Open Access Research

Semigroup approach for identification of the unknown diffusion coefficient in a linear parabolic equation with mixed output data

Ebru Ozbilge1* and Ali Demir2

Author Affiliations

1 Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No. 156, Balcova, Izmir, 35330, Turkey

2 Department of Mathematics, Kocaeli University, Umuttepe, Izmit, Kocaeli, 41380, Turkey

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Boundary Value Problems 2013, 2013:43  doi:10.1186/1687-2770-2013-43

Published: 1 March 2013

Abstract

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a> in the linear parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M2">View MathML</a> with mixed boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M4">View MathML</a>. The aim of this paper is to investigate the distinguishability of the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M6">View MathML</a> via semigroup theory. In this paper, we show that if the null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> consists of only zero function, then the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9">View MathML</a> have the distinguishability property. It is shown that the types of the boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability property of these mappings. Moreover, in the light of measured output data (boundary observations) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M10">View MathML</a> or/and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M11">View MathML</a>, the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13">View MathML</a> of the unknown diffusion coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a>, respectively, can be determined explicitly. In addition to these, the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M18">View MathML</a> of the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a>, respectively, are also determined via the input data. Furthermore, it is shown that measured output data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> can be determined analytically by an integral representation. Hence the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M25">View MathML</a> are given explicitly in terms of the semigroup.