# Identification of the unknown coefficient in a quasi-linear parabolic equation by a semigroup approach

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

**2013**:212

https://doi.org/10.1186/1029-242X-2013-212

© Ozbilge and Demir; licensee Springer 2013

**Received: **26 November 2012

**Accepted: **13 April 2013

**Published: **26 April 2013

## Abstract

This article presents a semigroup approach to the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient $a(x,t)$ in the quasi-linear parabolic equation ${u}_{t}(x,t)={u}_{xx}(x,t)+a(x,t)u(x,t)$ with Dirichlet boundary conditions $u(0,t)={\psi}_{0}$, $u(1,t)={\psi}_{1}$. It is shown that the unknown coefficient $a(x,t)$ can be approximately determined *via* the semigroup approach.

## 1 Introduction

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

(C1) $a(x,t)\in {H}^{1,2}({\mathrm{\Omega}}_{T})$;

(C2) $g(x)\in {H}^{3,2}[0,1]$, $g(0)={\psi}_{0}$, $g(1)={\psi}_{1}$.

Under these conditions, the initial boundary value problem (1) has the unique solution $u(x,t)\in {H}^{2,2}[0,1]\cap {H}^{1,2}[0,1]$ [1–3].

*the inverse problem*of determining the unknown coefficient $a(x,t)$ from the following observations at the boundary $x=0$:

Here $u=u(x,t)$ represents the solution of the parabolic problem (1). It is assumed that the function $f(t)$ is *noisy free measured output data*. In this context the parabolic problem (1) will be referred to as a *direct (forward) problem* with the *inputs* $g(x)$ and $a(x,t)$. The function $f(t)$ is assumed to belong to ${H}^{1,2}[0,T]$ and satisfy the consistency conditions $f(0)={g}^{\prime}(0)$. Moreover, it is assumed that the function $g(x)$ satisfies the consistency conditions $g(0)={\psi}_{0}$, $g(1)={\psi}_{1}$.

In inverse problems, mostly numerical methods are employed to determine the unknown function. Even using one or more measured output data in analytic methods does not help in determination of the unknown function exactly [4, 5].

The purpose of this paper is to determine the analytic representation of the approximate unknown coefficient $a(x,t)$ *via* a semigroup approach. We determine the approximate solution of the boundary value problem (1) without determining the unknown coefficient $a(x,t)$. In the determination of it, we just use the values $a(0,0)$, ${a}_{x}(0,0)$ which are obtained and the over measured data ${u}_{x}(0,t)=f(t)$. However, this cannot be possible for any kind of a boundary value problem with different type of boundary conditions. The semigroup approach for inverse problems for the identification of unknown coefficient in a quasi-linear parabolic equations was studied by Demir and Ozbilge [6, 7]. Moreover, the identification of the unknown diffusion coefficient in a linear parabolic equation was studied by Demir and Hasanov [8]. The study in this paper is based on similar philosophy to that in [6–9].

Recent studies [6–9] on the identification of the unknown function *via* a semigroup approach [10] show that the semigroup approach is the most effective analytical method which sheds more light on the unknown function. The weak point of this method is that you cannot apply this method to the initial boundary value problems with boundary conditions other than constant or zero. Overcoming this problem in some way makes this method more effective and valuable.

The paper is organized as follows. In Section 2, determination of the unknown coefficient $a(x,t)$ is given *via* the semigroup approach. Some concluding remarks are given in Section 3.

## 2 Determination of the unknown coefficient $a(x,t)$*via* the semigroup approach

*the inverse problem*of determining the explicit form of the unknown coefficient $a(x,t)$

*via*the semigroup approach. Let the parabolic equation be arranged as follows:

*i.e.*, $F(x,t)=a(x,t)u(x,t)$. In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, a new function needs to be defined to make the boundary values equal to zero:

We can represent the solution of this problem in terms of a semigroup of linear operators. Here, $A[\cdot ]:=-\frac{{d}^{2}[\cdot ]}{d{x}^{2}}$ 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.

*A*[5, 6]. Since the initial boundary value problem (4) possesses the property of uniqueness of its solutions, then the solution ${v}_{h}(x,t)$ of its homogeneous part can be written as follows:

*variations of parameters formula*, we can write the solution of the initial boundary value problem (3) as follows:

*A*. Hence, the following eigenvalue problem must first be considered:

*i.e.*, $N(T)=\{0\}$. The null space of the semigroup $T(t)$ of the linear operators can be defined as follows:

*t*yields

*x*yields

which leads us to the determination of the unknown coefficient $a=a(x,t)$ analytically.

**Lemma**

*Let*$T(t)$

*be the semigroup of the linear operators*,

*and let*$u(x,t)$

*be the solution of the equation*${u}_{t}(x,t)=Au(x,t)$,

*where*

*A*

*is the infinitesimal generator of*$T(t)$.

*Then the following identity holds*:

*for each* $t,s\in (0,T]$.

*Proof*First of all, observe that since eigenfunctions ${\varphi}_{n}(x)$ of the infinitesimal generator

*A*generate a complete orthogonal family, we can write $u(x,t)$ in terms of eigenfunctions in the following form:

*t*produces

*t*produces

does not hold. □

This lemma leads us to the solution of the parabolic problem (1) without knowing the unknown coefficient $a(x,t)$, but knowing $a(0,0)$ and ${a}_{x}(0,0)$. Then the solution enables us to determine the unknown coefficient $a(x,t)$ approximately.

*t*as follows:

Substituting the integration constant $C(x)$ into the identity (15), we obtain the approximate solution of the parabolic problem (1). Hence, by using this solution and the parabolic equation, we are able to determine the unknown coefficient $a(x,t)$ approximately.

## 3 Conclusion

The aim of this study was to determine the unknown coefficient $a(x,t)$ *via* the semigroup approach. Lemma in Section 2 plays a crucial role in the identification of the unknown coefficient $a(x,t)$. This lemma leads us to new conclusions about the identification problems *via* the semigroup approach. We will present these results in our future studies….

## 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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