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

DEMİR A. , Hasanov A.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.340, no.1, pp.5-15, 2008 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 340 Issue: 1
  • Publication Date: 2008
  • Doi Number: 10.1016/j.jmaa.2007.08.004
  • Page Numbers: pp.5-15


In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation u(t)(x, t)=(k(x)u(x)(x, t))(x), with Dirichlet boundary conditions u(0, t)=psi(0), u(1, t)=psi(1). Main goal of this study is to investigate the distinguishability of the input-output mappings Phi[.]:K -> C-1 [0, T], Psi[.]:K -> C-1[0, T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Phi[.] and Psi[.] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x = 0 and x = 1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f (t) := k(0)u(x)(0, t) or/and h(t) := k(1)u(x) (1, t). In addition to these, the values k'(0) and k'(1) of the unknown coefficient k(x) at x = 0 and x = 1, respectively, are also determined via the input data. Furthermore, it is shown that measured output data f (t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Phi[.] :K -> C-1 [0, T], Psi[.] :K -> C-1 [0, T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x = 0 and x=1. (c) 2007 Elsevier Inc. All rights reserved.