Late-Time Fractional-Order Identification in Caputo Diffusion Equation

We study late-time identification of the Caputo order in a linear diffusion equation generated by a strictly positive self-adjoint operator with compact resolvent. For signed scalar observations $M_{\alpha }(t)={\sum }_n{a}_n{E}_{\alpha ,1}(-{\lambda }_n{t}^{\alpha })$ satisfying ${\sum }_n\mathrm{\mid }{a}_n\mathrm{\mid }\mathrm{/}{\lambda }_n<\mathrm{\infty }$, we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment $S_m={\sum }_n{a}_n\mathrm{/}{\lambda }_n^m$. A uniform differentiated large-argument expansion of the Mittag-Leffler factor yields eventual strict monotonicity of $\alpha \mapsto M_{\alpha }(t)$ on admissible intervals avoiding the zeros of $1\mathrm{/}\mathrm{\Gamma }(1-m\alpha )$, hence uniqueness from one sufficiently late scalar measurement. For two measurements, $M_{\alpha }(\rho t)\mathrm{/}{M}_{\alpha }(t)={\rho }^{-m\alpha }(1+O(t^{-{\alpha }_0}))$, giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds. For bounded observations, $S_m=⟨{\mathcal{A}}^{-m}\phi ,h⟩$; for a finite rod, the leading point-sensor condition is $({\mathcal{A}}^{-1}\phi )(x_{\ast })\mathrm{\ne }0$. Counterexamples show the sharpness of the exclusions and noise interpretation.