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Internal layer solutions and coefficient recovery in time-periodic reaction-diffusion-advection equations

Dmitrii Chaikovskii, Ye Zhang, Aleksei Liubavin

TL;DR

This paper addresses inverse problems for a time-periodic nonlinear reaction-diffusion-advection equation with a small perturbation $\mu$. It develops a periodic matched asymptotics framework yielding inner and outer solutions and a moving front location $x_{t.p.}(t)$, and uses these results to formulate two asymptotic-based inverse methods for recovering either a spatially dependent $f(x)$ or a time-dependent $f(t)$. The authors prove well-posedness and derive convergence rates for the inverse problems, and they validate the approach with two numerical experiments showing accurate coefficient reconstruction under realistic noise levels. The work advances the mathematical understanding of singularly perturbed, non-stationary PDEs with moving internal layers and provides practical strategies for coefficient identification in applied settings.

Abstract

This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation (PDE) within a bounded spatial domain and an infinite temporal domain, subject to periodic temporal boundary conditions. A periodic asymptotic solution featuring an inner transition layer is proposed, advancing the mathematical modeling of reaction-diffusion-advection dynamics. Building on this asymptotic analysis, we develop a simple yet effective numerical algorithm to address ill-posed nonlinear inverse problems aimed at reconstructing coefficient functions that depend solely on spatial or temporal variables. Conditions ensuring the existence and uniqueness of solutions for both forward and inverse problems are established. The proposed method's effectiveness is validated through numerical experiments, demonstrating high accuracy in reconstructing coefficient functions under varying noise conditions.

Internal layer solutions and coefficient recovery in time-periodic reaction-diffusion-advection equations

TL;DR

This paper addresses inverse problems for a time-periodic nonlinear reaction-diffusion-advection equation with a small perturbation . It develops a periodic matched asymptotics framework yielding inner and outer solutions and a moving front location , and uses these results to formulate two asymptotic-based inverse methods for recovering either a spatially dependent or a time-dependent . The authors prove well-posedness and derive convergence rates for the inverse problems, and they validate the approach with two numerical experiments showing accurate coefficient reconstruction under realistic noise levels. The work advances the mathematical understanding of singularly perturbed, non-stationary PDEs with moving internal layers and provides practical strategies for coefficient identification in applied settings.

Abstract

This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation (PDE) within a bounded spatial domain and an infinite temporal domain, subject to periodic temporal boundary conditions. A periodic asymptotic solution featuring an inner transition layer is proposed, advancing the mathematical modeling of reaction-diffusion-advection dynamics. Building on this asymptotic analysis, we develop a simple yet effective numerical algorithm to address ill-posed nonlinear inverse problems aimed at reconstructing coefficient functions that depend solely on spatial or temporal variables. Conditions ensuring the existence and uniqueness of solutions for both forward and inverse problems are established. The proposed method's effectiveness is validated through numerical experiments, demonstrating high accuracy in reconstructing coefficient functions under varying noise conditions.

Paper Structure

This paper contains 19 sections, 10 theorems, 156 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem statement and assumptions
  3. Asymptotic analysis
  4. Asymptotic representation of the solution
  5. Outer functions
  6. The inner functions
  7. Main results
  8. Well-posedness of (DP)
  9. An efficient regularization method for (IP1)
  10. An efficient regularization method for (IP2)
  11. Simulations
  12. First example with coefficient function $f(x)$
  13. Forward problem
  14. Inverse problem
  15. Second example with coefficient function $f(t)$
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that $F(x,t)\in C^{n+3,n+2}$$([0,1], \mathbb{R})$, where $n$ is an order of asymptotic approximation, $u_{init} (x) \in C^{n+3}[0,1]$, $u^{0}(t),u^{1}(t) \in C^{n+2}( \mathbb{R})$, and $\mu \ll 1$. If Assumptions Cond1 - Cond3 are satisfied, then the boundary value problem InitProb has a uni

Figures (5)

  • Figure 1: Comparison of asymptotic and numerical solutions of equation \ref{['ExProb1']}.
  • Figure 2: Comparison of reconstructed coefficient functions for different $\delta$ values.
  • Figure 3: Comparison of asymptotic and numerical solutions of equation \ref{['ExProb2']}.
  • Figure 4: Function $x_{t.p.}$ numerically obtained from equation \ref{['barUatTP']}, representing the location of the transition layer.
  • Figure 5: Comparison of reconstructed coefficient functions for different $\delta$ values.

Theorems & Definitions (14)

  • Definition 1
  • Remark 1
  • Theorem 1: (Existence and asymptotic solution)
  • Theorem 2: (Asymptotic estimates)
  • Remark 2
  • Remark 3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 3
  • Lemma 4.3
  • ...and 4 more