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RESEARCH

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Analysis of the inverse problem in a time fractional parabolic equation with mixed boundary conditions Ebru Ozbilge1* and Ali Demir2 * Correspondence: [email protected] 1 Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No. 156, Balcova, Izmir, 35330, Turkey Full list of author information is available at the end of the article

Abstract This article deals with the mathematical analysis of the inverse coefficient problem of identifying the unknown coefficient k(x) in the linear time fractional parabolic equation Dαt u(x, t) = (k(x)ux )x , 0 < α ≤ 1, with mixed boundary conditions u(0, t) = ψ0 (t), ux (1, t) = ψ1 (t). By defining the input-output mappings [·] : K → C 1 [0, T] and  [·] : K → C[0, T], 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 [·] and  [·]. This work shows that the input-output mappings [·] and  [·] have the distinguishability property. Moreover, the value k(0) of the unknown diffusion coefficient k(x) at x = 0 can be determined explicitly by making use of measured output data (boundary observation) k(0)ux (0, t) = f (t), which brings greater restriction on the set of admissible coefficients. It is also shown that the measured output data f (t) and h(t) can be determined analytically by a series representation, which implies that the input-output mappings [·] : K → C 1 [0, T] and  [·] : K → C[0, T] can be described explicitly.

1 Introduction The inverse problem of determining an unknown coefficient in a linear parabolic equation by using over-measured data has generated an increasing amount of interest from engineers and scientist during the last few decades. This kind of problem plays a crucial role in engineering, physics and applied mathematics. The problem of recovering an unknown coefficient or coefficients in the mathematical model of physical phenomena is frequently encountered. Intensive study has been carried out on this kind of problem, and various numerical methods have been developed in order to overcome the problem of determining an unknown coefficient or coefficients [–]. The inverse problem of unknown coefficients in a quasi-linear parabolic equations was studied by Demir and Ozbilge [, ]. Moreover, the identification of the unknown diffusion coefficient in a linear parabolic equation was studied by Demir and Hasanov []. Fractional differential equations are generalizations of ordinary and partial differential equations to an arbitrary fractional order. By linear time-fractional parabolic equation, we mean a certain parabolic-like partial differential equation governed by master equations containing fractional derivatives in time [, ]. The research areas of fractional differential equations range from theoretical to applied aspects. The main goal of this study is ©2014 Ozbilge and Demir; licensee Springer. This is an Open Access article distributed under the terms of the Creativ