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The Regularized Solution Approximation of Forward/Backward Problems for a Fractional Pseudo-Parabolic Equation with Random Noise

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Abstract

This paper deals with the forward and backward problems for the nonlinear fractional pseudo-parabolic equation \({u_t} + {\left({- \Delta} \right)^{{s_1}}}{u_t} + \beta {\left({- \Delta} \right)^{{s_2}}}u = F\left({u,x,t} \right)\) subject to random Gaussian white noise for initial and final data. Under the suitable assumptions s1, s2 and β, we first show the ill-posedness of mild solutions for forward and backward problems in the sense of Hadamard, which are mainly driven by random noise. Moreover, we propose the Fourier truncation method for stabilizing the above ill-posed problems. We derive an error estimate between the exact solution and its regularized solution in an \(\mathbb{E}\,\,\left\| {\, \cdot \,} \right\|_{{H^{{s_2}}}}^2\) norm, and give some numerical examples illustrating the effect of above method.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Huafei Di & Weijie Rong

  2. Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006, China

    Huafei Di

Authors
  1. Huafei Di
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  2. Weijie Rong
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Correspondence to Huafei Di.

Additional information

This research is supported by the Natural Science Foundation of China (11801108), the Natural Science Foundation of Guangdong Province (2021A1515010314), and the Science and Technology Planning Project of Guangzhou City (202201010111).

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Di, H., Rong, W. The Regularized Solution Approximation of Forward/Backward Problems for a Fractional Pseudo-Parabolic Equation with Random Noise. Acta Math Sci 43, 324–348 (2023). https://doi.org/10.1007/s10473-023-0118-3

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  • DOI: https://doi.org/10.1007/s10473-023-0118-3

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