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

Abstract

In this work, an inverse problem in the fractional diffusion equation with random source is considered. Statistical moments are used of the realizations of single point observation u(x 0 , t, ω). We build the representation of the solution u in integral sense, then prove some theoretical results as uniqueness and stability. After that, we establish a numerical algorithm to solve the unknowns, where the mollification method is used.

Recovery of the time-dependent source term in the stochastic fractional diffusion equation with heterogeneous medium

“…Our data is the moments of the realizations of u on a single point x 0 ∈ D with the restriction x 0 / ∈ supp(f ), which is different from [31]. This condition will make the inverse problem more challenging in mathematics, but is meaningful in practical application.…”

“…Remark 1. In [31], the authors use integration by parts on the right side of ( 7) to deduce the following second kind Volterra equations,…”