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Space-Fractional Diffusion Equation with Variable Coefficients: Well-posedness and Fourier Pseudospectral Approximation

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Abstract

Multi-dimensional space-fractional diffusion equation with variable coefficients and fractional gradient is a difficult problem in theory and computation. As far as we know, there rarely exist well-posedness results and efficient numerical approaches for such equation. In this paper, we focus on this subject. First, we apply the commutator estimation method to prove the coercivity of the non-positive bilinear form for such equation in both continuous sense and discrete sense, and this is key for the later discussion. Then, we prove the well-posedness of the analytical solution and give the global error estimation of the numerical solution obtained by Crank–Nicolson Fourier pseudospectral scheme. Last, the numerical experiments are used to verify the main results of the theoretical analysis, and a model for the plume of solute through groundwater is exhibited to show the application of space-fractional diffusion theory.

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All data generated or analysed during this study are included in this published article.

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Authors and Affiliations

  1. School of Mathematics and Computational Science and Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, 411105, Hunan, China

    Xue-Yang Li & Ai-Guo Xiao

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  1. Xue-Yang Li
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  2. Ai-Guo Xiao
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Correspondence to Ai-Guo Xiao.

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This research is supported by the National Natural Science Foundation of China (No. 12071403), the Foundation of Hunan Educational Committee (No. 20B566), and the Scientific Research Fund of Science and Technology Department of Hunan Province in China (No. 2018WK4006).

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Li, XY., Xiao, AG. Space-Fractional Diffusion Equation with Variable Coefficients: Well-posedness and Fourier Pseudospectral Approximation. J Sci Comput 87, 28 (2021). https://doi.org/10.1007/s10915-021-01435-w

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  • DOI: https://doi.org/10.1007/s10915-021-01435-w

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