Distinguishability of a source function in a time fractional inhomogeneous parabolic equation with Robin boundary condition

Abstract

This article deals with the mathematical analysis of the inverse problem of identifying the distinguishability of input-output mappings in the linear time fractional inhomogeneous parabolic equation $D_{t}^{\alpha }u(x,t)=(k(x)u_{x})_{x}+F(x,t) \quad 0<\alpha \leq 1$, with Robin boundary conditions $u(0,t)=\psi _{0}(t)$, $u_{x}(1,t)=\gamma(u(1,t)-\psi _{1}(t))$. By defining the input-output mappings $\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T]$ and $\Psi [\cdot ]:\mathcal{K}\rightarrow C[0,T]$ 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 $\Phi[\cdot ]$ and $\Psi [\cdot ]$. Moreover, the measured output data $f(t)$ and $h(t)$ can be determined analytically by a series representation, which implies that the input-output mappings $\Phi [\cdot ]:\mathcal{K}\rightarrow C^1[0,T]$ and $\Psi [\cdot]:\mathcal{K}\rightarrow C[0,T]$ can be described explicitly.