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 \(M_\alpha(t)=\sum_n a_nE_{\alpha,1}(-\lambda_nt^\alpha)\) satisfying \(\sum_n|a_n|/\lambda_n<\infty\), we show that, after eigenspace grouping, every nontrivial observation has a finite first nonzero resolvent moment \(S_m=\sum_n a_n/\lambda_n^m\).

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 \(1/\Gamma(1-m\alpha)\), hence uniqueness from one sufficiently late scalar measurement.

For two measurements, \(M_\alpha(\rho t)/M_\alpha(t)=\rho^{-m\alpha}(1+O(t^{-\alpha_0}))\), giving a log-ratio estimator with asymptotic-bias and relative-noise error bounds.

For bounded observations, \(S_m=\langle\mathcal A^{-m}\varphi,h\rangle\); for a finite rod, the leading point-sensor condition is \((\mathcal A^{-1}\varphi)(x_*)\ne0\).

Counterexamples show the sharpness of the exclusions and noise interpretation.