Recovering an unknown source in a fractional diffusion problem

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Rundell , W & Zhang , Z 2018 , ' Recovering an unknown source in a fractional diffusion problem ' , Journal of Computational Physics , vol. 368 , pp. 299-314 . https://doi.org/10.1016/j.jcp.2018.04.046

Title: Recovering an unknown source in a fractional diffusion problem
Author: Rundell, William; Zhang, Zhidong
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2018-09-01
Language: eng
Number of pages: 16
Belongs to series: Journal of Computational Physics
ISSN: 0021-9991
URI: http://hdl.handle.net/10138/314608
Abstract: Abstract A standard inverse problem is to determine a source which is supported in an unknown domain D from external boundary measurements. Here we consider the case of a time-independent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω and the boundary of D has the star-like shape, i.e. ∂ D = { q ( θ ) ( cos ⁡ θ , sin ⁡ θ ) ⊤ : θ ∈ [ 0 , 2 π ] } . Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary ∂Ω. The case of a parabolic equation was considered in [6]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter α, the exponent of the fractional operator which controls the degree of anomalous behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.
Subject: Fractional diffusion equation
Inverse problem
Uniqueness
Unknown discontinuous source
Newton's method
Tikhonov regularization
EQUATIONS
114 Physical sciences
111 Mathematics
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