Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method

Boundary Value Problems20132013:2

DOI: 10.1186/1687-2770-2013-2

Received: 23 November 2012

Accepted: 17 December 2012

Published: 4 January 2013

Abstract

In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition u ( 1 , t ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq1_HTML.gif in the quasi-linear parabolic equation u t ( x , t ) = ( k ( u ( x , t ) ) u x ( x , t ) ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq2_HTML.gif, with Dirichlet boundary conditions u ( 0 , t ) = ψ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq3_HTML.gif, u ( 1 , t ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq1_HTML.gif, by making use of the over measured data u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif and u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif separately. The purpose of this study is to identify the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif by using the over measured data u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif and u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif. First, by using over measured data as a boundary condition, we define the problem on Ω T 0 = { ( x , t ) R 2 : 0 < x < x 0 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq8_HTML.gif, then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif, we reach the result. The main point here is the unique extensions of the solutions on [ 0 , x 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq10_HTML.gif to the closed interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif.

1 Introduction

Consider the following initial boundary value problem for quasilinear diffusion equation:
{ u t ( x , t ) = ( k ( u ( x , t ) ) u x ( x , t ) ) x , ( x , t ) Ω T , u ( x , 0 ) = g ( x ) , 0 < x < 1 , u ( 0 , t ) = ψ 0 , u ( 1 , t ) = f ( t ) , 0 < t < T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ1_HTML.gif
(1)

where Ω T = { ( x , t ) R 2 : 0 < x < 1 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq11_HTML.gif. The left boundary value ψ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq12_HTML.gif is assumed to be constant. The functions c 1 > k ( u ( x , t ) ) c 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq13_HTML.gif and g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq14_HTML.gif satisfy the following conditions:

(C 1) | k u ( u 1 ) k u ( u 2 ) | < d | u 1 u 2 | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq15_HTML.gif;

(C 2) g ( x ) C 2 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq16_HTML.gif, g ( 0 ) = ψ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq17_HTML.gif, g ( 1 ) = f ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq18_HTML.gif.

The initial boundary value problem (1) has a unique solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq19_HTML.gif satisfying u ( x , t ) H 2 , 2 [ 0 , 1 ] H 1 , 2 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq20_HTML.gif [14].

In physics, many applications of this problem can be found. The simple model of flame propagation and the spread of biological populations, where u = u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq21_HTML.gif denotes the temperature and density respectively, are given by the equation in the problem (1). Especially k = k ( u ( x , t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq22_HTML.gif represents the density-dependent coefficient in the problems of the spread of biological populations [59].

We consider the inverse problems [10] of determining boundary u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif in the problem (1) from Dirichlet type of measured output data at the boundaries x = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq23_HTML.gif
u ( x 0 , t ) = ψ 1 , t ( 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ2_HTML.gif
(2)
and from Neumann type of measured output data at the boundaries x = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq23_HTML.gif
u x ( x 0 , t ) = ψ 2 , t ( 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ3_HTML.gif
(3)

Here u = u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq21_HTML.gif is the solution of the parabolic problem (1). In this context, the parabolic problem (1) will be referred to as a direct (forward) problem with the inputs g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq14_HTML.gif, k ( u ( x , t ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq24_HTML.gif and f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq25_HTML.gif. It is assumed that the functions u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif and u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif respectively satisfy the consistency conditions ψ 1 = g ( x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq26_HTML.gif and ψ 2 = g ( x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq27_HTML.gif.

The semigroup approach [11] for inverse problems for the identification of an unknown coefficient in a quasi-linear parabolic equation was studied by Demir and Ozbilge [12]. The study in this paper is based on the philosophy similar to that in [1215].

The paper is organized as follows. In Section 2, the analysis of the semigroup approach is given for the inverse problem with the single measured output data u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif given at x = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq23_HTML.gif. The similar analysis is applied to the inverse problem with the single measured output data u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif given at the point x = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq23_HTML.gif in Section 3. Some concluding remarks are given in Section 4.

2 Analysis of the inverse problem of the boundary condition by Dirichlet type of over measured data u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif

Consider now the inverse problem with one measured output data u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif at x = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq23_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 ( u ( 0 , 0 ) ) u x ( x , t ) ) x = ( [ k ( u ) k ( u ( 0 , 0 ) ) ] u x ( x , t ) ) x , ( x , t ) Ω T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equa_HTML.gif
Then the initial boundary value problem (1) can be rewritten in the following form:
u t ( x , t ) k ( u ( 0 , 0 ) ) u x x ( x , t ) = ( ( k ( u ) k ( u ( 0 , 0 ) ) ) u x ( x , t ) ) x , ( x , t ) Ω T , u ( x , 0 ) = g ( x ) , 0 < x < 1 , u ( 0 , t ) = ψ 0 , u ( 1 , t ) = f ( t ) , 0 < t < T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ4_HTML.gif
(4)
In order to determine the unknown boundary condition u ( 1 , t ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq1_HTML.gif, we need to determine the solution of the following parabolic problem:
u t ( x , t ) k ( u ( 0 , 0 ) ) u x x ( x , t ) = ( ( k ( u ) k ( u ( 0 , 0 ) ) ) u x ( x , t ) ) x , ( x , t ) Ω T 0 , u ( x , 0 ) = g ( x ) , 0 < x < x 0 , u ( 0 , t ) = ψ 0 , u ( x 0 , t ) = ψ 1 , 0 < t < T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ5_HTML.gif
(5)
where Ω T 0 = { ( x , t ) R 2 : 0 < x < x 0 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq28_HTML.gif. To formulate the solution of the above problem in terms of a semigroup, we need to define a new function
v ( x , t ) = u ( x , t ) + ( ψ 0 ψ 1 ) x x 0 ψ 0 , x [ 0 , x 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ6_HTML.gif
(6)
which satisfies the following parabolic problem:
v t ( x , t ) + A [ v ( x , t ) ] = ( ( k ( v ( x , t ) + ψ 0 ( ψ 0 ψ 1 ) x x 0 ) v t ( x , t ) + A [ v ( x , t ) ] = k ( u ( 0 , 0 ) ) ) ( v x ( x , t ) ( ψ 0 ψ 1 ) x 0 ) ) x , ( x , t ) Ω T , v ( x , 0 ) = g ( x ) + ( ψ 0 ψ 1 ) x x 0 ψ 0 , 0 < x < x 0 , v ( 0 , t ) = 0 , v ( x 0 , t ) = 0 , 0 < t < T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ7_HTML.gif
(7)

Here, A [ ] : = k ( u ( 0 , 0 ) ) d 2 [ ] d x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq29_HTML.gif is a second-order differential operator and its domain is D A = { v ( x ) H 0 2 , 2 ( 0 , 1 ) H 0 3 , 2 [ 0 , 1 ] : v ( 0 ) = 0 = v ( 1 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq30_HTML.gif, where H 0 2 , 2 ( 0 , 1 ) = C 0 2 ( 0 , 1 ) ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq31_HTML.gif and H 0 1 , 2 [ 0 , 1 ] = C 0 1 [ 0 , 1 ] ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq32_HTML.gif are Sobolev spaces. Obviously, by completion g ( x ) D A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq33_HTML.gif, since the initial value function g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq14_HTML.gif belongs to C 3 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq34_HTML.gif. Hence, D A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq35_HTML.gif is dense in H 0 2 , 2 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq36_HTML.gif, which is a necessary condition for being an infinitesimal generator.

In the following, despite doing the calculations in the smooth function space, by completion they are valid in the Sobolev space.

Let us denote the semigroup of linear operators by T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq37_HTML.gif generated by the operator A [8, 9]. We can easily find the eigenvalues and eigenfunctions of the differential operator A. Moreover, the semigroup T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq37_HTML.gif can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generator A. Hence, we first consider the following eigenvalue problem:
A ϕ ( x ) = λ ϕ ( x ) , ϕ ( 0 ) = 0 ; ϕ ( x 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ8_HTML.gif
(8)
We can easily determine that the eigenvalues are λ n = k ( u ( 0 , 0 ) ) n 2 π 2 x 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq38_HTML.gif for all n = 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq39_HTML.gif and the corresponding eigenfunctions are ϕ n ( x ) = sin ( n π x x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq40_HTML.gif. In this case, the semigroup T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq37_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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ9_HTML.gif
(9)
where ϕ n ( x ) , U ( x , s ) = 0 1 ϕ n ( x ) U ( x , s ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq41_HTML.gif. Under this representation, the null space of the semigroup T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq37_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 = 0 , 1 , 2 , 3 , } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equb_HTML.gif

From the definition of the semigroup T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq37_HTML.gif, we can say that the null space of it consists of only zero function, i.e., N ( T ) = { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq42_HTML.gif. This result is very important for the uniqueness of the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif.

The unique solution of the initial-boundary value problem (7) in terms of the semigroup T ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq37_HTML.gif can be represented in the following form:
v ( x , t ) = T ( t ) v ( x , 0 ) + 0 t T ( t s ) ( ( k ( v ( x , s ) + ψ 0 ( ψ 0 ψ 1 ) x x 0 ) k ( u ( 0 , 0 ) ) ) ( v x ( x , s ) ( ψ 0 ψ 1 ) x 0 ) ) x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equc_HTML.gif
Now, by using the identity (6) and taking the initial value u ( x , 0 ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq43_HTML.gif into account, the integral equation for the solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq19_HTML.gif of the parabolic problem (5) in terms of a semigroup can be written in the following form:
u ( x , t ) = ψ 0 ( ψ 0 ψ 1 ) x x 0 + T ( t ) ( g ( x ) + ( ψ 0 ψ 1 ) x x 0 ψ 0 ) + 0 t T ( t s ) ( ( k ( u ( x , s ) ) k ( u ( 0 , 0 ) ) ) u x ( x , s ) ) x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ10_HTML.gif
(10)
In order to arrange the above integral equation, let us define the following:
ζ ( x ) = ( g ( x ) + ( ψ 0 ψ 1 ) x x 0 ψ 0 ) , ξ ( x , t ) = ( ( k ( u ( x , t ) ) k ( u ( 0 , 0 ) ) ) u x ( x , t ) ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equd_HTML.gif
Then we can rewrite the integral equation in terms of ζ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq44_HTML.gif and ξ ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq45_HTML.gif in the following form:
u ( x , t ) = ψ 0 ( ψ 0 ψ 1 ) x x 0 + ( T ( t ) ζ ( ) ) ( x , t ) + 0 t ( T ( t s ) ξ ( , s ) ) ( x , t , s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ11_HTML.gif
(11)

This is the integral representation of a solution of the initial-boundary value problem (5) on Ω T 0 = { ( x , t ) R 2 : 0 < x < x 0 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq28_HTML.gif. It is obvious from the eigenfunctions ϕ n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq46_HTML.gif, the domain of eigenfunctions can be extended to the closed interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif. Moreover they are continuous on [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif. Under this extension, the uniqueness of the solutions of the initial-boundary value problems (4) and (5) imply that the integral representation (11) becomes the integral representation of a solution of the initial-boundary value problem (4) on Ω T = { ( x , t ) R 2 : 0 < x < 1 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq47_HTML.gif.

At this stage, it is obvious that the solution of the inverse problem can easily be obtained by substituting x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif into the integral representation (11) of the solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq19_HTML.gif,
u ( 1 , t ) = f ( t ) = ψ 0 ( ψ 0 ψ 1 ) x 0 + ( T ( t ) ζ ( ) ) ( 1 , t ) + 0 t ( T ( t s ) ξ ( , s ) ) ( 1 , t , s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ12_HTML.gif
(12)

which implies that f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq25_HTML.gif can be determined analytically.

The right-hand side of the identity (12) defines the semigroup representation of the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif.

3 Analysis of the inverse problem of the boundary condition by Neumann type of over measured data u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif

Consider now the inverse problem with one measured output data u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif at x = x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq23_HTML.gif. In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, we arrange the parabolic equation as follows:
u t ( x , t ) ( k ( u ( 0 , 0 ) ) u x ( x , t ) ) x = ( [ k ( u ) k ( u ( 0 , 0 ) ) ] u x ( x , t ) ) x , ( x , t ) Ω T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Eque_HTML.gif
Then the initial boundary value problem (1) can be rewritten in the following form:
u t ( x , t ) k ( u ( 0 , 0 ) ) u x x ( x , t ) = ( ( k ( u ) k ( u ( 0 , 0 ) ) ) u x ( x , t ) ) x , ( x , t ) Ω T , u ( x , 0 ) = g ( x ) , 0 < x < 1 , u ( 0 , t ) = ψ 0 , u ( 1 , t ) = f ( t ) , 0 < t < T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ13_HTML.gif
(13)
In order to determine the unknown boundary condition u ( 1 , t ) = f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq1_HTML.gif, we need to determine the solution of the following parabolic problem:
u t ( x , t ) k ( u ( 0 , 0 ) ) u x x ( x , t ) = ( ( k ( u ) k ( u ( 0 , 0 ) ) ) u x ( x , t ) ) x , ( x , t ) Ω T 0 , u ( x , 0 ) = g ( x ) , 0 < x < x 0 , u ( 0 , t ) = ψ 0 , u x ( x 0 , t ) = ψ 2 , 0 < t < T , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ14_HTML.gif
(14)

where Ω T 0 = { ( x , t ) R 2 : 0 < x < x 0 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq28_HTML.gif.

To formulate the solution of the above problem in terms of a semigroup, we need to define a new function
v ( x , t ) = u ( x , t ) ψ 2 x ψ 0 , x [ 0 , x 0 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ15_HTML.gif
(15)
which satisfies the following parabolic problem:
v t ( x , t ) + B [ v ( x , t ) ] = ( ( k ( v ( x , t ) + ψ 0 + ψ 2 x ) k ( u ( 0 , 0 ) ) ) ( v x ( x , t ) + ψ 2 ) ) x , ( x , t ) Ω T , v ( x , 0 ) = g ( x ) ψ 2 x ψ 0 , 0 < x < x 0 , v ( 0 , t ) = 0 , v x ( x 0 , t ) = 0 , 0 < t < T . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ16_HTML.gif
(16)

Here B [ ] : = k ( u ( 0 , 0 ) ) d 2 [ ] d x 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq48_HTML.gif is a second-order differential operator, its domain is D B = { v C 2 ( 0 , x 0 ) C 1 [ 0 , x 0 ] : v ( 0 ) = v x ( x 0 ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq49_HTML.gif. It is clear from the definition of D B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq50_HTML.gif that D B C 2 [ 0 , x 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq51_HTML.gif.

Let us denote the semigroup of linear operators by S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq52_HTML.gif generated by the operator B [8, 9]. We can easily find the eigenvalues and eigenfunctions of the differential operator B. Moreover, the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq52_HTML.gif can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generator B. Hence, we first consider the following eigenvalue problem:
B ϕ ( x ) = λ ϕ ( x ) , ϕ ( 0 ) = 0 ; ϕ x ( x 0 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ17_HTML.gif
(17)
We can easily determine that the eigenvalues are λ n = k ( u ( 0 , 0 ) ) ( 2 n + 1 ) 2 π 2 4 x 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq53_HTML.gif for all n = 0 , 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq54_HTML.gif and the corresponding eigenfunctions are ϕ n ( x ) = sin ( ( 2 n + 1 ) π x 2 x 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq55_HTML.gif. In this case, the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq52_HTML.gif can be represented in the following way:
S ( t ) U ( x , s ) = n = 0 ϕ n ( x ) , U ( x , s ) e λ n t ϕ n ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ18_HTML.gif
(18)
where ϕ n ( x ) , U ( x , s ) = 0 1 ϕ n ( x ) U ( x , s ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq56_HTML.gif. Under this representation, the null space of the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq52_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 = 0 , 1 , 2 , 3 , } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equf_HTML.gif

From the definition of the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq52_HTML.gif, we can say that the null space of it consists of only zero function, i.e., N ( S ) = { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq57_HTML.gif. This result is very important for the uniqueness of the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif.

The unique solution of the initial-boundary value problem (16) in terms of the semigroup S ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq52_HTML.gif can be represented in the following form:
v ( x , t ) = S ( t ) v ( x , 0 ) + 0 t S ( t s ) ( ( k ( v ( x , s ) + ψ 0 + ψ 2 x ) k ( u ( 0 , 0 ) ) ) ( v x ( x , s ) + ψ 2 ) ) x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equg_HTML.gif
Now, by using the identity (15) and taking the initial value u ( x , 0 ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq43_HTML.gif into account, the integral equation for the solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq19_HTML.gif of the parabolic problem (14) in terms of a semigroup can be written in the following form:
u ( x , t ) = ψ 0 + ψ 2 x + S ( t ) ( g ( x ) ψ 0 ψ 2 x ) + 0 t S ( t s ) ( ( k ( u ( x , s ) ) k ( u ( 0 , 0 ) ) ) u x ( x , s ) ) x d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ19_HTML.gif
(19)
In order to arrange the above integral equation, let us define the following:
ζ ( x ) = ( g ( x ) ψ 0 ψ 2 x ) , ξ ( x , t ) = ( ( k ( u ( x , t ) ) k ( u ( 0 , 0 ) ) ) u x ( x , t ) ) x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equh_HTML.gif
Then we can rewrite the integral equation in terms of ζ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq44_HTML.gif and ξ ( x , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq45_HTML.gif in the following form:
u ( x , t ) = ψ 0 + ψ 2 x + ( S ( t ) ζ ( ) ) ( x , t ) + 0 t ( S ( t s ) ξ ( , s ) ) ( x , t , s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ20_HTML.gif
(20)

This is the integral representation of a solution of the initial-boundary value problem (14) on Ω T 0 = { ( x , t ) R 2 : 0 < x < x 0 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq28_HTML.gif. It is obvious from the eigenfunctions ϕ n ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq46_HTML.gif, the domain of eigenfunctions can be extended to the closed interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif. Moreover, they are continuous on [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif. Under this extension, the uniqueness of the solutions of the initial-boundary value problems (13) and (14) imply that the integral representation (20) becomes the integral representation of a solution of the initial-boundary value problem (13) on Ω T = { ( x , t ) R 2 : 0 < x < 1 , 0 < t T } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq47_HTML.gif.

Substituting x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif into the integral representation (20) of the solution u ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq19_HTML.gif yields
u ( 1 , t ) = f ( t ) = ψ 0 + ψ 2 + ( S ( t ) ζ ( ) ) ( 1 , t ) + 0 t ( S ( t s ) ξ ( , s ) ) ( 1 , t , s ) d s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_Equ21_HTML.gif
(21)

which implies that f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq25_HTML.gif can be determined analytically.

The right-hand side of the identity (21) defines the semigroup representation of the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif. …

4 Conclusion

The goal of this study is to identify the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif by using the over measured data u ( x 0 , t ) = ψ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq4_HTML.gif and u x ( x 0 , t ) = ψ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq5_HTML.gif. The key point here is the unique extensions of solutions on [ 0 , x 0 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq10_HTML.gif to the closed interval [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq9_HTML.gif which are implied by the uniqueness of the solutions. This key point leads to the integral representation of the unknown boundary condition u ( 1 , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq6_HTML.gif at x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-2/MediaObjects/13661_2012_Article_259_IEq7_HTML.gif obtained analytically. …

Declarations

Acknowledgements

Dedicated to my father and mother Yusuf/Sevim Ozbilge.

The research was supported by parts by the Scientific and Technical Research Council (TUBITAK) of Turkey and Izmir University of Economics. …

Authors’ Affiliations

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

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© Ozbilge; licensee Springer. 2013

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