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Simultaneous uniqueness for a coefficient inverse problem in one-dimensional fractional diffusion equation from an interior point measurement

Xiaohua Jing, Zhiyuan Li, Masahiro Yamamoto

Abstract

This article is concerned with an inverse problem of simultaneously determining a spatially varying coefficient and a Robin coefficient for a one-dimensional fractional diffusion equation with a time-fractional derivative of order $α\in(0,1)$. We prove the uniqueness for the inverse problem by observation data at one interior point over a finite time interval, provided that a coefficient is known on a subinterval. Our proof is based on the uniqueness in the inverse spectracl problem for a Sturm-Liouville problem by means of the Weyl $m$-function and the spectral representation of the solution to an initial-boundary value problem for the fractional diffusion equation.

Simultaneous uniqueness for a coefficient inverse problem in one-dimensional fractional diffusion equation from an interior point measurement

Abstract

This article is concerned with an inverse problem of simultaneously determining a spatially varying coefficient and a Robin coefficient for a one-dimensional fractional diffusion equation with a time-fractional derivative of order . We prove the uniqueness for the inverse problem by observation data at one interior point over a finite time interval, provided that a coefficient is known on a subinterval. Our proof is based on the uniqueness in the inverse spectracl problem for a Sturm-Liouville problem by means of the Weyl -function and the spectral representation of the solution to an initial-boundary value problem for the fractional diffusion equation.

Paper Structure

This paper contains 7 sections, 9 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Main results and outline
  3. Preliminaries
  4. Auxiliary functions
  5. Solution representation of forward problem
  6. Proof of Theorems 1 and 2
  7. Conclusions

Key Result

Theorem 1

Let $d\in (0,1)$ and $x_0 \in (0,1)$ be given. Let $u_j(x, t)$ be the solutions of eq-gov' with $(q_j,h_j,H) \in\mathcal{A}$, $j=1, 2$. We assume that $\eta\in H_{\alpha}(0,T)$ does not vanish identically and $q_{1}(x)=q_{2}(x)$, $x\in (d,1)$. Suppose that Then $u_{1}(x_{0},t) = u_{2}(x_{0},t)$, $0<t<T$, implies

Theorems & Definitions (17)

  • Theorem 1
  • Remark 2.1
  • Theorem 2
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 7 more