Estimating parameters of the diffusion model via asymptotic expansions

A broad class of inverse problems deals with determining certain parameters, from measurement data, in models which are associated to certain partial differential equations. In this work we focus on the heat equation on a finite interval and we determine the dimensionless diffusion parameter from a single measurement. Our results extend to estimating additional parameters of the initial-boundary value problem, such as the length of the interval and/or the time required for the solution to achieve a specific state. Our approach relies on the asymptotic solution of an integral equation: The formulation of this integral equation is based on the solution of the direct problem via the Fokas method; the solution of this equation is achieved through the asymptotic evaluation of the associated integrals which yield an effective approximate solution, supported by numerical verifications. We apply these approximations to well-established problems in soil science and we compare our results with existing ones, displaying clear improvement.

💡 Research Summary

The manuscript tackles the inverse problem of determining diffusion‑type parameters from a single observation of the solution to the one‑dimensional heat equation on a finite interval. By employing the unified transform (Fokas method), the authors obtain an exact integral representation of the forward solution that features a continuous spectral parameter λ and a contour C in the complex plane. This representation replaces the traditional Fourier series and allows the formulation of a nonlinear integral equation of the form I(a)=c, where the auxiliary variable a encodes the physical parameters (diffusivity, interval length, observation time) and the normalized measurement appears as c∈(0,1).

Two concrete inverse problems from soil engineering are examined. The first concerns the estimation of subsurface drain spacing L from a prescribed water‑table height at the far end of the domain; the second concerns the estimation of the soil moisture diffusivity D₀ from a moisture measurement at mid‑depth. In both cases the forward solution is expressed via the Fokas integral, leading to integral equations (9) and (17) that share the same kernel I(a) defined by \