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Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions

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 2014, 2014:134  doi:10.1186/1687-2770-2014-134

Published: 27 May 2014

Abstract

This article deals with the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M1">View MathML</a> in the linear time fractional parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M3">View MathML</a>, with mixed boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M5">View MathML</a>. By defining the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M7">View MathML</a>, the inverse problem is reduced to the problem of their invertibility. Hence the main purpose of this study is to investigate the distinguishability of the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M9">View MathML</a>. This work shows that the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M9">View MathML</a> have the distinguishability property. Moreover, the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M12">View MathML</a> of the unknown diffusion coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M1">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M14">View MathML</a> can be determined explicitly by making use of measured output data (boundary observation) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M15">View MathML</a>, which brings greater restriction on the set of admissible coefficients. It is also shown that the measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M17">View MathML</a> can be determined analytically by a series representation, which implies that the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/134/mathml/M7">View MathML</a> can be described explicitly.