Inverse source problem for coupled time-fractional diffusion equations: Stability and Bayesian ensemble Kalman reconstruction

This paper investigates the inverse problem of simultaneously identifying the spatial components of source terms in a system of coupled time-fractional diffusion equations. The forward model comprises N coupled equations involving Caputo fractional derivatives of order 0 < α < 1, with spatially varying diffusion and coupling coefficients. Using a series representation of the solution, we establish the uniqueness of the inverse problem and derive a conditional stability estimate, extending classical results known for single-equation cases to the coupled system setting. To address the inherent ill-posedness and account for measurement uncertainty, we adopt a Bayesian inference framework, under which we rigorously prove the well-posedness of the posterior distribution. For efficient numerical reconstruction, we employ the iterative regularizing ensemble Kalman method, which provides an accurate approximation of the Bayesian solution while incorporating a built-in regularization mechanism. Numerical experiments demonstrate the accuracy, stability, and robustness of the proposed approach in recovering spatially distributed sources from noisy final-time data.