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 (partial derivative(alpha)(t) - triangle)u(x, t) = rho(t)g(x) (0 < alpha <= 1) by observation at a single point x(0). We are mainly concerned with the situation of x(0) is not an element of supp g, which is practically important but has not been well investigated in literature. Assuming finite sign changes of. 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.