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

## TL;DR

This paper develops a stable method for reconstructing a time-dependent source term in a time-fractional diffusion equation from single-point observations, especially when the observation point is outside the source support.

## Contribution

It introduces a novel stability analysis and a convergent fixed point iteration for numerical reconstruction of the source term in fractional diffusion equations with limited observation data.

## Key findings

- Established multiple logarithmic stability for the inverse problem.
- Developed a convergent fixed point iterative reconstruction method.
- Validated the approach with numerical examples.

## Abstract

In this article, we consider the reconstruction of $\rho(t)$ in the (time-fractional) diffusion equation $(\partial_t^\alpha-\triangle)u(x,t)=\rho(t)g(x)$ ($0<\alpha \le 1$) by the observation at a single point $x_0$. We are mainly concerned with the situation of $x_0 \notin$ 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.