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A fast linearized numerical method for nonlinear time-fractional diffusion equations

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Abstract

In this paper, we study a fast linearized numerical method for solving nonlinear time-fractional diffusion equations. A new weighted method is proposed to construct linearized approximation, which enables the unconditional convergence to be established when the nonlinearity f(u) is only locally Lipschitz continuous. In order to reduce the computational cost, the sum-of-exponentials (SOE) technique is employed to evaluate the kernel function in the Caputo derivative. By using the complementary discrete kernels for the coefficients of the refined fast weighted discretization, the proposed method is shown to be unconditionally convergent with respect to the discrete H1-norm. The fast linearized method can also be extended to nonlinear multi-term and distributed-order time-fractional diffusion equations. Numerical examples with different types of nonlinear functions are provided to demonstrate the behavior of proposed methods for both smooth and weakly singular solutions.

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Funding

The first author is supported by the Fundamental Research Funds for the Central Universities (JBK2001002, JBK200535). The second author is funded by The Science and Technology Development Fund, Macau SAR (File no. 0005/2019/A) and is funded by University of Macau (File no. MYRG2018-00047-FST, MYRG2017-00098-FST).

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Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China

    Pin Lyu

  2. Department of Mathematics, University of Macau, Macao, China

    Seakweng Vong

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  1. Pin Lyu
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  2. Seakweng Vong
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Correspondence to Seakweng Vong.

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Lyu, P., Vong, S. A fast linearized numerical method for nonlinear time-fractional diffusion equations. Numer Algor 87, 381–408 (2021). https://doi.org/10.1007/s11075-020-00971-0

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  • DOI: https://doi.org/10.1007/s11075-020-00971-0

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