Generalized Hyers-Ulam stability of the time-fractional diffusion equation

초록

This paper establishes the generalized Hyers–Ulam stability of the time-fractional diffusion equation CDtαu=Δu for α ∈ (0, 1] in Rn with n ≥ 3, where CDtα denotes the Caputo fractional derivative. Unlike classical diffusion equations, the non-local memory effects inherent to the Caputo derivative introduce significant analytical challenges, requiring a combination of Fourier and Laplace transform techniques together with the asymptotic theory of the Fox H-function via Mellin–Barnes representations. The stability bounds derived here reveal a two-rate spatial decay structure that is absent from all existing integer-order stability results, reflecting the genuinely richer behavior of fractional diffusion. Explicit dependence of the stability constant on the fractional order α, the spatial dimension n, and the perturbation decay rate is quantified in terms of the Euler Beta and Gamma functions. The theoretical estimates are validated by numerical experiments in three dimensions. © 2026 Elsevier Inc.