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Identification of the Unknown Diffusion Coefficient in a Linear Parabolic Equation by the Semigroup Approach by Ali Demir; Alemdar Hasanov is a Mathematics article available to read on EtoBox.

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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 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 Φ[•] and Ψ [•] 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 Φ 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.

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Author

Ali Demir; Alemdar Hasanov

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-247X)

Published

2008

Language

EN

Field

Mathematics (Physical Sciences)