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Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation

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Abstract.

In this article, a Galerkin finite element method combined with second-order time discrete scheme for finding the numerical solution of nonlinear time fractional Cable equation is studied and discussed. At time \( t_{k-\frac{\alpha}{2}}\) , a second-order two step scheme with \( \alpha\) -parameter is proposed to approximate the first-order derivative, and a weighted discrete scheme covering second-order approximation is used to approximate the Riemann-Liouville fractional derivative, where the approximate order is higher than the obtained results by the L1-approximation with order ( \( 2-\alpha\) in the existing references. For the spatial direction, Galerkin finite element approximation is presented. The stability of scheme and the rate of convergence in \( L^2\) -norm with \( O(\Delta t^2+(1+\Delta t^{-\alpha})h^{m+1})\) are derived in detail. Moreover, some numerical tests are shown to support our theoretical results.

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, 010021, Hohhot, China

    Yajun Wang, Yang Liu, Hong Li & Jinfeng Wang

  2. School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, 010070, Hohhot, China

    Jinfeng Wang

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  1. Yajun Wang
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  2. Yang Liu
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  3. Hong Li
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  4. Jinfeng Wang
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Correspondence to Yang Liu.

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Wang, Y., Liu, Y., Li, H. et al. Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation. Eur. Phys. J. Plus 131, 61 (2016). https://doi.org/10.1140/epjp/i2016-16061-3

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