| Abstract: |
| We address forward analysis and inverse coefficient recovery for reaction-diffusion-advection models with time-periodic forcing in the singularly perturbed setting where diffusion is small and solutions develop a sharp moving front. Using matched asymptotic expansions, we construct a uniformly valid periodic approximation that resolves both the outer solution away from the front and the inner transition layer governing the front dynamics. The approximation yields explicit formulas for the leading-order profile and for the front location, and we provide rigorous error bounds. Building on this asymptotic surrogate, we develop two fast identification strategies: one for spatially heterogeneous coefficients and one for time-varying coefficients. In both cases, the inverse problem is reduced to a regularized least-squares fit involving readily measurable features (e.g., local gradients and front motion), eliminating iterative PDE-constrained optimization and significantly reducing computational cost. We prove stability and obtain convergence rates. Computational experiments confirm that the method recovers unknown coefficients accurately and robustly across a range of parameter regimes and noise levels, making it suitable for rapid calibration in environmental, biological, and engineering applications. |
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