HTML.gif as an intersection:
K 2 : = K 0 K 1 = { k K : k ( 0 ) = ψ 0 ψ 0 + ψ 1 + z ( 0 , 0 ) , k ( 1 ) = h ( 0 ) ψ 0 / k ( 0 ) + z 1 ( 1 , 0 ) , k ( 0 ) = k 2 ( 0 ) g ( 0 ) k ( 0 ) f ( 0 ) g ( 0 ) , k ( 1 ) = k 2 ( 1 ) g ( 1 ) h ( 0 ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equal_HTML.gif

On this set, both input-output mappings Φ [ k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq63_HTML.gif and Ψ [ k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq109_HTML.gif have distinguishability property.

Corollary 4.1 The input-output mappings Φ [ ] : K 2 H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq128_HTML.gif and Ψ [ ] : K 2 H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq129_HTML.gif distinguish any two functions k 1 ( x ) k 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq130_HTML.gif from the set K 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq127_HTML.gif, i.e.,
k 1 ( x ) , k 2 ( x ) K 2 , k 1 ( x ) k 2 ( x ) , Φ [ k 1 ] Φ [ k 2 ] , Ψ [ k 1 ] Ψ [ k 2 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_Equam_HTML.gif

5 Conclusion

The aim of this study was to analyze distinguishability properties of the input-output mappings Φ [ ] : K 2 H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq128_HTML.gif and Ψ [ ] : K 2 H 1 , 2 [ 0 , T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq129_HTML.gif which are naturally determined by the measured output data. In this paper we show that if the null spaces of the semigroups T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif and S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif include only zero function then the corresponding input-output mappings Φ [ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq8_HTML.gif and Ψ [ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq9_HTML.gif have distinguishability property.

This study shows that boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability of the input-output mappings Φ [ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq8_HTML.gif and Ψ [ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq9_HTML.gif since these key elements determine the structure of the semigroups T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq7_HTML.gif and S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-43/MediaObjects/13661_2013_Article_301_IEq101_HTML.gif of linear operators and their null spaces.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The research was supported in part by the Scientific and Technical Research Council (TUBITAK) and Izmir University of Economics.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics
(2)
Department of Mathematics, Kocaeli University, Umuttepe

References

  1. DuChateau P: Monotonicity and invertibility of coefficient-to-data mappings for parabolic inverse problems. SIAM J. Math. Anal. 1995, 26: 1473-1487. 10.1137/S0036141093259257MathSciNetView Article
  2. Isakov V: On uniqueness in inverse problems for quasilinear parabolic equations. Arch. Ration. Mech. Anal. 1993, 124: 1-13. 10.1007/BF00392201MathSciNetView Article
  3. Pilant MS, Rundell W: A uniqueness theorem for conductivity from overspecified boundary data. J. Math. Anal. Appl. 1988, 136: 20-28. 10.1016/0022-247X(88)90112-6MathSciNetView Article
  4. Renardy M, Rogers R: An Introduction to Partial Differential Equations. Springer, New York; 2004.
  5. Cannon JR: The One-Dimensional Heat Equation. Addison-Wesley, Reading; 1984.View Article
  6. DuChateau P, Thelwell R, Butters G: Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. Inverse Probl. 2004, 20: 601-625. 10.1088/0266-5611/20/2/019MathSciNetView Article
  7. Showalter R: Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations. Am. Math. Soc., Providence; 1997.
  8. Hasanov A, Demir A, Erdem A: Monotonicity of input-output mappings in inverse coefficient and source problem for parabolic equations. J. Math. Anal. Appl. 2007, 335: 1434-1451. 10.1016/j.jmaa.2007.01.097MathSciNetView Article
  9. Hasanov A, DuChateau P, Pektas B: An adjoint approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equations. J. Inverse Ill-Posed Probl. 2006, 14: 435-463. 10.1515/156939406778247615MathSciNetView Article
  10. DuChateau P, Gottlieb J: Introduction to Inverse Problems in Partial Differential Equations for Engineers, Physicists and Mathematicians. Kluwer Academic, Dordrecht; 1996.
  11. Ashyralyev A, San ME: An approximation of semigroup method for stochastic parabolic equations. Abstr. Appl. Anal. 2012., 2012: Article ID 684248. doi:10.1155/2012/684248
  12. Demir A, Ozbilge E: Semigroup approach for identification of the unknown diffusion coefficient in a quasi-linear parabolic equation. Math. Methods Appl. Sci. 2007, 30: 1283-1294. 10.1002/mma.837MathSciNetView Article
  13. Ozbilge E: Identification of the unknown diffusion coefficient in a quasi-linear parabolic equation by semigroup approach with mixed boundary conditions. Math. Methods Appl. Sci. 2008, 31: 1333-1344. 10.1002/mma.974MathSciNetView Article
  14. Demir A, Ozbilge E: Analysis of a semigroup approach in the inverse problem of identifying an unknown coefficient. Math. Methods Appl. Sci. 2008, 31: 1635-1645. 10.1002/mma.989MathSciNetView Article
  15. Demir A, Hasanov A: Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach. J. Math. Anal. Appl. 2008, 340: 5-15. 10.1016/j.jmaa.2007.08.004MathSciNetView Article

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