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 $D_{t}^{\alpha}u(x,t)=(k(x)u_{x})_{x}+qu_{x}(x,t)+p(t)u(x,t)$ , $0<\alpha\leq1$ , 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 $\Phi[\cdot]:\mathcal {K}\rightarrow C[0,T]$ and $\Psi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]$ 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 $\Phi[\cdot]$ and $\Psi[\cdot]$ . This work shows that the input-output mappings $\Phi[\cdot]$ and $\Psi [\cdot]$ 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 $\Phi [\cdot]: \mathcal{K}\rightarrow C[0,T]$ and $\Psi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]$ can be described explicitly, where $\Phi[k]=u(x,t;k)|_{x=0}$ and $\Psi[k]=k(x)u_{x}(x,t;k)|_{x=1}$ .