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

Yajun Wang; Yang Liu; Hong Li; Jinfeng Wang

doi:10.1140/epjp/i2016-16061-3

EPJ

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

Yajun Wang¹, Yang Liu¹^*, Hong Li¹ and Jinfeng Wang¹^,2

¹ School of Mathematical Sciences, Inner Mongolia University, 010021, Hohhot, China
² School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, 010070, Hohhot, China

Received: 24 November 2015
Accepted: 25 January 2016
Published online: 21 March 2016

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.

15th European Conference on Atoms Molecules and Photons (ECAMP)
June 30 to July 4, 2025
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EPJ

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

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