TIME-SPLITTING SCHEMES FOR FRACTIONAL DIFFERENTIAL EQUATIONS I: SMOOTH SOLUTIONS

被引：28

作者：

Cao, Wanrong ^{[1
]}

Zhang, Zhongqiang ^{[2
]}

Karniadakis, George Em ^{[3
]}

机构：

[1] Southeast Univ, Dept Math, Nanjing 210096, Jiangsu, Peoples R China

[2] Worcester Polytech Inst, Dept Math Sci, Worcester, MA 01609 USA

[3] Brown Univ, Div Appl Math, Providence, RI 02912 USA

来源：

SIAM JOURNAL ON SCIENTIFIC COMPUTING | 2015年 / 37卷 / 04期

关键词：

time-fractional derivatives; multirate systems; stiff systems; PREDICTOR-CORRECTOR APPROACH; VOLTERRA INTEGRAL-EQUATIONS; COMPACT ADI SCHEME; DIFFUSION EQUATION; INTEGRODIFFERENTIAL EQUATIONS; SUBDIFFUSION EQUATION; MULTISTEP METHODS; ERROR ANALYSIS; 2ND KIND; STABILITY;

D O I：

10.1137/140996495

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We propose three time-splitting schemes for nonlinear time-fractional differential equations with smooth solutions, where the order of the fractional derivative is 0 < alpha < 1. While one of the schemes is of order a, the other two schemes are of order 1 + alpha and 2 - alpha and thus they can be combined to provide flexible numerical methods with convergence order no less than 3/2. We prove the convergence and stability of the proposed schemes. Numerical examples illustrate the flexibility and the efficiency of these time-splitting schemes and show that they work for multirate and stiff time-fractional differential systems effectively.

引用

页码：A1752 / A1776

页数：25

共 44 条

[1] The heterogeneous multiscale method [J].

Abdulle, Assyr ;

Weinan, E. ;

Engquist, Bjoern ;

Vanden-Eijnden, Eric .

ACTA NUMERICA, 2012, 21 :1-87

[2]

[Anonymous], 2010, LECT NOTES MATH

[3]

[Anonymous], 2007, FRACT CALC APPL ANAL

[4]

[Anonymous], 1999, FRACTIONAL DIFFERENT

[5] Numerical simulation for the three-dimension fractional sub-diffusion equation [J].

Chen, J. ;

Liu, F. ;

Liu, Q. ;

Chen, X. ;

Anh, V. ;

Turner, I. ;

Burrage, K. .

APPLIED MATHEMATICAL MODELLING, 2014, 38 (15-16) :3695-3705

[6] Natural Volterra Runge-Kutta methods [J].

Conte, Dajana ;

D'Ambrosio, Raffaele ;

Izzo, Giuseppe ;

Jackiewicz, Zdzislaw .

NUMERICAL ALGORITHMS, 2014, 65 (03) :421-445

[7] A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces [J].

Cuesta, E ;

Palencia, C .

APPLIED NUMERICAL MATHEMATICS, 2003, 45 (2-3) :139-159

[8] Compact finite difference method for the fractional diffusion equation [J].

Cui, Mingrong .

JOURNAL OF COMPUTATIONAL PHYSICS, 2009, 228 (20) :7792-7804

[9] A new predictor-corrector method for fractional differential equations [J].

Daftardar-Gejji, Varsha ;

Sukale, Yogita ;

Bhalekar, Sachin .

APPLIED MATHEMATICS AND COMPUTATION, 2014, 244 :158-182

[10] Short memory principle and a predictor-corrector approach for fractional differential equations [J].

Deng, Weihua .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2007, 206 (01) :174-188

← 1 2 3 4 5 →