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 Dtαu(x,t)=(k(x)ux)x+r(t)F(x,t)\documentclass[12pt]{minimal}
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\begin{document}$D_{t}^{\alpha}u(x,t)=(k(x)u_{x})_{x}+r(t)F(x,t)$\end{document}, 0<α≤1\documentclass[12pt]{minimal}
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\begin{document}$0<\alpha\leq 1$\end{document}, with mixed boundary conditions u(0,t)=ψ0(t)\documentclass[12pt]{minimal}
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\begin{document}$u(0,t)=\psi_{0}(t)$\end{document}, ux(1,t)=ψ1(t)\documentclass[12pt]{minimal}
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\begin{document}$u_{x}(1,t)=\psi_{1}(t)$\end{document}. By defining the input-output mappings Φ[⋅]:K→C1[0,T]\documentclass[12pt]{minimal}
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\begin{document}$\Phi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]$\end{document} and Ψ[⋅]:K→C[0,T]\documentclass[12pt]{minimal}
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\begin{document}$\Psi[\cdot]:\mathcal{K}\rightarrow C[0,T]$\end{document} 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 Φ[⋅]\documentclass[12pt]{minimal}
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\begin{document}$\Phi[\cdot]$\end{document} and Ψ[⋅]\documentclass[12pt]{minimal}
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\begin{document}$\Psi[\cdot]$\end{document}. Moreover, the measured output data f(t)\documentclass[12pt]{minimal}
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\begin{document}$f(t)$\end{document} and h(t)\documentclass[12pt]{minimal}
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\begin{document}$h(t)$\end{document} can be determined analytically by a series representation, which implies that the input-output mappings Φ[⋅]:K→C1[0,T]\documentclass[12pt]{minimal}
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\begin{document}$\Phi[\cdot]:\mathcal{K}\rightarrow C^{1}[0,T]$\end{document} and Ψ[⋅]:K→C[0,T]\documentclass[12pt]{minimal}
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\begin{document}$\Psi[\cdot]:\mathcal{K}\rightarrow C[0,T]$\end{document} can be described explicitly, where Φ[r]=k(x)ux(x,t;r)|x=0\documentclass[12pt]{minimal}
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\begin{document}$\Phi[r]=k(x)u_{x}(x,t;r)|_{x=0}$\end{document} and Ψ[r]=u(x,t;r)|x=1\documentclass[12pt]{minimal}
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\begin{document}$\Psi [r]=u(x,t;r)|_{x=1}$\end{document}. Also, numerical tests using finite difference scheme combined with an iterative method are presented.
