Some second-order 𝜃 schemes combined with finite element method for nonlinear fractional cable equation

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.

[1]  Hong Li,et al.  A new fully discrete finite difference/element approximation for fractional cable equation , 2016 .

[2]  Yingjun Jiang,et al.  Moving finite element methods for time fractional partial differential equations , 2013 .

[3]  Bangti Jin,et al.  The Galerkin finite element method for a multi-term time-fractional diffusion equation , 2014, J. Comput. Phys..

[4]  Meng Li,et al.  Galerkin finite element method for nonlinear fractional Schrödinger equations , 2017, Numerical Algorithms.

[5]  S. Wearne,et al.  Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions , 2009, Journal of mathematical biology.

[6]  Anatoly A. Alikhanov,et al.  A new difference scheme for the time fractional diffusion equation , 2014, J. Comput. Phys..

[7]  Yang Liu,et al.  A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative , 2015, Comput. Math. Appl..

[8]  Cui-Cui Ji,et al.  A High-Order Compact Finite Difference Scheme for the Fractional Sub-diffusion Equation , 2014, Journal of Scientific Computing.

[9]  Hong Li,et al.  A two-grid finite element approximation for a nonlinear time-fractional Cable equation , 2015, 1512.08082.

[10]  Bo Yu,et al.  Numerical Identification of the Fractional Derivatives in the Two-Dimensional Fractional Cable Equation , 2016, J. Sci. Comput..

[11]  Yangquan Chen,et al.  Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion , 2011, Comput. Math. Appl..

[12]  Changpin Li,et al.  A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application , 2017 .

[13]  Fawang Liu,et al.  The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation , 2013, SIAM J. Sci. Comput..

[14]  Fawang Liu,et al.  Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations , 2016, Appl. Math. Lett..

[15]  M. Zaky,et al.  Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation , 2014, Nonlinear Dynamics.

[16]  Yang Liu,et al.  Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation , 2017, J. Comput. Phys..

[17]  Yang Liu,et al.  Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation , 2016 .

[18]  K. Burrage,et al.  A new fractional finite volume method for solving the fractional diffusion equation , 2014 .

[19]  Zhi-Zhong Sun,et al.  Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence , 2015, J. Comput. Phys..

[20]  Santos B. Yuste,et al.  A finite difference method with non-uniform timesteps for fractional diffusion equations , 2011, Comput. Phys. Commun..

[21]  Danping Yang,et al.  A Petrov–Galerkin finite element method for variable-coefficient fractional diffusion equations , 2015 .

[22]  Zhibo Wang,et al.  Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation , 2013, J. Comput. Phys..

[23]  Xianjuan Li,et al.  Finite difference/spectral approximations for the fractional cable equation , 2010, Math. Comput..

[24]  Yunqing Huang,et al.  Developing Finite Element Methods for Maxwell's Equations in a Cole-Cole Dispersive Medium , 2011, SIAM J. Sci. Comput..

[25]  Han Zhou,et al.  A class of second order difference approximations for solving space fractional diffusion equations , 2012, Math. Comput..

[26]  Jiye Yang,et al.  Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations , 2015, J. Comput. Phys..

[27]  Yang Liu,et al.  High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation , 2017, Comput. Math. Appl..

[28]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[29]  Yang Liu,et al.  Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem , 2015, Comput. Math. Appl..

[30]  Fawang Liu,et al.  Galerkin finite element method and error analysis for the fractional cable equation , 2015, Numerical Algorithms.

[31]  Hong Sun,et al.  Some temporal second order difference schemes for fractional wave equations , 2016 .

[32]  Fawang Liu,et al.  Finite element method for space-time fractional diffusion equation , 2015, Numerical Algorithms.

[33]  Santos B. Yuste,et al.  Corrigendum to "A finite difference method with non-uniform timesteps for fractional diffusion equations" [Computer Physics Communications 183 (12) (2012) 2594-2600] , 2014, Comput. Phys. Commun..

[34]  Fawang Liu,et al.  Galerkin finite element approximation of symmetric space-fractional partial differential equations , 2010, Appl. Math. Comput..