 首页 |  期刊介绍 |  编委会 |  投稿指南 |  期刊订阅 |  下载中心 |  留言板 |  联系我们 |  重点论文 |  在线办公 |
 计算数学 2016, Vol. 38 Issue (1): 1-24    DOI:
 综述 最新目录 | 下期目录 | 过刊浏览 | 高级检索 |  Next Articles THEORETICAL AND NUMERICAL INVESTIGATION OF FRACTIONAL DIFFERENTIAL EQUATIONS
Lin Shimin, Xu Chuanju
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientic Computing, Xiamen University, Xiamen 361005, Fujian, China
 全文: PDF (5784 KB)   HTML (1 KB)   输出: BibTeX | EndNote (RIS)      背景资料

 服务 把本文推荐给朋友 加入我的书架 加入引用管理器 E-mail Alert RSS 作者相关文章

Abstract： The study of the fractional differential equations has a very long history, and is attracting increasing attention in recent years. As compared to the very limit theoretical work, signi cant progress has been made on numerical investigations. Several research groups have contributed to this progress. This paper has the objective to review the recent progress made in the theoretical and numerical studies of the fractional differential equations. We particularly focus on the development of high order numerical methods. The main content of the paper is to discuss the progress made in recent ten years on theoretical and numerical investigation of the three basic fractional equations: time fractional di usion equation, space fractional di usion equation, and time-space fractional di usion equation. We also provide some illustrative numerical examples to verify the accuracy and effciency of some selected numerical methods.

 引用本文: . 分数阶微分方程的理论和数值方法研究[J]. 计算数学, 2016, 38(1): 1-24. . THEORETICAL AND NUMERICAL INVESTIGATION OF FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Mathematica Numerica Sinica, 2016, 38(1): 1-24.

  Agrawal O. Formulation of Euler-Lagrange equations for fractional variational problems[J]. J. Math. Anal. Appl., 2002, 272:368-379. Agrawal O. A general formulation and solution scheme for fractional optimal control problems[J]. Nonlinear. Dynam., 2004, 38:191-206. Benson D, Schumer R, Meerschaert M, Wheatcraft S. Fractional dispersion, Lévy motion, and the MADE tracer tests[J]. Transp. Por. Media., 2001, 42(1/2):211-240. Benson D, Wheatcraft S, Meerschaert M. Application of a fractional advection-dispersion equation[J]. Water. Resour. Res., 2006, 36:1403-1412.  Benson D, Wheatcraft S, Meerschaert M. The fractional-order governing equation of Lévy motion[J]. Water. Resour. Res., 2006, 36:1413-1423.  Ga ychuk V, Datsko B, Meleshko V. Mathematical modeling of time fractional reactiondi usion systems[J]. J. Math. Anal. Appl., 2008, 220(1-2):215-225.  Gorenflo R, Mainardi F, Scalas E, Raberto M. Fractional calculus and continuous-time nance Ⅲ:the di usion limit[G]. In Mathematical nance, pages 171-180. Springer, 2001.  Koeller R. Applcation of fractional calculus to the theory of viscoelasticity[J]. J. Appl. Mech., 1984, 51:229-307. Kusnezov D, Bulgac A, Dang G. Quantum levy processes and fractional kinetics[J]. Phys. Rev. Lett., 1999, 82:1136-1139. Meerschaert M, Scalas E. Coupled continuous time random walks in nance[J]. Phys. A., 2006, 390:114-118.  Podlubny I. Fractional di erential equations[M]. Academic Press, 1998.  Raberto M, Scalas E, Mainardi F. Waiting-times and returns in high-frequency nanical data:An empirical study[J]. Phys. A., 2002, 314:749-755. Carlson G E. Investigation of fractional capacitor approximation by means of regular Newton processes[D]. PhD thesis, 1964.  Sugimoto N. Burgers equation with a fractional derivative; hereditary effects on nonlinear acoustic waves[J]. J. Fluid. Mech., 1991, 225:631-653. Oustaloup A, Coi et P. Syst emes asservis lin eaires d'ordre fractionnaire:th eorie et pratique[M]. Masson, 1983.  Oustaloup A, Mathieu B. La commande CRONE[M]. Hermes Science Publishing Paris, 1999.  Mainardi F. Fractional calculus:some basic problems in countinuum and statistical mechanics, 291-348[J]. Fract. Fract. Calc. Continuum. Mech., 378.  Rossikhin Y A, Shitikova M V. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids[J]. Appl. Mech. Rev., 1997, 50(1):15-67. Ichise M, Nagayanagi Y, Kojima T. An analog simulation of non-integer order transfer functions for analysis of electrode processes[J]. J. Electroanal. Chem., 1971, 33(2):253-265. Sun H, Abdelwahab A, Onaral B. Linear approximation of transfer function with a pole of fractional power[J]. Ieee. T. Automat. Contr., 1984, 29(5):441-444. Mainardi F. Fractional relaxation-oscillation and fractional di usion-wave phenomena[J]. Chaos. Soliton. Fract., 1996, 7(9):1461-1477. Deng W. Generalized synchronization in fractional order systems[J]. Phys. Rev. E., 2007, 75(5):056201. Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional di usion equation[J]. J. Comput. Phys., 2007, 225(2):1533-1552. Müller H P, Kimmich R, Weis J. NMR ow velocity mapping in random percolation model objects:Evidence for a power-law dependence of the volume-averaged velocity on the probevolume radius[J]. Phys. Rev. E., 1996, 54:5278-5285. Amblard F, Maggs A C, Yurke B, Pargellis A N, Leibler S. Subdi usion and anomalous Local viscoelasticity in actin networks[J]. Phys. Rev. Lett., 1996, 77:4470. Mainardi F. Fractional di usive waves in viscoelastic solids[J]. Nonlinear. Waves. Solids., 1995, pages 93-97.  Hughes B D. Random walks and random environments[J]. 1996.  Gu Q, Schi E, Grebner S, Wang F, Schwarz R. Non-Gaussian transport measurements and the Einstein relation in amorphous silicon[J]. Phys. Rev. Lett., 1996, 76(17):3196. Klemm A, Müller H P, Kimmich R. NMR microscopy of pore-space backbones in rock, sponge, and sand in comparison with random percolation model objects[J]. Phys. Rev. E., 1997, 55(4):4413. Klammler F, Kimmich R. Geometrical Restrictions of Incoherent Transport of Water by Di usion in Protein of Silica Fineparticle Systems and by Flow in a Sponge. A Study of Anomalous Properties Using an NMR Field-Gradient Technique[J]. Croat. Chem. Acta., 1992, 65(2):455-470.  Weber H W, Kimmich R. Anomalous segment di usion in polymers and nmr relaxation spectroscopy[J]. Macromolecules., 1993, 26(10):2597-2606. Porto M, Bunde A, Havlin S, Roman H E. Structural and dynamical properties of the percolation backbone in two and three dimensions[J]. Phys. Rev. E., 1997, 56(2):1667. Luedtke W, Landman U. Slip di usion and Levy ights of an adsorbed gold nanocluster[J]. Phys. Rev. Lett., 1999, 82(19):3835. Shlesinger M, West B, Klafter J. Lévy dynamics of enhanced di usion:Application to turbulence[J]. Phys. Rev. Lett., 1987, 58(11):1100. Bychuk O V, O'Shaughnessy B. Anomalous di usion at liquid surfaces[J]. Phys. Rev. Lett., 1995, 74(10):1795. Klafter J, Blumen A, Zumofen G, Shlesinger M. Lévy walk approach to anomalous di usion[J]. Physica. A., 1990, 168(1):637-645.  Schau er S, Schleich W, Yakovlev V. Scaling and asymptotic laws in subrecoil laser cooling[J]. Epl-Europhys. Lett., 1997, 39(4):383. Zumofen G, Klafter J. Spectral random walk of a single molecule[J]. Chem. Phys. Lett., 1994, 219(3):303-309. Diethelm K, Ford N J. Analysis of fractional di erential equations[J]. J. Math. Anal. Appl., 2002, 265(2):229-248. Ca arelli L, Vasseur A. Drift di usion equations with fractional di usion and the quasi-geostrophic equation[J]. Ann. Math., 2010, 171(3):1903-1930. Kilbas A A A, Srivastava H M, Trujillo J J. Theory and applications of fractional di erential equations[M], volume 204. Elsevier Science Limited, 2006.  Li X, Xu C. Existence and uniqueness of the weak solution of the space-time fractional di usion equation and a spectral method approximation[J]. Commun. Comput. Phys., 2010, 8(5):1016-1051.  Ervin V J, Roop J P. Variational formulation for the stationary fractional advection dispersion equation[J]. Numer. Meth. Part. D. E., 2006, 22(3):558-576. Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional di usion-wave equations and applications to some inverse problems[J]. J. Math. Anal. Appl., 2011, 382(1):426-447. Lubich C. Discretized fractional calculus[J]. Siam. J. Math. Anal., 1986, 17(3):704-719. Diethelm K, Walz G. Numerical solution of fractional order di erential equations by extrapolation[J]. Numer. Algorithms., 1997, 16(3-4):231-253. Diethelm K, Ford N J, Freed A D. A predictor-corrector approach for the numerical solution of fractional di erential equations[J]. Nonlinear. Dynam., 2002, 29(1-4):3-22. Diethelm K, Ford N J, Freed A D. Detailed error analysis for a fractional Adams method[J]. Numer. Algorithms., 2004, 36(1):31-52. Liu F, Shen S, Anh V, Turner I. Analysis of a discrete non-Markovian random walk approximation for the time fractional di usion equation[J]. Anziam. J., 2005, 46:488-504.  Langlands T, Henry B. The accuracy and stability of an implicit solution method for the fractional di usion equation[J]. J. Comput. Phys., 2005, 205(2):719-736. Zhang Y, Sun Z. Alternating direction implicit schemes for the two-dimensional fractional subdi usion equation[J]. J. Comput. Phys., 2011, 230(24):8713-8728. Wang H, Wang K. An O(N log2 N) alternating-direction nite difference method for twodimensional fractional di usion equations[J]. J. Comput. Phys., 2011.  Sun H, Chen W, Li C, Chen Y. Finite difference schemes for variable-order time fractional di usion equation[J]. Int. J. Bifurcat. Chaos., 2012, 22(04):1250085. Li X, Xu C. A space-time spectral method for the time fractional di usion equation.[J]. Siam. J. Numer. Anal., 2009, 47(3):2108-2131. Lin Y, Li X, Xu C. Finite difference/Spectral approximations for the fractional cable equation[J]. Math. Comput., 2011, 80(275):1369-1396.  Lv C, Xu C. Improved error estimates of a nite difference/spectral method for time-fractional di usion equations[J]. Int. J. Numer. Anal. Mod., 2015, 12(2):384-400.  Ford N J, Simpson A C. The numerical solution of fractional di erential equations:speed versus accuracy[J]. Numer. Algorithms., 2001, 26(4):333-346. Diethelm K, Freed A D. An effcient algorithm for the evaluation of convolution integrals[J]. Comput. Math. Appl., 2006, 51(1):51-72. Cao J, Xu C,Wang Z. A high order nite difference/spectral approximations to the time fractional di usion equations[J]. Adv. Mater. Res., 2014, 875:781-785.  Lv C, Xu C. Error analysis of a high order method for time-fractional di usion equations[J]. submitted, 2015.  Cao J, Xu C. A high order schema for the numerical solution of the fractional ordinary di erential equations[J]. J. Comput. Phys., 2013, 238(2):154-168. Meerschaert M M, Tadjeran C. Finite difference approximations for fractional advection-dispersion ow equations[J]. J. Comput. Appl. Math., 2004, 172(1):65-77. Meerschaert M M, Tadjeran C. Finite difference approximations for two-sided space-fractional partial di erential equations[J]. Appl. Numer. Math., 2006, 56(1):80-90. Liu F, Anh V, Turner I. Numerical solution of the space fractional Fokker-Planck equation[J]. J. Comput. Appl. Math., 2004, 166:209-219. Jin B, Lazarov R, Zhou Z. Error estimates for a semidiscrete nite element method for fractional order parabolic equations[J]. Siam. J. Numer. Anal., 2013, 51(1):445-466. Meerschaert M, Sche er H, Tadjeran C. Finite difference methods for two dimensional fractional dispersion equation[J]. J. Comput. Phys., 2006, 211:249-261. Tadjeran C, Meerschaert M. A second-order accurate numerical method for the two-dimensional di usion equation[J]. J. Comput. Phys., 2007, 220:813-823. Sousa E. Numerical approximations for fractional di usion equations via splines[J]. Comput. Math. Appl., 2011, 62:938-944. Liu F, Zhuang P, Anh V, Turner I, Burrage K. Stability and convergence of the difference methods for the space-time fractional advection-di usion equation[J]. Appl. Math. Comput., 2007, 191(1):12-20. Yang Q, Liu F, Turner I. Numerical methods for fractional partial di erential equations with Riesz space fractional derivatives[J]. Appl. Math. Model., 2010, 34(1):200-218. Deng W. Finite element method for the space and time fractional Fokker-Planck equation[J]. Siam. J. Numer. Anal., 2008, 47:204-226.  Wang H, Du N. Fast alternating-direction nite difference methods for three-dimensional spacefractional di usion equations[J]. J. Comput. Phys., 2013, 258:305-318.  Song F, Xu C. Spectral direction splitting methods for two-dimensional space fractional di usion equations[J]. J. Comput. Phys., 2015, 299:196-214. Chen S, Shen J, Wang L. Generalized Jacobi functions and their applications to fractional di erential equations[J]. arXiv preprint arXiv:1407.8303, 2014.  Zayernouri M, Karniadakis G E. Fractional Sturm-Liouville eigen-problems:theory and numerical approximation[J]. J. Comput. Phys., 2013, 252:495-517. Zayernouri M, Karniadakis G E. Exponentially accurate spectral and spectral element methods for fractional ODEs[J]. J. Comput. Phys., 2014, 257:460-480. 隆璐帆, 李晓, 张辉. 一维分子束外延方程线性部分求解及数值模拟[J]. 计算数学, 2015, 36(3): 225-233.  单炜琨, 李会元. 任意三角形Laplace特征值问题谱方法的数值对比研究[J]. 计算数学, 2015, 36(2): 113-131.  王俊杰, 王连堂. 一类二次KdV类型水波方程的多辛Fourier拟谱方法[J]. 计算数学, 2014, 35(4): 241-254.  王自强, 曹俊英. 时间分数阶扩散方程的一个新的高阶数值格式[J]. 计算数学, 2014, 35(4): 277-288.  席钧, 曹建文. 美式期权定价的分数阶偏微分方程组及其数值离散方法[J]. 计算数学, 2014, 35(3): 229-240.  乔海军, 李会元. 二维各向同性湍流直接数值模拟的六边形谱方法及其GPU实现和优化[J]. 计算数学, 2013, 34(2): 147-160.  王文强, 李东方. 线性变系数中立型变延迟微分方程谱方法的收敛性[J]. 计算数学, 2012, 34(1): 68-80.  高建芳, 张艳英, 唐黎明. 种群动力系统的数值解的振动性分析[J]. 计算数学, 2011, 33(4): 357-366.  周天孝. 弹塑性力学的典则变分原理——特此纪念冯康先生九十周年诞辰[J]. 计算数学, 2011, 32(1): 1-7.  周婷, 向新民. 无界域上半线性强阻尼波动方程的全离散有理谱逼近[J]. 计算数学, 2009, 31(4): 335-348.  张稳, 马和平. 线性---二次型最优控制问题的Chebyshev--Legendre拟谱方法[J]. 计算数学, 2009, 30(2): 100-112.  刘霖雯,刘超,江成顺,. 一类非线性伪抛物型方程的伪谱解法[J]. 计算数学, 2007, 29(1): 99-12.  郑华盛,赵宁,成娟. 一维高精度离散GDQ方法[J]. 计算数学, 2004, 26(3): 293-302.  徐阳,赵景军,刘明珠. 二阶延迟微分方程θ-方法的TH-稳定性[J]. 计算数学, 2004, 26(2): 189-192.  叶兴德,程晓良. Cahn—Hilliard方程的Legendre谱逼近[J]. 计算数学, 2003, 25(2): 157-170.
 Copyright 2008 计算数学 版权所有 中国科学院数学与系统科学研究院 《计算数学》编辑部 北京2719信箱 (100190) Email: gxy@lsec.cc.ac.cn 本系统由北京玛格泰克科技发展有限公司设计开发 技术支持: 010-62662699 E-mail:support@magtech.com.cn 京ICP备05002806号-10