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

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 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.