Recovering an unknown source in a fractional diffusion problem

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-dependent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω. 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 [5]. 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 behavior of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.

On the Identification of Source Term in the Heat Equation from Sparse Data

“…In experiment (e2), a discontinuous, star-like supported exact solution p(x) is considered, where the radius function is r(θ ) = 0.5+0.2 cos 2θ . We can see this p is out of Assumption 2.1, so the iteration (17) may not be appropriate here and in fact, we use the Levenberg-Marquardt algorithm to recover the radius function r(θ ), see [11] for details. The numerical results are presented in Figures 3, 4 and 5, in which the blue dotted line and the red dashed line mean the boundaries of supp(p) and supp(p j ), respectively, and the black bullets are the locations of observation points.…”

Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

“…For diffusion equations corresponding to the case α = 1 with time independent source terms, several authors investigated the conditional stability (e.g. [5,37,38] [15,16,17,19,23,31] where some inverse coefficient problems and some related results have been considered.…”

Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

“…, which will be used to construct the discretized scheme for equation (1). For the discretization on time, the L 1 -stepping scheme is used, which can be seen in [23,51]. Set the discrete time mesh 0 = t 0 < t 1 < • • • < t N = T and denote the time step size as ∆t = T /N .…”