Inverse Problem for a Time-Fractional Parabolic Equation

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) + qu(x)(x,t) + p(t)u(x,t), 0 <= alpha <= 1, with mixed boundary conditions k(0)u(x)(0,t) = psi(0)(t), u(1,t) = psi(1)(t). By defining the input-output mappings Phi[.] : K -> C[0, T] and psi [.] : K -> C-1[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 psi[.]. This work shows that the input-output mappings Phi[.] and psi[.] have distinguishability property. Moreover, the value k(1) of the unknown diffusion coefficient k(x) at x = 1 can be determined explicitly by making use of measured output data (boundary observation) k(1)u(x)(1, t) = h(t), which brings about a 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. Hence the input-output mappings Phi[.] : K -> C[0, T] and psi [.] : K -> C-1[0, T] can be described explicitly, where Phi[k] = u(x,t;k)|(x=0) and psi[k] = k(x)u(x)(x,t;k)vertical bar(x=1).