Strong approximation of the time-fractional Cahn–Hilliard equation driven by a fractionally integrated additive noise

In this article, we present a numerical scheme for solving a time-fractional stochastic Cahn–Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order α∈(0,1) and a fractional time-integral noise of order γ∈[0,1]. Our numerical approach combines a piecewise linear finite element method in space with a convolution quadrature in time, designed to handle both time-fractional operators, along with the L2-projection for the noise. We conduct a detailed analysis of both spatially semidiscrete and fully discrete schemes, deriving strong convergence rates through energy-based arguments. The solution's temporal Hölder continuity played a key role in the error analysis. Unlike the stochastic Allen–Cahn equation, the inclusion of the unbounded elliptic operator in front of the cubic nonlinearity in our model added complexity and challenges to the error analysis. To address these challenges, we introduce novel techniques and refined error estimates. We conclude with numerical examples that validate our theoretical findings.