This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. Based on the expression of the solution for the direct problem, the inverse problem for searching the space source term is converted into solving the first kind of Fredholm integral equation. The conditional stability for the inverse source problem is investigated. The fractional Landweber method is used to deal with this inverse problem and the regularized solution is also obtained. Furthermore, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Several numerical examples are given to show the proposed method is efficient and stable.

本文致力于解决一个反问题，该问题用于识别非局部边界上带有分数拉普拉斯算子的时间分数阶非齐次扩散方程的源项。根据直接问题解的表达式，将搜索空间源项的反问题转换为求解第一类Fredholm积分方程。研究了反源问题的条件稳定性。使用分数Landweber方法处理该反问题，并获得正则解。此外，可以通过使用先验的正则化参数选择规则和后验参数选择规则来证明正则化解的收敛速度。数值算例表明了该方法的有效性和稳定性。