Reconstruction of the Temporal Component in the Source Term of a (Time-Fractional) Diffusion Equation
This addresses a challenging inverse problem for mathematicians working on time-fractional diffusion equations, but the stability result is only logarithmic and the method is incremental.
The authors tackle the reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation from a single-point observation, achieving multiple logarithmic stability under finite sign changes and an extra observation interval, with a convergent fixed-point iteration demonstrated numerically.
In this article, we consider the reconstruction of $ρ(t)$ in the (time-fractional) diffusion equation $(\partial_t^α-\triangle)u(x,t)=ρ(t)g(x)$ ($0<α\le 1$) by the observation at a single point $x_0$. We are mainly concerned with the situation of $x_0 \notin$ supp g, which is practically important but has not been well investigated in literature. Assuming the finite sign changes of $ρ$ and an extra observation interval, we establish the multiple logarithmic stability for the problem based on a reverse convolution inequality and a lower estimate for positive solutions. Meanwhile, we develop a fixed point iteration for the numerical reconstruction and prove its convergence. The performance of the proposed method is illustrated by several numerical examples.