Numerical Approximation of Stochastic Time-Fractional Diffusion

2018/10/03 by Jin, Bangti, Yan, Yubin, Zhou, Zhi
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1810.01822

Abstract

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order α∈(0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ∈[0,1] in the front). The numerical scheme approximates the model in space by the Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method, and the noise by the L²-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the deterministic counterpart. Numerical results are presented to support the theoretical findings.

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