An efficient numerical method to the stochastic fractional heat equation with random coefficients and fractionally integrated multiplicative noise

This paper studies the stochastic time-fractional heat diffusion equation involving a Caputo derivative in time of order \(\alpha \in (\frac{1}{2},1]\), driven simultaneously by a random diffusion coefficient field and fractionally integrated multiplicative noise. First, the well-posedness of the underlying problem is established by proving the existence, uniqueness, and stability of the mild solution. Then a spatio-temporal discretization method based on a Milstein exponential integrator scheme and finite element method is constructed and analyzed. The strong convergence rate of the fully discrete solution is derived. Numerical experiments are finally reported to confirm the theoretical result.