On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:500

Article

Kaltenbacher, Barbara¹  Rundell, William² 

[1] Alpen Adria Univ Klagenfurt, Dept Math, Klagenfurt, Austria

[2] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA

关键词: Inverse problem;    Nonlinear diffusion;    Reconstruction algorithms;   

DOI  :  10.1016/j.jmaa.2021.125145

来源: Elsevier

【 摘 要 】

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as the diffusion coefficient ain u(t) - del(a del u) = f. In this paper we seek the unknown aassuming that a = a(u) depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of a(u): the value of a time trace u(x(0), t) for some fixed point x(0) on the boundary of the region Omega; or the value of uon an interior curve Sigma lying within Omega. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of Omega taken at a fixed time T> 0. In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of a. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper. (c) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free