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Identification of the unknown diffusion coefficient in a quasi‐linear parabolic equation by semigroup approach with mixed boundary conditions
Author(s) -
Ozbilge Ebru
Publication year - 2008
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.974
Subject(s) - mathematics , semigroup , boundary (topology) , zero (linguistics) , dirichlet boundary condition , heat equation , parabolic partial differential equation , mathematical analysis , function (biology) , diffusion , diffusion equation , space (punctuation) , pure mathematics , discrete mathematics , partial differential equation , economics , service (business) , thermodynamics , philosophy , linguistics , evolutionary biology , biology , physics , economy
In this article, a semigroup approach is presented for the mathematical analysis of the inverse coefficient problems of identifying the unknown diffusion coefficient k ( u ( x, t )) in the quasi‐linear parabolic equation u t ( x, t )=( k ( u ( x, t )) u x ( x, t )) x , with Dirichlet boundary conditions u x (0, t )=ψ 0 , u (1, t )=ψ 1 . The main purpose of this work is to analyze the distinguishability of the input–output mappings Φ[·] : → C 1 [0, T ], Ψ[·] : → C 1 [0, T ] using semigroup theory. In this article, it is shown that if the null space of semigroups T ( t ) and S ( t ) consists of only a zero function, then the input–output mappings Φ[·] and Ψ[·] have the distinguishability property. Copyright © 2008 John Wiley & Sons, Ltd.