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.

本文研究了随机时间分数式热扩散方程，涉及阶数为\(\alpha \in (\frac{1}{2},1]\)的时间 Caputo 导数，由随机扩散系数场和分数阶同时驱动首先，通过证明温和解的存在性、唯一性和稳定性来确定基础问题的适定性，然后构造基于米尔斯坦指数积分器方案和有限元方法的时空离散化方法。并进行了分析，最终得出了全离散解的强收敛速度，以证实理论结果。