The MFS for the forward and inverse problems of the time-fractional diffusion equation

The method of fundamental solutions for the time-fractional diffusion equation is developed, with emphasis on both the forward simulation and the reconstruction of a space-dependent source from terminal measurement data. The objective is to provide a meshless approach that can handle the nonlocal temporal dependence of the model while remaining applicable to irregular domains. For the forward problem, a collocation scheme based on the fundamental solution is employed. For the inverse problem, the source reconstruction is reformulated as a homogeneous fractional diffusion problem with an additional terminal time constraint, and the resulting ill-conditioned linear system is stabilized through regularization. Numerical experiments on both circular and non-circular domains show that the proposed method produces accurate forward solutions and successfully recovers the main spatial features of the source under small to moderate noise levels. The results further illustrate the effects of source point placement, terminal time, and regularization parameters on the reconstruction quality. The proposed framework provides a simple meshless alternative to conventional domain-based discretization methods and offers a practical foundation for inverse source problems governed by time-fractional diffusion models.