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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]