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

- Ebru Ozbilge
^{1}Email author

**2013**:2

https://doi.org/10.1186/1687-2770-2013-2

© Ozbilge; licensee Springer. 2013

**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 $u(1,t)=f(t)$ in the quasi-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. The purpose of this study is to identify the unknown boundary condition $u(1,t)$ at $x=1$ by using the over measured data $u({x}_{0},t)={\psi}_{1}$ and ${u}_{x}({x}_{0},t)={\psi}_{2}$. First, by using over measured data as a boundary condition, we define the problem on ${\mathrm{\Omega}}_{{T}_{0}}=\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$, then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval $[0,1]$, we reach the result. The main point here is the unique extensions of the solutions on $[0,{x}_{0}]$ to the closed interval $[0,1]$ which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition $u(1,t)$ at $x=1$.

## 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 ${c}_{1}>k(u(x,t))\ge {c}_{0}>0$ and $g(x)$ satisfy the following conditions:

(*C* 1) $|{k}_{u}({u}_{1})-{k}_{u}({u}_{2})|<d|{u}_{1}-{u}_{2}|$;

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

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

In physics, many applications of this problem can be found. The simple model of flame propagation and the spread of biological populations, where $u=u(x,t)$ denotes the temperature and density respectively, are given by the equation in the problem (1). Especially $k=k(u(x,t))$ 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 the problem (1) from Dirichlet type of measured output data at the boundaries $x={x}_{0}$

Here $u=u(x,t)$ 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* $g(x)$, $k(u(x,t))$ and $f(t)$. 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 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 [12–15].

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 $u({x}_{0},t)={\psi}_{1}$ given at $x={x}_{0}$. The 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, $A[\cdot ]:=-k(u(0,0))\frac{{d}^{2}[\cdot ]}{d{x}^{2}}$ is a second-order differential operator and its domain is ${D}_{A}=\{v(x)\in {H}_{0}^{2,2}(0,1)\cap {H}_{0}^{3,2}[0,1]:v(0)=0=v(1)\}$, where ${H}_{0}^{2,2}(0,1)=\overline{{C}_{0}^{2}(0,1)}$ and ${H}_{0}^{1,2}[0,1]=\overline{{C}_{0}^{1}[0,1]}$ 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*[8, 9]. 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 definition of the semigroup $T(t)$, we can say that the null space of it consists of only zero function, *i.e.*, $N(T)=\{0\}$. This result is very important for 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 ${\mathrm{\Omega}}_{{T}_{0}}=\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$. It is obvious from the eigenfunctions ${\varphi}_{n}(x)$, the domain of eigenfunctions can be extended to the closed interval $[0,1]$. Moreover they are continuous on $[0,1]$. 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 ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$.

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

The right-hand side of the identity (12) defines *the semigroup representation of the unknown boundary condition* $u(1,t)$ *at* $x=1$.

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

where ${\mathrm{\Omega}}_{{T}_{0}}=\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$.

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

*B*[8, 9]. 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:

From the definition of the semigroup $S(t)$, we can say that the null space of it consists of only zero function, *i.e.*, $N(S)=\{0\}$. This result is very important for the uniqueness of the unknown boundary condition $u(1,t)$.

This is the integral representation of a solution of the initial-boundary value problem (14) on ${\mathrm{\Omega}}_{{T}_{0}}=\{(x,t)\in {R}^{2}:0<x<{x}_{0},0<t\le T\}$. It is obvious from the eigenfunctions ${\varphi}_{n}(x)$, the domain of eigenfunctions can be extended to the closed interval $[0,1]$. Moreover, they are continuous on $[0,1]$. 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 ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$.

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

The right-hand side of the identity (21) defines *the semigroup representation of the unknown boundary condition* $u(1,t)$ *at* $x=1$. …

## 4 Conclusion

The goal of this study is to identify the unknown boundary condition $u(1,t)$ at $x=1$ by using the over measured data $u({x}_{0},t)={\psi}_{1}$ and ${u}_{x}({x}_{0},t)={\psi}_{2}$. The key point here is the unique extensions of solutions on $[0,{x}_{0}]$ to the closed interval $[0,1]$ which are implied by the uniqueness of the solutions. This key point leads to the integral representation of the unknown boundary condition $u(1,t)$ at $x=1$ 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

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