AN UNDETERMINED TIME-DEPENDENT COEFFICIENT IN A FRACTIONAL DIFFUSION EQUATION

摘要

In this work, we consider a FDE (fractional diffusion equation) ~CD_t~α u(x, t)-a(t)Lu(x, t) = F (x, t) with a time-dependent diffusion coefficient a(t). This is an extension of [13], which deals with this FDE in one-dimensional space. For the direct problem, given an a(t), we establish the existence, uniqueness and some regularity properties with a more general domain Ω and right-hand side F (x, t). For the inverse problem-recovering a(t), we introduce an operator K one of whose fixed points is a(t) and show its monotonicity, uniqueness and existence of its fixed points. With these properties, a reconstruction algorithm for a(t) is created and some numerical results are provided to illustrate the theories.