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Error Analysis for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise

Author

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  • Xiaolei Wu

    (Department of Mathematics and Artificial Intelligence, Lvliang University, Lvliang 033000, China
    These authors contributed equally to this work.)

  • Yubin Yan

    (School of Computer and Engineering Sciences, University of Chester, Chester CH1 4BJ, UK
    These authors contributed equally to this work.)

Abstract

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter H ∈ ( 0 , 1 ) . The covariance operator Q of the stochastic fractional Wiener process satisfies ∥ A − ρ Q 1 / 2 ∥ H S < ∞ for some ρ ∈ [ 0 , 1 ) , where ∥ · ∥ H S denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings.

Suggested Citation

  • Xiaolei Wu & Yubin Yan, 2024. "Error Analysis for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise," Mathematics, MDPI, vol. 12(22), pages 1-28, November.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:22:p:3579-:d:1522049
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    References listed on IDEAS

    as
    1. Mijena, Jebessa B. & Nane, Erkan, 2015. "Space–time fractional stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3301-3326.
    2. Chen, Zhen-Qing & Kim, Kyeong-Hun & Kim, Panki, 2015. "Fractional time stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1470-1499.
    3. Chen, Le & Hu, Yaozhong & Nualart, David, 2019. "Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5073-5112.
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