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In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L 2 - 1 σ formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements. [ABSTRACT FROM AUTHOR]

In this paper, a high order finite difference scheme for a two dimensional nonlinear fractional Klein–Gordon equation in polar coordinates is proposed. The difficulty caused by the variable coefficient and the nonlinear term in the equation is carefully handled. Stability and convergence are analyzed using the matrix form of the scheme. Numerical examples are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a high order compact (HOC) scheme for time fractional Fokker–Planck equations with variable convection is constructed. The scheme is studied using its matrix form by the energy method. We find that the difficulty arising from the variable coefficient can be overcome by simple modifications of the coefficient matrices. The scheme is shown to be stable and convergent with order τ 2 − α + h 4 which is higher than some recently studied schemes. Numerical examples are given to justify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical method for time-fractional KdV–Burgers' equation with initial singularity. The famous L 2 - 1 σ formula on graded meshes is adopted to approximate the Caputo derivative. Meanwhile, a nonlinear finite difference method on uniform grids is deduced for spatial discretization. The proposed method is second order in time and first order in space. With the help of the fractional Grönwall inequality, the unconditional stability and convergence of the current scheme are analyzed based on some skills. To raise the accuracy in spatial direction, a second order method is then carefully deduced. At last, theoretical results are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the recently established fast L 2 - 1 σ formula and a weighted approach. Then we apply the discretization to construct a fully fast linearized discrete scheme for the nonlinear problem under consideration. The nonlinear term, which just fulfills the Lipschitz condition, will be evaluated on the previous time level. Therefore only linear systems are needed to be solved for obtaining numerical solutions. The proposed scheme is shown to have second-order unconditional convergence with respect to the discrete H 1 -norm. The proposed fast linearized method can be directly extended to solve the nonlinear distributed-order time-fractional wave problem. Numerical examples are provided to justify the efficiency. [ABSTRACT FROM AUTHOR]