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Numerical approximation of stochastic time-fractional diffusion

Jin, B; Yan, Y; Zhou, Z; (2019) Numerical approximation of stochastic time-fractional diffusion. ESAIM: Mathematical Modelling and Numerical Analysis , 53 (4) pp. 1245-1265. 10.1051/m2an/2019025. Green open access

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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 standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Gr¨unwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the L 2 -projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.

Type: Article
Title: Numerical approximation of stochastic time-fractional diffusion
Open access status: An open access version is available from UCL Discovery
DOI: 10.1051/m2an/2019025
Publisher version: https://doi.org/10.1051/m2an/2019025
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10070473
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