带周期边界的时间分数阶扩散方程的差分格式

张会琴; 汪志波

doi:10.12052/gdutxb.180122

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带周期边界的时间分数阶扩散方程的差分格式

综合研究 | 更新时间：2026-05-27

- 带周期边界的时间分数阶扩散方程的差分格式
- Finite Difference Schemes for Time Fractional Diffusion Equations with Periodic Boundary Conditions
- 广东工业大学学报 2019年36卷第3期页码：74-79
- 作者机构：
  
  广东工业大学　应用数学学院,广东,广州,510520
- 作者简介：
  
  张会琴(1992–)，女，硕士研究生，主要研究方向为微分方程数值解
  汪志波(1987–)，男，副教授，主要研究方向为微分方程数值解. E-mail：wzbmath@gdut.edu.cn
- 基金信息：
  
  国家自然科学基金资助项目(11701103);广东省自然科学基金资助项目(2017A030310538);广东省青年拔尖人才项目(2017GC010379);广东省高校特色创新项目(2017KTSCX062);广东工业大学基金资助项目(220413131，220413550)
- DOI：10.12052/gdutxb.180122
  中图分类号： 35R11;65M06;65M12;65M15
- 网络首发：2019-05-09，
  
  纸质出版：2019
- 稿件说明：
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张会琴, 汪志波. 带周期边界的时间分数阶扩散方程的差分格式[J]. 广东工业大学学报, 2019, 36(3): 74-79.

Zhang Hui-qin, Wang Zhi-bo. Finite Difference Schemes for Time Fractional Diffusion Equations with Periodic Boundary Conditions[J]. 2019, 36(3): 74-79.
张会琴, 汪志波. 带周期边界的时间分数阶扩散方程的差分格式[J]. 广东工业大学学报, 2019, 36(3): 74-79. DOI： 10.12052/gdutxb.180122.

Zhang Hui-qin, Wang Zhi-bo. Finite Difference Schemes for Time Fractional Diffusion Equations with Periodic Boundary Conditions[J]. 2019, 36(3): 74-79. DOI： 10.12052/gdutxb.180122.

摘要

鉴于分数阶方程的解析解实难求得

本文主要研究了带周期边界的时间分数阶扩散方程的有限差分方法

时间方向采用

2-1

离散公式

空间方向采用二阶差分格式离散

数值格式整体可达到二阶精度. 随后利用Fourier方法证明了有限差分格式的唯一可解性、稳定性和收敛性. 最后用 MATLAB语言对具体的模型进行了数值求解

数值实验能很好地印证理论结果.

Abstract

Generally

the analytic solution for fractional differential equations is hard to obtain. The difference scheme of time fractional diffusion equations with periodic boundaries is mainly studied

adopting the

2-1

formula in temporal direction

and the second order difference approximation in spatial direction. The proposed scheme can obtain second order accuracy globally. The unique solvability

stability and convergence of the proposed scheme are proved by the Fourier method. Finally

the specific fractional models are solved numerically by MATLAB language. Numerical experiments are carried out to support the theoretical results.

关键词

Keywords

references

NIGMATULLIN R R. To the theoretical explanation of the "universal response" [J]. Physica Status Solidi, 1984, 123(2): 739-745.

GORENFLO R, MAINARDI F, SCALAS E, et al . Fractional calculus and continuous-time finance III: the diffusion limit [J ] . Physica A Statistical Mechanics Its Applications, 2000, 284(1-4): 376-384.

MEI L Q, LI R, LI Z. Numerical solutions of partial differential equations for trivariate option pricing [J]. Journal of Xi'an Jiaotong University, 2006, 40(4): 484-487.

ZHANG H, LIU F, PHANIKUMAR M S, et al . A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model [J ] . Computers Mathematics with Applications, 2013, 66(5): 693-701.

CAIN G, MEYER G H. Separation of variables for partial differential equations[M]. New York: Chapman and Hall, 2006.

PODLUBNY I. Fractional differential equations[M]. San Diego: Academic Press, 1999.

SUN Z, WU X. A fully discrete difference scheme for a diffusion-wave system [J]. Applied Numerical Mathematics, 2006, 56(2): 193-209.

LIN Y, XU C. Finite difference/spectral approximations for the time-fractional diffusion equation [J]. Journal of Computational Physics, 2007, 225(2): 1533-1552.

CUI M. Compact finite difference method for the fractional diffusion equation [J]. Journal of Comp-utational Physics, 2009, 228(20): 7792-7804.

REN J, SUN Z, ZHAO X. Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions [J]. Journal of Computational Physics, 2013, 232(1): 456-467.

WANG Z, VONG S. A compact difference scheme for a two dimensional nonlinear fractional Klein-Gordon equation in polar coordinates [J]. Computers Mathematics with Applications, 2016, 71(12): 2524-2540.

FENG Q, MENG F. Finite difference scheme with spatial fourth-order accuracy for a class of time fractional parabolic equations with variable coefficient [J]. Advances in Difference Equations, 2016, 2016(1): 305.

YUSTE S B, ACEDO L. An explicit finite difference method and a new von neumann-type stability analysis for fractional diffusion equations [J]. SIAM Journal on Numerical Analysis, 2005, 42(5): 1862-1874.

ZHUANG P, LIU F, ANH V, et al . New solution and analytical techniques of the implicit numerical method for the anomalous sub-diffusion equation [J ] . SIAM Journal on Numerical Analysis, 2008, 46(2): 1079-1095.

ALIKHANOV A A. A new difference scheme for the time fractional diffusion equation [J]. Journal of Computational Physics, 2015, 280(C): 424-438.

VONG S, LYU P, CHEN X, et al . High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives [J ] . Numerical Algorithms, 2016, 72(1): 195-210.

ZHAO X, XU Q. Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient [J]. Applied Mathematical Modelling, 2014, 38(15-16): 3848-3859.

VONG S, WANG Z. A high order compact finite difference scheme for time fractional Fokker-Planck equations [J]. Applied Mathematics Letters, 2015, 43: 38-43.

CUI M. Compact exponential scheme for the time fractional convection-diffusion reaction equation with variable coefficients [J]. Journal of Computational Physics, 2015, 280(2): 143-163.

DU R L, LIANG Z Q. Interpolation approximation of the right side of the caputo fractional derivative [J]. Journal of Jimei University (Natural Science), 2017, 22(4): 68-74.

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