Late-Time Fractional-Order Identification in Caputo Diffusion Equation

Abstract

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 \(M_α(t)=\sum_n a_nE_{α,1}(-λ_nt^α)\) satisfying \(\sum_n|a_n|/λ_n<\infty\), we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment \(S_m=\sum_n a_n/λ_n^m\). A uniform differentiated large-argument expansion of the Mittag-Leffler factor yields eventual strict monotonicity of \(α\mapsto M_α(t)\) on admissible intervals avoiding the zeros of \(1/Γ(1-mα)\), hence uniqueness from one sufficiently late scalar measurement. For two measurements, \(M_α(ρt)/M_α(t)=ρ^{-mα}(1+O(t^{-α_0}))\), giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds. For bounded observations, \(S_m=\langle\mathcal A^{-m}φ,h\rangle\); for a finite rod, the leading point-sensor condition is \((\mathcal A^{-1}φ)(x_*)\ne0\). Counterexamples show the sharpness of the exclusions and noise interpretation.