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

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].

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.

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.

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

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

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.

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.

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….