Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation
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In this article, some second-order time discrete schemes covering parameter 𝜃 combined with Galerkin finite element (FE) method are proposed and analyzed for looking for the numerical solution of nonlinear cable equation with time fractional derivative. At time tk−𝜃, some second-order 𝜃 schemes combined with weighted and shifted Grünwald difference (WSGD) approximation of fractional derivative are considered to approximate the time direction, and the Galerkin FE method is used to discretize the space direction. The stability of second-order 𝜃 schemes is derived and the second-order time convergence rate in L2-norm is proved. Finally, some numerical calculations are implemented to indicate the feasibility and effectiveness for our schemes.
KeywordsSecond-order 𝜃 scheme Nonlinear fractional cable equation Finite element algorithm Stability Error estimates
Mathematics Subject Classification (2010)65M60 65N15 65N30
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The authors thank the reviewers and editor very much for their insightful comments for improving our work.
This work is supported by the National Natural Science Fund (11661058,11761053,11772046), Australian Research Council (ARC) via the Discovery Project (DP180103858) and Natural Science Fund of Inner Mongolia Autonomous Region (2016MS0102, 2017MS0107).
- 10.Ding, H.F., Li, C.P.: A novel second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and application. arXiv:1605.02177 (2016)