Abstract
This article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p(t) and q in the linear parabolic equation u t (x,t)= u x x +q u x (x,t)+p(t)u(x,t), with mixed boundary conditions u x (0,t)= ψ 0 , k(1)u(1,t)= ψ 1 . The main purpose of this paper is to investigate the distinguishability of the input-output mapping Φ[⋅]:P→ H 1 , 2 [0,T], via semigroup theory. In this paper, it is shown that if the nullspace of the semigroup T(t) consists of only the zero function, then the input-output mapping Φ[⋅] has the distinguishability property. It is also shown that both types of boundary conditions and also the region in which the problem is defined play an important role in the distinguishability property of the input-output mapping. Moreover, the input data can be used to determine the unknown parameter p(t) at (x,t)=(0,0) and also the unknown coefficient q. Furthermore, it is shown that measured output data f(t) can be determined analytically by an integral representation. Hence the input-output mapping Φ[⋅]:P→ H 1 , 2 [0,T] is given explicitly in terms of the semigroup.
Highlights
1 Introduction The inverse problem of determining parameter in a linear parabolic equation by using over-measured data has generated an increasing interest from engineers and scientist in recent years because such problems play a crucial role in engineering, physics and applied mathematics
The inverse problem of unknown coefficients in a quasi-linear parabolic equations was studied by Demir and Ozbilge [ – ]
We conclude that the semigroup with a trivial null space, i.e., N(T) = { }, plays a crucial role in the distinguishability of the input-output mappings
Summary
The inverse problem of determining parameter in a linear parabolic equation by using over-measured data has generated an increasing interest from engineers and scientist in recent years because such problems play a crucial role in engineering, physics and applied mathematics. Intensive study has been done on these kind of problems and various numerical methods developed which are used to overcome the problem of determining unknown parameter or parameters [ – ]. The purpose of this study is to investigate the inverse problem of determining unknown parameters q and p(t) in a one-dimensional parabolic equation using a semigroup approach. The inverse problem of unknown coefficients in a quasi-linear parabolic equations was studied by Demir and Ozbilge [ – ]. The identification of the unknown diffusion coefficient in a linear parabolic equation was studied by Demir and Hasanov [ ]. The study in this article is based on a philosophy similar to that used in [ – ]
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