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A DufortFrankel Difference Scheme for TwoDimensional SineGordon EquationLiang, Zongqi; Yan, Yubin; Cai, Guorong; University of Chester (Hindawi Publishing Corporation, 20141029)A standard CrankNicolson finitedifference scheme and a DufortFrankel finitedifference scheme are introduced to solve twodimensional damped and undamped sineGordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictorcorrector techniques are applied in the numerical methods. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Error estimates of a continuous Galerkin time stepping method for subdiffusion problemYan, Yubin; Yan, Yuyuan; Liang, Zongqi; Egwu, Bernard; Jimei University; University of Chester (Springer, 20210729)A continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and Ltype methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

HighOrder Numerical Methods for Solving Time Fractional Partial Differential EquationsLi, Zhiqiang; Liang, Zongqi; Yan, Yubin; Luliang University, P. R. China, Jimei University, P. R. China, University of Chester, UK (Springer Link, 20161115)In this paper we introduce a new numerical method for solving time fractional partial differential equation. The time discretization is based on Diethelm’s method where the Hadamard finitepart integral is approximated by using the piecewise quadratic interpolation polynomials. The space discretization is based on the standard finite element method. The error estimates with the convergence order O(τ^(3−α) +h^2 ),0

A highorder scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equationDu, Ruilian; Yan, Yubin; Liang, Zongqi; Jimei University; University of Chester (Elsevier, 20181005)A new highorder finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k1}) \big ), k=1, 2, \dots, N, $ with the convergence order $O(\Delta t^{4\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Optimal convergence rates for semidiscrete finite element approximations of linear spacefractional partial differential equations under minimal regularity assumptionsLiu, Fang; Liang, Zongqi; Yan, Yubin; Luliang University; Jimei University; University of Chester (Elsevier, 20181217)We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear spacefractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear spacefractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear spacefractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.