Open Access Research

Determination of the unknown boundary condition of the inverse parabolic problems via semigroup method

Ebru Ozbilge

Author Affiliations

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

Boundary Value Problems 2013, 2013:2  doi:10.1186/1687-2770-2013-2

Published: 4 January 2013

Abstract

In this article, a semigroup approach is presented for the mathematical analysis of inverse problems of identifying the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1">View MathML</a> in the quasi-linear parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M2">View MathML</a>, with Dirichlet boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M1">View MathML</a>, by making use of the over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a> separately. The purpose of this study is to identify the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a> by using the over measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M6">View MathML</a>. First, by using over measured data as a boundary condition, we define the problem on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M11">View MathML</a>, then the integral representation of this problem via a semigroup of linear operators is obtained. Finally, extending the solution uniquely to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a>, we reach the result. The main point here is the unique extensions of the solutions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M13">View MathML</a> to the closed interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M12">View MathML</a> which are implied by the uniqueness of the solutions. This point leads to the integral representation of the unknown boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M7">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/2/mathml/M8">View MathML</a>.