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Subjectsfinite element method (2)Caputo fractional derivative (1)Dufort-Frankel difference scheme (1)error estimates (1)Finite difference method (1)finite difference method (1)fractional diffusion wave equation (1)optimal convergence (1)SIne-Gordon equation (1)space fractional derivative (1)View MoreJournalDiscrete Dynamics in Nature and Society (1)Journal of Computational and Applied Mathematics (1)Journal of Computational Physics (1)Journal of Scientific Computing (1)Authors

Liang, Zongqi (4)

Yan, Yubin (4)

Cai, Guorong (1)Du, Ruilian (1)Li, Zhiqiang (1)Liu, Fang (1)TypesArticle (4)

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A high-order scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equation

Du, Ruilian; Yan, Yubin; Liang, Zongqi (Elsevier, 2018-10-05)

A new high-order 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_{k-1}) \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.

A Dufort-Frankel Difference Scheme for Two-Dimensional Sine-Gordon Equation

Liang, Zongqi; Yan, Yubin; Cai, Guorong (Hindawi Publishing Corporation, 2014-10)

A standard Crank-Nicolson finite-difference scheme and a Dufort-Frankel finite-difference scheme are introduced to solve two-dimensional damped and undamped sine-Gordon equations. The stability and convergence of the numerical methods are considered. To avoid solving the nonlinear system, the predictor-corrector techniques are applied in the numerical methods. 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 space-fractional partial differential equations under minimal regularity assumptions

Liu, Fang; Liang, Zongqi; Yan, Yubin (Elsevier, 2018-12-17)

We consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear space-fractional 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 space-fractional 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 space-fractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

Li, Zhiqiang; Liang, Zongqi; Yan, Yubin (Springer Link, 2016-11-15)

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 finite-part 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

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