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Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations

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Abstract

Recently, numerous numerical schemes for solving linear time fractional diffusion-wave equations have been developed. However, most of these methods require relatively high smoothness in time and need extensive computational work and large storage due to the nonlocal property of fractional derivatives. In this paper, an efficient scheme and an alternating direction implicit (ADI) scheme are constructed for one-dimensional and two-dimensional nonlinear time fractional diffusion-wave equations based on their equivalent partial integro-differential equations. The proposed methods require weaker smoothness in time compared to the methods based on discretizing fractional derivative directly. They are proved to be unconditionally stable and convergent with first-order of accuracy in time and second order of accuracy in space. Fast implementations of the proposed methods are presented by the sum-of-exponentials (SOE) approximation for the kernel t− 2+α on the interval [τ,T], where 1 < α < 2. Finally, numerical experiments are carried out to illustrate the theoretical results of our direct schemes and demonstrate their powerful computational performances.

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Funding

This research is financially supported by the National Natural Science Foundation of China (Grant Nos. 11701502 and 11426141).

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Authors and Affiliations

  1. College of Mathematical Sciences, Yangzhou University, Yangzhou, 225002, China

    Jianfei Huang

  2. School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, China

    Dandan Yang

  3. Department of Mathematics, 14 MacLean Hall, The University of Iowa, Iowa City, IA, 52242, USA

    Laurent O. Jay

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  1. Jianfei Huang
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  2. Dandan Yang
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  3. Laurent O. Jay
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Correspondence to Dandan Yang.

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Huang, J., Yang, D. & Jay, L.O. Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations. Numer Algor 85, 375–397 (2020). https://doi.org/10.1007/s11075-019-00817-4

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