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# Identification of the unknown boundary condition in a linear parabolic equation

- Ali Demir
^{1}and - Ebru Ozbilge
^{2}Email author

**2013**:96

https://doi.org/10.1186/1029-242X-2013-96

© Demir and Ozbilge; licensee Springer 2013

**Received:**27 December 2012**Accepted:**18 February 2013**Published:**7 March 2013

## Abstract

In this article, a semigroup approach is presented for the mathematical analysis of the inverse problems of identifying the unknown boundary condition $u(1,t)=f(t)$ in a linear parabolic equation ${u}_{t}(x,t)={(k(u(x,t)){u}_{x}(x,t))}_{x}$ with Dirichlet boundary conditions $u(0,t)={\psi}_{0}$, $u(1,t)=f(t)$ by making use of the over measured data $u({x}_{0},t)={\psi}_{1}$ and ${u}_{x}({x}_{0},t)={\psi}_{2}$ separately.

## Keywords

- Integral Equation
- Inverse Problem
- Integral Representation
- Null Space
- Initial Boundary

## 1 Introduction

where ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$. The left boundary value ${\psi}_{0}$ is assumed to be constant. The functions ${c}_{1}>k(x)\ge {c}_{0}>0$ and $g(x)$ satisfy the following conditions:

(C1) $k(x)\in {H}^{1,2}[0,1]$;

(C2) $g(x)\in {H}^{2,2}[0,1]$, $g(0)={\psi}_{0}$, $g(1)=f(0)$.

The initial boundary value problem (1) has the unique solution $u(x,t)$ satisfying $u(x,t)\in {H}^{2,2}[0,1]\cap {H}^{1,2}[0,1]$ [1–4] under these conditions.

In physics, many applications of this problem can be found. The simple model of flame propagation and the spread of a biological populations, where $u=u(x,t)$, $k(x)$ denote the temperature and density respectively, are given by the equation in problem (1). Especially $k=k(x)$ represents the density-dependent coefficient in the problems of the spread of biological populations [5–9].

*the inverse problems*[10] of determining boundary $u(1,t)$ at $x=1$ in problem (1) from Dirichlet type of measured output data at the boundaries $x={x}_{0}$:

Here the solution of the parabolic problem (1) is denoted by $u=u(x,t)$. In this context, the parabolic problem (1) will be referred to as a *direct (forward) problem* with the *inputs* $g(x)$, $k(x)$ and $f(t)$. Notice that $u({x}_{0},0)={\psi}_{1}$ and ${u}_{x}({x}_{0},0)={\psi}_{2}$. Therefore it is assumed that the functions $u({x}_{0},t)={\psi}_{1}$ and ${u}_{x}({x}_{0},t)={\psi}_{2}$ respectively satisfy the consistency conditions ${\psi}_{1}=g({x}_{0})$ and ${\psi}_{2}={g}^{\prime}({x}_{0})$.

The purpose of this paper is to determine the boundary function $u(1,t)$ at $x=1$ via the semigroup approach which is studied by [11–15].

The paper is organized as follows. In Section 2, an analysis of the semigroup approach is given for the inverse problem with the single measured output data $u({x}_{0},t)={\psi}_{1}$ given at $x={x}_{0}$. A similar analysis is applied to the inverse problem with the single measured output data ${u}_{x}({x}_{0},t)={\psi}_{2}$ given at the point $x={x}_{0}$ 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 $u({x}_{0},t)={\psi}_{1}$

Here, $A[\cdot ]:=-k(0)\frac{{d}^{2}[\cdot ]}{d{x}^{2}}$ is a second-order differential operator and its domain is ${D}_{A}=\{v\in {H}^{2,2}(0,{x}_{0})\cap {H}^{1,2}[0,{x}_{0}]:v(0)=v({x}_{0})=0\}$, where ${H}_{0}^{2,2}(0,{x}_{0})=\overline{{C}_{0}^{2}(0,{x}_{0})}$ and ${H}_{0}^{1,2}[0,{x}_{0}]=\overline{{C}_{0}^{1}[0,{x}_{0}]}$ are Sobolev spaces. Obviously, by completion $g(x)\in {D}_{A}$ since the initial value function $g(x)$ belongs to ${C}^{3}[0,1]$. Hence ${D}_{A}$ is dense in ${H}_{0}^{2,2}[0,1]$, 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.

*A*[9, 10]. We can easily find the eigenvalues and eigenfunctions of the differential operator

*A*. Moreover, the semigroup $T(t)$ can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generator

*A*. Hence we first consider the following eigenvalue problem:

From the construction of the semigroup $T(t)$, it can be concluded that the null space of it consists of only zero function, *i.e.*, $N(T)=\{0\}$. This result implies the uniqueness of the unknown boundary condition $u(1,t)$.

This is the integral representation of a solution of the initial-boundary value problem (5) on ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$. It is obvious from the eigenfunctions ${\varphi}_{n}(x)$, the domain of eigenfunctions can be extended to a closed interval $[0,1]$. Moreover, they are continuous on $[0,1]$. Under this extension, the uniqueness 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 ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$.

which implies that $f(t)$ can be determined analytically. The right-hand side of identity (12) defines *the semigroup representation of the unknown boundary condition* $u(1,t)$ *at* $x=1$.

which is the initial condition we have.

## 3 Analysis of the inverse problem of the boundary condition by Neumann type of over measured data ${u}_{x}({x}_{0},t)={\psi}_{2}$

Here $B[\cdot ]:=-k(0)\frac{{d}^{2}[\cdot ]}{d{x}^{2}}$ is a second-order differential operator, its domain is ${D}_{B}=\{v\in {H}^{2,2}(0,{x}_{0})\cap {H}^{1,2}[0,{x}_{0}]:v(0)={v}_{x}({x}_{0})=0\}$. It is clear from the definition of ${D}_{B}$ that ${D}_{B}\subset {H}^{2,2}[0,{x}_{0}]$.

*B*[9, 10]. We can easily find the eigenvalues and eigenfunctions of the differential operator

*B*. Moreover, the semigroup $S(t)$ can be easily constructed by using the eigenvalues and eigenfunctions of the infinitesimal generator

*B*. Hence we first consider the following eigenvalue problem:

As in the previous section, the construction of the semigroup $S(t)$ implies that the null space of it consists of only zero function, *i.e.*, $N(S)=\{0\}$. The uniqueness of the unknown boundary condition $u(1,t)$ follows from this result.

This is the integral representation of a solution of the initial-boundary value problem (15) on ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$. It is obvious from the eigenfunctions ${\varphi}_{n}(x)$, the domain of eigenfunctions can be extended to a closed interval $[0,1]$. Moreover, they are continuous on $[0,1]$. Under this extension, the uniqueness of the initial-boundary value problems (15) and (16) imply that integral representation (22) becomes the integral representation of a solution of the initial-boundary value problem (16) on ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$.

which implies that $f(t)$ can be determined analytically.

The right-hand side of identity (23) defines *the semigroup representation of the unknown boundary condition* $u(1,t)$ *at* $x=1$.

## 4 Conclusion

The purpose of this study is to identify the unknown boundary condition $u(1,t)$ at $x=1$ via the semigroup approach by using the over measured data $u({x}_{0},t)={\psi}_{1}$ and ${u}_{x}({x}_{0},t)={\psi}_{2}$. The crucial point here is the unique extensions of the solutions on $\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$ to $\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$ which are implied by the uniqueness of the solutions. This key leads to the integral representation of the unknown boundary condition $u(1,t)$ at $x=1$ obtained analytically.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

## Authors’ Affiliations

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