Numerical Approximation of Stochastic Fractional Differential Equations

Stochastic fractional differential equations have gained significant attention in recent years because of their potential applications in various fields of science and engineering. These equations have proven to be efficient in describing complex phenomena with long memory processes. The focus of this project is to analyze and develop numerical methods for solving stochastic time-fractional diffusion equations that are driven by additive fractionally integrated Gaussian noise. The model includes two nonlocal time terms: a Caputo fractional derivative and a Riemann-Liouville fractional integral operator applied to a Gaussian noise. Our approach will involve discretizing the model using a standard piecewise linear finite element method in space and a convolution quadrature method in time for both time-fractional operators. We will investigate spatially semi-discrete and fully discrete schemes and establish optimal error estimates for solutions with smooth and non-smooth data by exploiting the temporal Holder continuity property of the exact solution. Our methodology will rely on a semi-group type approach. Furthermore, extensive numerical tests will be conducted to confirm and illustrate the convergence results.