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Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods

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Abstract

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear time-fractional Tricomi-type equation (TFTTE), which is obtained from the standard one-dimensional linear Tricomi-type equation by replacing the first-order time derivative with a fractional derivative (of order α, with 1 < α ≤ 2). The proposed LDG is based on LDG finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the numerical solution converges to the exact one with order O(h k + 1 + τ 2), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The comparison of the LDG results with the exact solutions is made, numerical experiments reveal that the LDG is very effective.

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Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, People’s Republic of China

    Xindong Zhang & Yinnian He

  2. College of Mathematics Sciences, Xinjiang Normal University, Urumqi, Xinjiang, 830054, People’s Republic of China

    Xindong Zhang & Juan Liu

  3. Center for Computational Geosciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shanxi, 710049, People’s Republic of China

    Juan Wen, Bo Tang & Yinnian He

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  1. Xindong Zhang
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  2. Juan Liu
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  3. Juan Wen
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Correspondence to Xindong Zhang.

Additional information

This work is supported by the Key Project of Chinese Ministry of Education (211202), the NSF of China (Nos. 10971166 and 61163027) and the National High Technology Research and Development Program of China (863 Program, No. 2009AA01A135).

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Zhang, X., Liu, J., Wen, J. et al. Analysis for one-dimensional time-fractional Tricomi-type equations by LDG methods. Numer Algor 63, 143–164 (2013). https://doi.org/10.1007/s11075-012-9617-3

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