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Estimating parameters of the diffusion model via asymptotic expansions

Konstantinos Kalimeris, Leonidas Mindrinos

TL;DR

This work develops a unified framework to estimate diffusion-model parameters on a bounded domain from a single measurement by converting the forward diffusion problem, solved with the Fokas method, into an integral equation $I(a)=c$. It then derives comprehensive asymptotic expansions for $I(a)$ in both large-$a$ and small-$a$ regimes, and couples them with inversion techniques (Lagrange–Bürmann, perturbative refinements, erfc-based inversions) to obtain practical, closed-form approximations for the inverse problem variables. The authors assemble these into composite, uniformly accurate formulas that cover the full data range with controllable relative error, and validate them through drainage and soil-moisture applications, showing markedly improved accuracy over classical approaches. The results offer a tractable pathway for real-time parameter estimation in hydrological contexts and underscore the value of integral-equation reformulations for IBVPs on bounded domains.

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.

Estimating parameters of the diffusion model via asymptotic expansions

TL;DR

This work develops a unified framework to estimate diffusion-model parameters on a bounded domain from a single measurement by converting the forward diffusion problem, solved with the Fokas method, into an integral equation . It then derives comprehensive asymptotic expansions for in both large- and small- regimes, and couples them with inversion techniques (Lagrange–Bürmann, perturbative refinements, erfc-based inversions) to obtain practical, closed-form approximations for the inverse problem variables. The authors assemble these into composite, uniformly accurate formulas that cover the full data range with controllable relative error, and validate them through drainage and soil-moisture applications, showing markedly improved accuracy over classical approaches. The results offer a tractable pathway for real-time parameter estimation in hydrological contexts and underscore the value of integral-equation reformulations for IBVPs on bounded domains.

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.

Paper Structure

This paper contains 14 sections, 68 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction and Problem formulation
  2. Estimating drain spacing from water table observations
  3. Determining soil properties from moisture data
  4. Asymptotic approximation of the solution
  5. Integral estimation for large values of $a$
  6. Series inversion
  7. Further approximations
  8. Integral estimation for small $a>0$
  9. Main Results
  10. Numerical experiments and physical interpretation
  11. Drain spacing
  12. Parameter estimation
  13. Proof of \ref{['eq_K']}
  14. Proof of \ref{['a-sol-small-2']}

Figures (6)

  • Figure 1: The values of $c:$ real (red) and reconstructed (blue) with $a$ given by \ref{['eq:solve_a_large']}. The right plot is a zoomed-in view of the left.
  • Figure 2: The values of $c:$ real (red) and reconstructed with $a$ given by \ref{['sol_fourierN']} for different values of $N$. The right plot is a zoomed-in view of the left.
  • Figure 3: The values of $c:$ real (red) and reconstructed (blue) with $a$ given by \ref{['a-sol-2nd']}. The right plot is a zoomed-in view of the left.
  • Figure 4: The values of $c:$ real (red) and reconstructed (blue) with $a$ given by \ref{['sol_erf']} (left) and by \ref{['a-sol-small-1']} (right).
  • Figure 5: Left: The real (red) and the reconstructed (blue) $c$ values with $a$ given by \ref{['a-sol-small-2']}. Right: The relative errors using \ref{['a-sol-small-1']} (blue) and \ref{['a-sol-small-2']} (red).
  • ...and 1 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2