Reconstruction of the Temporal Component in the Source Term of a (Time-Fractional) Diffusion Equation

In this article, we consider the reconstruction of $\rho (t)$ in the (time-fractional) diffusion equation $({\mathrm{\partial }}_t^{\alpha }-\mathrm{△})u(x,t)=\rho (t)g(x)$ ($0<\alpha \le 1$) by the observation at a single point $x_0$. We are mainly concerned with the situation of $x_0\mathrm{\notin }$ supp g, which is practically important but has not been well investigated in literature. Assuming the finite sign changes of $\rho$ 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.