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


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Ozbilge E., DEMİR A.

BOUNDARY VALUE PROBLEMS, 2014 (SCI İndekslerine Giren Dergi) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası:
  • Basım Tarihi: 2014
  • Doi Numarası: 10.1186/1687-2770-2014-134
  • Dergi Adı: BOUNDARY VALUE PROBLEMS

Özet

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 D-t(alpha) u(x, t) = (k(x)u(x))(x), 0 < alpha <= 1, with mixed boundary conditions u(0, t) = psi(0)(t), u(x)(1, t) = psi(1)(t). By defining the input-output mappings Phi[.] : kappa -> C-1[0, T] and psi[.] : kappa -> 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 Phi[.] and Phi[.]. This work shows that the input-output mappings Phi[.] and Phi[.] 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 Phi[.] : kappa -> C1[0, T] and Phi[.] : kappa -> C[0, T] can be described explicitly.