Finite Element Method Combined With Second-order Time Approximation Scheme For Partial Fractional Differential Equations

In this article,two finite element methods combined with second-order time ap-proximation scheme for studying nonlinear time fractional Cable equation and nonlin-car spacce-time fractional differcential equation numerically.For fractional Cable equa-tion,a weighted discrete scheme covering second-order accuracy is used to approximate the Riemann-Liouville fractional derivative,then form the second-order discrete scheme,where the approximate order is higher than the obtained results by the Ll-approximation with order(2-?)in general.For space-time fractional differential equation.the dis-crete idea of WSGD operator combining Galerkin finite element method is used to deal it numerically,and the second-order accuracy can be arrived.Firstly at time tk-?/2,a second-order two step scheme with ?-parameter is proposed to approximate the first-order derivative for nonlinear time fractional Cable equation.By applying a weighted discrete scheme covering second-order accuracy to approximate the Riemann-Liouville fractional derivative,and Galerkin finite element method is em-ployed in spatial direction,then form the finite element method with second-order time approximation numerically.Moreover,the numerical theory is provided,and the sta-bility of the scheme is demonstrated,and the rate of convergence in L2-norm with O(?t2 +(1+?t-?)hm+1)is arrived at the same time.And some numerical tests are shown to support our theoretical results.Secondly,a second-order WSGD approximation combined with finite element numeri-cal method to solve space-time fractional differential equation,and the detailed calculation is analysed.And we provide three type of numerical examples to verify the effectiveness of two order approximation accuracy.