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.

中文翻译：

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由分数阶积分加性噪声驱动的时间分数 Cahn-Hilliard 方程的强近似


在本文中，我们提出了一种数值方案，用于求解由加法分数阶积分高斯噪声驱动的时间分数随机 Cahn-Hilliard 方程。该模型涉及 α∈ 阶时间的 Caputo 分数阶导数 （0,1） 和 γ∈ 阶 [0,1] 的分数时间积分噪声。我们的数值方法将空间中的分段线性有限元方法与时间卷积正交相结合，旨在处理时间分数算子以及噪声的 L2 投影。我们对空间半离散和完全离散方案进行了详细分析，通过基于能量的论点得出了强大的收敛率。该解决方案的时间 Hölder 连续性在误差分析中起着关键作用。与随机 Allen-Cahn 方程不同，在我们的模型中，在三次非线性之前包含无界椭圆算子增加了误差分析的复杂性和挑战。为了应对这些挑战，我们引入了新技术和改进的误差估计。我们以验证我们的理论发现的数值示例作为结论。