求解分数阶微分方程问题的几类数值方法
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摘要
近年来，分数阶微分方程问题在现代科学技术领域获得了日益广泛的应用。由于其勿庸置疑的重要性，国内外对于分数阶微分方程问题数值方法的研究正在蓬勃兴起。因此，开展分数阶微分方程数值方法的研究具有重要理论意义和实用价值。
本文分为四个部分。在第二章中给出了求解分数阶微分方程问题的三类数值方法，即分数阶欧拉方法，分数阶BDF方法以及分数阶降阶BDF方法。第三章进行了方法的稳定性分析，获得了方法的稳定域。第四章给出了方法的相容阶。第五章对文中所提出的数值方法进行了数值试验，特别就计算生物学中的两个分数阶微分方程问题进行测试，数值结果表明方法是有效的。文中最后一部分对论文所做的主要工作进行了总结并对今后的工作提出展望。
In recent years, the extensive application of the fractional differential equation in many modern scientific technique realms makes the research of its numerical method in and abroad prospers. Therefore, the research of numerical method of the fractional differential equation is of important theory significance and practical value.
This thesis consists of five chapters. Chapter 2 introduces the linear multi-step method of the fractional differential equation which includes Fractional Order Euler method, Fractional Order BDF method, and Fractional Order decline order BDF method. Chapter 3 A stability analysis that carried on the method, acquired stable area of the method, Chapter 4 give method of consist order. Chapter 5 to text the numerical method put forward carries on the numerical experiment, special two fractional differential equation problem within biologies carries on the test, the number result enunciation method is valid. The article last part we carry on summary's combine to this text to put forward the outlook to the work of the future.
本文分为四个部分。在第二章中给出了求解分数阶微分方程问题的三类数值方法，即分数阶欧拉方法，分数阶BDF方法以及分数阶降阶BDF方法。第三章进行了方法的稳定性分析，获得了方法的稳定域。第四章给出了方法的相容阶。第五章对文中所提出的数值方法进行了数值试验，特别就计算生物学中的两个分数阶微分方程问题进行测试，数值结果表明方法是有效的。文中最后一部分对论文所做的主要工作进行了总结并对今后的工作提出展望。
In recent years, the extensive application of the fractional differential equation in many modern scientific technique realms makes the research of its numerical method in and abroad prospers. Therefore, the research of numerical method of the fractional differential equation is of important theory significance and practical value.
This thesis consists of five chapters. Chapter 2 introduces the linear multi-step method of the fractional differential equation which includes Fractional Order Euler method, Fractional Order BDF method, and Fractional Order decline order BDF method. Chapter 3 A stability analysis that carried on the method, acquired stable area of the method, Chapter 4 give method of consist order. Chapter 5 to text the numerical method put forward carries on the numerical experiment, special two fractional differential equation problem within biologies carries on the test, the number result enunciation method is valid. The article last part we carry on summary's combine to this text to put forward the outlook to the work of the future.
引文
[1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[2] R. Hilfer(ed.) Applications of fractional calculus in physics, World Scientific, Singapore 2000.
[3] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[4] R. Kopelman, Fractal reaction kinetics, Science, 241(1988), 1620-1626.
[5] M. A. savageau, Michaelis-Menten mechanism reconsidered: Implications of fractal kinetics, J. Theoretical Biology, 176(1995) 115-124.
[6] S. B. Yuste and K. Lindenberg, Subdiffusion limited A+A reactions, Phys. Rev. Lett., 87, (2001), 1-4.
[7] S. B. Yuste and K. Lindenberg, Subdiffusion-limited reactions, Chem. Phys., 284, (2002), 169-180.
[8] S. B. Yuste, L. Acedo and K. Lindenberg, Reaction front in an A+B→ C reaction- subdiffnsion process, Phys. Rev., E69, (2004), 036126.
[9] C. Lubich, Discretized fractional calculus, SIAM J. Math.Anal., 17, 3, (1986), pp. 704-716.
[10] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal., 5, (1997), 1-6.
[11] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numerical Algorithms, 16, (1997), 231-253.
[12] N. J. Ford and C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy, Numerical Algorithms, 26, (2001), pp. 333-346.
[13] N. J. Ford and C. Simpson, Numerical approaches to the solution of some fractional differential equations, Proceedings of Hercma, (2001), pp. 100-107.
[14] K.Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265(2002), 229-248.
[15] K. Diethelm and N.J. Ford, Numerical solution of the Bagley-Torvik equation, BIT, 42(2002), 490-507.
[16] K. Diethelm, N. J. Ford and A. D. Freed, A Predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, (2002), pp. 3-22.
[17] K. Diethelm, N.J. Ford and A.D. Freed, Detailed error analysis for a fractional Adams method, Numerical Algorithms, 36, (2004), pp. 31-52.
[18] K. Diethelm, N.J. Ford, A.D. Freed and Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg, 194, (2005), pp.743-773.
[19] E. Cuesta and C. Palencia, A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces, Appl. Mumer. Math., 45(2003), 139-159.
[20] R. Lin, F.Liu, a high order approximation of fractional order ordinary differential equation initial value problem，厦门大学学报(自科版)，43(1)，2004，25-30．
[21] R. Lin, F. Liu, Analysis of fractional-order numerical method for the fractional relaxation equation, Comput. Mech., (CD-ROM), ID-362, 2004.
[22] S. Shen and F.Liu, A computational effective method for fractional order Bagley-Torvik equation，厦门大学学报(自科版)，45(3)，2004，306-311．
[23] Y. Hu and F. Liu, Numerical methods for a fractional order control system, 厦门大学学报(自科版)，45(3)，2005，313-317．
[24] 杨晨航，刘发旺，分数阶Relaxation-Oscillation方程的一种分数阶预估-校正方法，厦门大学学报(自科版)，45(3)，2005，761-765．
[25] 卢旋珠，刘发旺，时间分数阶扩散反应方程．高等学校计算数学学报，2005，27(3)：267-273．
[26] X. N. Cao, K. Burrage, F. Abdullah, Variable stepsize implementation for fractional differen- tial equations.
[27] J. M. Sanz-Serna, A Numerical Method for a partial integro-differential equation, SIAM. J. Numer.Anal., 25, 1988, 319-327.
[28] C. Chen, V. Thomee and L.B.Wablbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math.Comp.58, 1992, 587-602.
[29] Huang Yueqing, Time discretization scheme for an integro-differential equation of parabolic type, J.Comp.Math., 12(3), 1994, 259-263.
[30] Xu Da, The global behaviour of time discretization for an abstract Volterra equation in Hilbert space, CALCOLD, 34, 1997, 71-104.
[31] F. Liu, V. Anh, I. Turner and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math.computing, 13, (2003), pp. 233-245.
[32] X. Lu and F. Liu, The explicit and implicit finite difference approximations for a space fractional advection diffusion equation, Computational Mechanics, ID-120, 2004.
[33] S. Shen and F. Liu, A fully discrete difference approximation for the time fractional diffusion equation, Computational Mechanics, ID-79, 2004.
[34] F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, (2004), pp. 209-219.
[35] F.Liu, V.Anh, I.Turner et al, Numerical simulation for solute transport in fractalporous media, ANZIAM J., 2004, 45(E): 461-473.
[36] F. Liu, S. Shen, V. Anh and I. Turner, Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46(E), 2005, 488-504.
[37] S. Shen and F. Liu, Error analysis of an explicit finite difference approximation for the space fractional diffusion, ANZIAM J., 46(E), 2005, 871-887.
[38] S. B. Yuste and L. Acedo, An explicit finite difference method and a new yon neumann-type stability analysis for fractional differential equation, SIAM J. Numer. Anal., 42, (2005), 1862-1874.
[39] 陈红斌，陈传淼，徐大，一类偏积分微分方程二阶差分全离散格式，计算数学，28(2)，2006，141-154．
[40] E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave euations, Math. Comput., 254, (2006), 673-696.
[41] Q.Yu and F.Liu, Implicit difference Approximation for the Time-Fractional Order Reaction-Diffusion Equation，厦门大学学报(自科版)，2006，45(3)：315-319
[42] Liu F.W. fractional differential equations and its application. Australia, 2006.
[43] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time Fractional Diffusion: A Discrete Random Walk Approach, J, Nonlinear Dynamics, 29, (2000), 129-143.
[44] 李荣华，冯果忱等，微分方程数值解法[M]，高等教育出版社，1979年12月．
[45] 汤怀民，胡健伟，微分方程数值方法[M]，南开大学出版社，1990年1月．
[46] 李寿佛，刚性微分方程算法理论[M]，湖南科学技术出版社，1996年11月．
[2] R. Hilfer(ed.) Applications of fractional calculus in physics, World Scientific, Singapore 2000.
[3] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[4] R. Kopelman, Fractal reaction kinetics, Science, 241(1988), 1620-1626.
[5] M. A. savageau, Michaelis-Menten mechanism reconsidered: Implications of fractal kinetics, J. Theoretical Biology, 176(1995) 115-124.
[6] S. B. Yuste and K. Lindenberg, Subdiffusion limited A+A reactions, Phys. Rev. Lett., 87, (2001), 1-4.
[7] S. B. Yuste and K. Lindenberg, Subdiffusion-limited reactions, Chem. Phys., 284, (2002), 169-180.
[8] S. B. Yuste, L. Acedo and K. Lindenberg, Reaction front in an A+B→ C reaction- subdiffnsion process, Phys. Rev., E69, (2004), 036126.
[9] C. Lubich, Discretized fractional calculus, SIAM J. Math.Anal., 17, 3, (1986), pp. 704-716.
[10] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal., 5, (1997), 1-6.
[11] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numerical Algorithms, 16, (1997), 231-253.
[12] N. J. Ford and C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy, Numerical Algorithms, 26, (2001), pp. 333-346.
[13] N. J. Ford and C. Simpson, Numerical approaches to the solution of some fractional differential equations, Proceedings of Hercma, (2001), pp. 100-107.
[14] K.Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265(2002), 229-248.
[15] K. Diethelm and N.J. Ford, Numerical solution of the Bagley-Torvik equation, BIT, 42(2002), 490-507.
[16] K. Diethelm, N. J. Ford and A. D. Freed, A Predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, (2002), pp. 3-22.
[17] K. Diethelm, N.J. Ford and A.D. Freed, Detailed error analysis for a fractional Adams method, Numerical Algorithms, 36, (2004), pp. 31-52.
[18] K. Diethelm, N.J. Ford, A.D. Freed and Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg, 194, (2005), pp.743-773.
[19] E. Cuesta and C. Palencia, A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces, Appl. Mumer. Math., 45(2003), 139-159.
[20] R. Lin, F.Liu, a high order approximation of fractional order ordinary differential equation initial value problem，厦门大学学报(自科版)，43(1)，2004，25-30．
[21] R. Lin, F. Liu, Analysis of fractional-order numerical method for the fractional relaxation equation, Comput. Mech., (CD-ROM), ID-362, 2004.
[22] S. Shen and F.Liu, A computational effective method for fractional order Bagley-Torvik equation，厦门大学学报(自科版)，45(3)，2004，306-311．
[23] Y. Hu and F. Liu, Numerical methods for a fractional order control system, 厦门大学学报(自科版)，45(3)，2005，313-317．
[24] 杨晨航，刘发旺，分数阶Relaxation-Oscillation方程的一种分数阶预估-校正方法，厦门大学学报(自科版)，45(3)，2005，761-765．
[25] 卢旋珠，刘发旺，时间分数阶扩散反应方程．高等学校计算数学学报，2005，27(3)：267-273．
[26] X. N. Cao, K. Burrage, F. Abdullah, Variable stepsize implementation for fractional differen- tial equations.
[27] J. M. Sanz-Serna, A Numerical Method for a partial integro-differential equation, SIAM. J. Numer.Anal., 25, 1988, 319-327.
[28] C. Chen, V. Thomee and L.B.Wablbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math.Comp.58, 1992, 587-602.
[29] Huang Yueqing, Time discretization scheme for an integro-differential equation of parabolic type, J.Comp.Math., 12(3), 1994, 259-263.
[30] Xu Da, The global behaviour of time discretization for an abstract Volterra equation in Hilbert space, CALCOLD, 34, 1997, 71-104.
[31] F. Liu, V. Anh, I. Turner and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math.computing, 13, (2003), pp. 233-245.
[32] X. Lu and F. Liu, The explicit and implicit finite difference approximations for a space fractional advection diffusion equation, Computational Mechanics, ID-120, 2004.
[33] S. Shen and F. Liu, A fully discrete difference approximation for the time fractional diffusion equation, Computational Mechanics, ID-79, 2004.
[34] F. Liu, V. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, (2004), pp. 209-219.
[35] F.Liu, V.Anh, I.Turner et al, Numerical simulation for solute transport in fractalporous media, ANZIAM J., 2004, 45(E): 461-473.
[36] F. Liu, S. Shen, V. Anh and I. Turner, Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46(E), 2005, 488-504.
[37] S. Shen and F. Liu, Error analysis of an explicit finite difference approximation for the space fractional diffusion, ANZIAM J., 46(E), 2005, 871-887.
[38] S. B. Yuste and L. Acedo, An explicit finite difference method and a new yon neumann-type stability analysis for fractional differential equation, SIAM J. Numer. Anal., 42, (2005), 1862-1874.
[39] 陈红斌，陈传淼，徐大，一类偏积分微分方程二阶差分全离散格式，计算数学，28(2)，2006，141-154．
[40] E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave euations, Math. Comput., 254, (2006), 673-696.
[41] Q.Yu and F.Liu, Implicit difference Approximation for the Time-Fractional Order Reaction-Diffusion Equation，厦门大学学报(自科版)，2006，45(3)：315-319
[42] Liu F.W. fractional differential equations and its application. Australia, 2006.
[43] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, Time Fractional Diffusion: A Discrete Random Walk Approach, J, Nonlinear Dynamics, 29, (2000), 129-143.
[44] 李荣华，冯果忱等，微分方程数值解法[M]，高等教育出版社，1979年12月．
[45] 汤怀民，胡健伟，微分方程数值方法[M]，南开大学出版社，1990年1月．
[46] 李寿佛，刚性微分方程算法理论[M]，湖南科学技术出版社，1996年11月．