Estimating parameters of the diffusion model via asymptotic expansions

TL;DR

This paper introduces a novel asymptotic expansion method for estimating parameters like the diffusion coefficient and domain size in heat equations from minimal measurement data, improving accuracy over existing techniques.

Contribution

It develops an asymptotic solution approach for inverse parameter estimation in heat equations using the Fokas method, extending to multiple parameters and validated with numerical and real-world soil science data.

Findings

Accurately estimates diffusion parameters from a single measurement.

Provides improved parameter estimates compared to existing methods.

Demonstrates applicability to soil science problems.

Abstract

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.