Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unique Determination of Variable Order in Subdiffusion from a Single Measurement

Jiho Hong, Bangti Jin, Yavar Kian

Abstract

We study the inverse problem of recovering a spatially dependent variable order in a time-fractional diffusion model from the boundary flux measurement generated by a single boundary excitation. It arises in the identification of heterogeneous media in anomalous diffusion processes. In this work, we establish several new uniqueness results for the inverse problem in the case of piecewise constant variable orders, without any monotonicity condition. The analysis follows a new approach that combines properties of harmonic functions, a linearization technique in the Laplace domain, and tools from complex, asymptotic, and geometrical analysis. In addition, we weaken the regularity assumptions on the problem data and extend the analysis of previous contributions to higher-dimensional settings.

Unique Determination of Variable Order in Subdiffusion from a Single Measurement

Abstract

We study the inverse problem of recovering a spatially dependent variable order in a time-fractional diffusion model from the boundary flux measurement generated by a single boundary excitation. It arises in the identification of heterogeneous media in anomalous diffusion processes. In this work, we establish several new uniqueness results for the inverse problem in the case of piecewise constant variable orders, without any monotonicity condition. The analysis follows a new approach that combines properties of harmonic functions, a linearization technique in the Laplace domain, and tools from complex, asymptotic, and geometrical analysis. In addition, we weaken the regularity assumptions on the problem data and extend the analysis of previous contributions to higher-dimensional settings.
Paper Structure (13 sections, 25 theorems, 137 equations, 3 figures)

This paper contains 13 sections, 25 theorems, 137 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results and discussions
  3. Spherical inclusions
  4. Polygonal inclusions
  5. Preliminary results
  6. Spherical inclusions
  7. Proof of Theorem \ref{['theorem:balls']}
  8. Proofs of Lemmas \ref{['lemma:Bessel']} and \ref{['lemma:main:linindep:general']}
  9. Polygonal inclusions
  10. Proof of Theorem \ref{['theorem:simplex']}
  11. Proof of Theorem \ref{['theorem:2D:convex']}
  12. Rectangular inclusions
  13. Multiple polygonal inclusions

Key Result

Theorem 2.1

Let Assumption ass:g:exptype hold. Let $U=U^i$ be the solution to eq:2D3D:ibvp with $\alpha=\alpha^i$ for $i=1,2$ satisfying Assumption ass:alpha. If $\partial_\nu U^1(t,x)=\partial_\nu U^2(t,x)$ for all $(t,x)\in I\times\partial\Omega$, then we have $\alpha^1=\alpha^2$.

Figures (3)

  • Figure 2.1: Examples of $\alpha$ defined on $\Omega=B_1(0)$ with (a) $\alpha_{\rm in}$, (b) and (c) candidate $\alpha$ profiles satisfying the requirements of Theorems \ref{['theorem:balls']} and \ref{['theorem:2D:convex']}, respectively.
  • Figure 2.2: A schematic illustration of Assumption \ref{['ass:alpha:simplices']} on the variable order $\alpha$.
  • Figure 2.3: The differences $\alpha^1-\alpha^i$ ($i=2,3$) in Fig. \ref{['fig:examples']}. The pair $(\alpha^1,\alpha^2)$ satisfies Assumption \ref{['ass:alpha:simplices']}, whereas the pair $(\alpha^1,\alpha^3)$ violates Assumption \ref{['ass:alpha:simplices']}. In (c), the regions $\operatorname{supp}(\alpha^1-\alpha^2)$ and $\operatorname{supp}(\alpha^1-\alpha^3)$ are shaded in green, and the blue and red line segments indicate the necessary and optional choices for the decomposition by nondegenerate simplices, respectively.

Theorems & Definitions (56)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.1
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 46 more