An inverse time-dependent source problem for a time-fractional diffusion equation
type: Item
instance of: scholarly article
title: An inverse time-dependent source problem for a time-fractional diffusion equation
zbMATH Open document ID: 1351.65072
DOI: 10.1088/0266-5611/32/8/085003
published in: Inverse Problems
publication date: 2016-09-09T00:00:00Z
Mathematics Subject Classification ID: 65M32
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 6625824
zbMATH Keywords: inverse source problem
zbMATH Keywords: fractional diffusion equation
zbMATH Keywords: uniqueness
zbMATH Keywords: stability
zbMATH Keywords: Tikhonov regularization method
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1088/0266-5611/32/8/085003
OpenAlex ID: W2420651851
author: Ting Wei
author: X. L. Li
author: Y. S. Li
review text: Let \(\Omega\) be a bounded domain in \(\mathbb R^d\), \(d\geq 1\) with sufficiently smooth boundary \(\partial \Omega\). The authors consider the time-fractional diffusion equation \(\partial^{\alpha}_{0+} u(x,t)=\Delta u(x,t)+f(x) p(t)\), \(x\in \Omega\), \(t\in (0,T]\); \(\alpha\in (0,1)\). Here, \(\partial^{\alpha}_{0+}\) denotes the Caputo fractional left derivative of order \(\alpha\). The initial and boundary conditions are \(u(x,0)=\psi(x)\), \(x\in \Omega\) and \(\partial u/\partial n (x,t)=0\), \(x\in \partial \Omega\), \(t\in (0,T]\), and \(n\) is the unit outward normal to \(\partial \Omega\). The inverse problem under investigation is to determine the source term \(p(t)\) from the additional lateral data \(u(x,t)=g(x,t)\), \(x\in \Gamma\), \(t\in (0,T]\), where \(\Gamma\) is a nonempty part of \(\partial \Omega\). The authors establish the uniqueness and the stability estimate for the inverse problem. Numerical examples with the Tikhonov regularization method in 1D and 2D problems are discussed.
reviewed by: Mikhail Yu. Kokurin
description: scientific article; zbMATH DE number 6625824