Abstract
In this article, a semigroup approach is presented for the mathematical analysis of
inverse problems of identifying the unknown boundary condition
1 Introduction
Consider the following initial boundary value problem for quasilinear diffusion equation:
where
(C1)
(C2)
The initial boundary value problem (1) has a unique solution
In physics, many applications of this problem can be found. The simple model of flame
propagation and the spread of biological populations, where
We consider the inverse problems[10] of determining boundary
and from Neumann type of measured output data at the boundaries
Here
The semigroup approach [11] for inverse problems for the identification of an unknown coefficient in a quasilinear 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
2 Analysis of the inverse problem of the boundary condition by Dirichlet type of over
measured data
u
(
x
0
,
t
)
=
ψ
1
Consider now the inverse problem with one measured output data
Then the initial boundary value problem (1) can be rewritten in the following form:
In order to determine the unknown boundary condition
where
which satisfies the following parabolic problem:
Here,
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
We can easily determine that the eigenvalues are
where
From the definition of the semigroup
The unique solution of the initialboundary value problem (7) in terms of the semigroup
Now, by using the identity (6) and taking the initial value
In order to arrange the above integral equation, let us define the following:
Then we can rewrite the integral equation in terms of
This is the integral representation of a solution of the initialboundary value problem
(5) on
At this stage, it is obvious that the solution of the inverse problem can easily be
obtained by substituting
which implies that
The righthand side of the identity (12) defines the semigroup representation of the unknown boundary condition
3 Analysis of the inverse problem of the boundary condition by Neumann type of over
measured data
u
x
(
x
0
,
t
)
=
ψ
2
Consider now the inverse problem with one measured output data
Then the initial boundary value problem (1) can be rewritten in the following form:
In order to determine the unknown boundary condition
where
To formulate the solution of the above problem in terms of a semigroup, we need to define a new function
which satisfies the following parabolic problem:
Here
Let us denote the semigroup of linear operators by
We can easily determine that the eigenvalues are
where
From the definition of the semigroup
The unique solution of the initialboundary value problem (16) in terms of the semigroup
Now, by using the identity (15) and taking the initial value
In order to arrange the above integral equation, let us define the following:
Then we can rewrite the integral equation in terms of
This is the integral representation of a solution of the initialboundary value problem
(14) on
Substituting
which implies that
The righthand side of the identity (21) defines the semigroup representation of the unknown boundary condition
4 Conclusion
The goal of this study is to identify the unknown boundary condition
Competing interests
The author declares that they have no competing interests.
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. …
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