Journal of Guangdong University of Technology ›› 2019, Vol. 36 ›› Issue (03): 74-79.doi: 10.12052/gdutxb.180122

Previous Articles     Next Articles

Finite Difference Schemes for Time Fractional Diffusion Equations with Periodic Boundary Conditions

Zhang Hui-qin, Wang Zhi-bo   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2018-09-20 Online:2019-05-09 Published:2019-04-04

HTML

PDF (PC)

2838

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 L2-1σ 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.

Key words: fractional diffusion equation, Fourier method, variable coefficients, stability, convergence

CLC Number: 

  • 35R11
[1] NIGMATULLIN R R. To the theoretical explanation of the "universal response"[J]. Physica Status Solidi, 1984, 123(2):739-745
[2] 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
[3] 梅立泉, 李瑞, 李智. 三元期权定价问题的偏微分方程数值解[J]. 西安交通大学学报, 2006, 40(4):484-487 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
[4] 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
[5] CAIN G, MEYER G H. Separation of variables for partial differential equations[M]. New York:Chapman and Hall, 2006.
[6] PODLUBNY I. Fractional differential equations[M]. San Diego:Academic Press, 1999.
[7] SUN Z, WU X. A fully discrete difference scheme for a diffusion-wave system[J]. Applied Numerical Mathematics, 2006, 56(2):193-209
[8] LIN Y, XU C. Finite difference/spectral approximations for the time-fractional diffusion equation[J]. Journal of Computational Physics, 2007, 225(2):1533-1552
[9] CUI M. Compact finite difference method for the fractional diffusion equation[J]. Journal of Comp-utational Physics, 2009, 228(20):7792-7804
[10] 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
[11] 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
[12] 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
[13] 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
[14] 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
[15] ALIKHANOV A A. A new difference scheme for the time fractional diffusion equation[J]. Journal of Computational Physics, 2015, 280(C):424-438
[16] 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
[17] 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
[18] 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
[19] 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
[20] 杜瑞连, 梁宗旗. 右侧Caputo分数阶导数的L2-1插值逼近[J]. 集美大学学报(自然科学版), 2017, 22(4):68-74 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
[1] Wu Yi, Mei Ke-qi, Ding Shi-hong, Ge Qun-hui, Wang Wei-zhi. An Extended State Observer Based Continuous Twisting Control for PMSM Speed Regulation [J]. Journal of Guangdong University of Technology, 2022, 39(05): 112-119.
[2] Cen Da-kang, Wang Zhi-bo. A Variational Iteration Method for Fractional Predator-Prey Model [J]. Journal of Guangdong University of Technology, 2022, 39(02): 62-65.
[3] Liu Xue, Liu Zhong-ming, Xi Yue-bin, Wang Shou-juan, Kong Fan-gong. A Study of Preparation and Application Performance of Lignin-based Superhydrophobic Coatings [J]. Journal of Guangdong University of Technology, 2022, 39(01): 34-40,134.
[4] Tang Hao-yi, Peng Hong-yun. Global Weak Solutions and Asymptotics of a 3D Chemotaxis System [J]. Journal of Guangdong University of Technology, 2022, 39(01): 93-98.
[5] Chen Ci, Xie Li-hua. A Data-Driven Prescribed Convergence Rate Design for Robust Tracking of Discrete-Time Systems [J]. Journal of Guangdong University of Technology, 2021, 38(06): 29-34.
[6] Zhou Min-hua, Zhang Wen-jin, Pan Chun-yang. Preparation and Properties of CsPbBr3 Quantum Dots Doped with Gallium Ions [J]. Journal of Guangdong University of Technology, 2021, 38(02): 83-87.
[7] Huang Hui-min, Guo Cheng-jun. Stability of Solutions for a Class of Impulsive Stochastic Differential Equations [J]. Journal of Guangdong University of Technology, 2020, 37(06): 56-62.
[8] Liang Shi-hua, Chen Jun-tao, Lin Huan-sheng, Feng De-luan, Gong Xing, Luo Qing-zi. An Experimental Study of Using Cement Stabilized Silt as Filling Material [J]. Journal of Guangdong University of Technology, 2020, 37(02): 102-106.
[9] Liu Chen-lin, Zheng San-qiang, Han Xiao-zhuo. A Spatio-temporal Dynamic Analysis of a Predation-competition System with Allee Effect [J]. Journal of Guangdong University of Technology, 2019, 36(06): 38-44.
[10] Wang Xiang, Liu Li-fan, Wang Zhi-hong, Qi Wen-guang. Quality Stability in Long-distance Water Conveyance after Sudden Pollution of Quality Surface Water [J]. Journal of Guangdong University of Technology, 2019, 36(03): 99-102,110.
[11] Liu Ming-wei, Yu Zhi-tao, He Shao-hua. A Stability Analysis of Steel Tube Concrete Arch Bridge Based on Unified Theory [J]. Journal of Guangdong University of Technology, 2018, 35(06): 63-68.
[12] Liu Ya-dong, Yang Xue-qiang, Xu Lei, Lin Yao-kang, Gong Xing. A Stability Analysis of Slope Considering Strength Anisotropy in Soils [J]. Journal of Guangdong University of Technology, 2018, 35(06): 57-62.
[13] Zhuang Xiao-lan, Wang Qi. Dynamics of Euler Method for the First Order Delay Differential Equation [J]. Journal of Guangdong University of Technology, 2018, 35(01): 46-49.
[14] Liu Yi, Zhang Yun. Convergence Condition of Value-iteration Based Adaptive Dynamic Programming [J]. Journal of Guangdong University of Technology, 2017, 34(05): 10-14.
[15] Zhang Zhi-zhong, Peng Shi-guo. Input-to-state Stability for Stochastic Neural Networks with Time-Varying Delay [J]. Journal of Guangdong University of Technology, 2017, 34(04): 84-88.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright © 2017 Journal of Guangdong University of Technology
Add：729 East Dongfeng RD., Guangzhou PRC. Postcode: 510090
Tel：020-37626653,020-37626139,020-37626396
Email：xbzrb@gdut.edu.cn
Supported:Beijing Magtech