分数阶微分方程线性多步法的研究
|
摘要
本文主要研究分数阶微分方程的数值处理及稳定性的分析,分为两个部分:第一,研究了用显隐式分数阶后退的差分格式,考虑实验方程数值解的性质及稳定性分析;第二,讨论了分数阶线性多步法相容格式的零稳定性和收敛性,分析其可能的最大稳定域的估计。
本论文的结构安排如下:
第一章首先回顾了分数阶微分方程的产生和近几十年来的发展,介绍了分数阶微分方程线性多步法的起源及其优越性,详细讨论了分数阶导数定义发展完善的过程。
第二章重点讨论分数阶向后差分格式,并将整数阶时的数值方法的有关定义推广到分数阶的情况。分为两个方面:首先,在满足相容性条件下,对右端函数用两点和三点格式进行离散,对所得到的显式格式应用于分数阶实验方程,分析其数值解的正性和单调性,从而很好的保持了其精确解Mittage-Leffler函数的性质;利用伴随矩阵的非负不可约性,结合Perron-Frobenius定理和笛卡尔符号准则等,分析其绝对稳定性,给出显式线性多步方法的绝对稳定区间。其次,在满足相容性条件下,适当约束生成多项式的系数,对所得到的隐式格式数值解的正性和单调性进行分析,并给出其绝对稳定区间的讨论。最后,通过具体的数值算例验证了其理论的有效性及良好的可行性。
第三章分析对几类低阶分数阶线性多步法稳定域的讨论。首先,研究了分数阶线性多步法相容格式的零稳定性和收敛性;其次引入分数阶线性多步法的稳定多项式,分析其与实轴的交点,通过理论证明,估算出该方法绝对稳定区间的最大可能长度;最后,给出几类具体的低阶显隐式分数阶线性多步法,并计算出其稳定区间的最大可能长度的估计式及局部阶段误差的估计式。
This paper concerns with numerical methods and stability properties for the treatment of differential equations of fractional order, comprises of two parts: Firstly, our attention is concentrated on fractional multistep methods of both implicit and explicit types, for which numerical solution properties and stability properties are investigated. Secondly, we mainly discuss zero-stability and convergence for fractional multistep methods. Furthermore, an analytical estimation for the interval of absolute stability is discussed.
The thesis is organized as follow.
In Chapter One, we first present the origin of differential equations of fractional order and their development during these years. And then, we introduce the origin of the linear fractional differential equations of multistep method, discuss their advantages and research the development of the definition of fractional derivative in detail.
In Chapter Two, we mainly discuss fractional backward differentiation formulae and extend the definition of integer order numerical in the fractional case. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which numerical solution properties and stability properties are discuss. Firsty, on the condition of the consistency, we discreate the right function with Two-point and Three-point, apply the explicit schemes to the linear test equation, analyze their numerical solution. The positive and monotonic character of the Mittage-Leffler function is preserved. Then we use the nonnegative and irreducible prperities of the companion matrix, by combining the Perron-Frobenius theorem and the Descartes’rule of signs we can analyze their absolute stability regions. We can deal with the implicit type in the same way. Finally, numerical experiments demonstrate that the theory about numerical solution properties is effective and feasible.
In Chapter Three, we consider several types of low-order fractional multistep methods and their stability domains are investigated. We mainly discuss the zero-stability and convergence for fractional multistep methods. Furthermore, an analytical estimation for the interval of absolute stability is discussed. Finally, specific examples which include estimation for the interval of absolute stability and the local truncation consolidate the conclusions about stability domain properties.
本论文的结构安排如下:
第一章首先回顾了分数阶微分方程的产生和近几十年来的发展,介绍了分数阶微分方程线性多步法的起源及其优越性,详细讨论了分数阶导数定义发展完善的过程。
第二章重点讨论分数阶向后差分格式,并将整数阶时的数值方法的有关定义推广到分数阶的情况。分为两个方面:首先,在满足相容性条件下,对右端函数用两点和三点格式进行离散,对所得到的显式格式应用于分数阶实验方程,分析其数值解的正性和单调性,从而很好的保持了其精确解Mittage-Leffler函数的性质;利用伴随矩阵的非负不可约性,结合Perron-Frobenius定理和笛卡尔符号准则等,分析其绝对稳定性,给出显式线性多步方法的绝对稳定区间。其次,在满足相容性条件下,适当约束生成多项式的系数,对所得到的隐式格式数值解的正性和单调性进行分析,并给出其绝对稳定区间的讨论。最后,通过具体的数值算例验证了其理论的有效性及良好的可行性。
第三章分析对几类低阶分数阶线性多步法稳定域的讨论。首先,研究了分数阶线性多步法相容格式的零稳定性和收敛性;其次引入分数阶线性多步法的稳定多项式,分析其与实轴的交点,通过理论证明,估算出该方法绝对稳定区间的最大可能长度;最后,给出几类具体的低阶显隐式分数阶线性多步法,并计算出其稳定区间的最大可能长度的估计式及局部阶段误差的估计式。
This paper concerns with numerical methods and stability properties for the treatment of differential equations of fractional order, comprises of two parts: Firstly, our attention is concentrated on fractional multistep methods of both implicit and explicit types, for which numerical solution properties and stability properties are investigated. Secondly, we mainly discuss zero-stability and convergence for fractional multistep methods. Furthermore, an analytical estimation for the interval of absolute stability is discussed.
The thesis is organized as follow.
In Chapter One, we first present the origin of differential equations of fractional order and their development during these years. And then, we introduce the origin of the linear fractional differential equations of multistep method, discuss their advantages and research the development of the definition of fractional derivative in detail.
In Chapter Two, we mainly discuss fractional backward differentiation formulae and extend the definition of integer order numerical in the fractional case. Our attention is concentrated on fractional multistep methods of both implicit and explicit type, for which numerical solution properties and stability properties are discuss. Firsty, on the condition of the consistency, we discreate the right function with Two-point and Three-point, apply the explicit schemes to the linear test equation, analyze their numerical solution. The positive and monotonic character of the Mittage-Leffler function is preserved. Then we use the nonnegative and irreducible prperities of the companion matrix, by combining the Perron-Frobenius theorem and the Descartes’rule of signs we can analyze their absolute stability regions. We can deal with the implicit type in the same way. Finally, numerical experiments demonstrate that the theory about numerical solution properties is effective and feasible.
In Chapter Three, we consider several types of low-order fractional multistep methods and their stability domains are investigated. We mainly discuss the zero-stability and convergence for fractional multistep methods. Furthermore, an analytical estimation for the interval of absolute stability is discussed. Finally, specific examples which include estimation for the interval of absolute stability and the local truncation consolidate the conclusions about stability domain properties.
引文
1 R. Grenflo, S. Vessella. Abel Integral Equations. Analysis and Applications. Lecture Notes in Mathematics. Springer-Verlag. Berlin, 1991:10~15
2 A. Y. Rossikhin, M. V. Shitikova. Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids. Appl. Mech. Rev. 1997, 50(1):16~67
3 D. A. Benson, S. W. Wheatcraft. Application of a Fractional Advection Despersion Equation. Water Resour. Res. 2000, 36(6):1403~1412
4 W. Wyss. The Fractional Black-Scholes Equation. Fractional Calculus Appl. Anal. 2000, (3):51~61
5 I. Podlubny. Fractional Differential Equations. Academic Press, 1999:183~192
6 R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific. Singaopore, 2000:125~127
7 R. E. Scalas, F. Mainardi. Fractional Calculus and Continuous-Time Finance. Physica A. 2000, (24):376~384
8 W. Wyss. The Fractional Black-Scholes Equations. Fractional Calculus Appl. Anal. 2000, 3:51~61
9 F. Huang, F. Liu. The Time Fractional Diffusion Equation and Advection-Dispersion Equation. ANZIAM J. 2005, 46(E):317~330
10 N. Su, G. Sander, F. Liu. Similarity Solution of Fokker-Planck Equation with Time and Scale-Dependent Dispersivity for Solute Ttransport in Fractal Porous Media. Applied Mathematical Modelling. 2005, 2:852~870
11 F. Huang, F. Liu. The Space-Time Fractional Diffusion Equation with Caputo Derivatives. J. Appl. Math. Comp. 2005, (19):233~245
12 F. Huang, F. Liu. The Fundamental Solution of The Space-Time Fractional aDvection Equation. J. Appl. Math. Comp. 2005, 18:339~350
13 R. Gorenflo, F. Maniardi, D. Moretti. Descrete Random Walk Models for Space-Time Fractional Duffusion. Chemical Physics. 2002, 284:521~541
14陈世平.用变量分离法求解分数阶微分方程.泉州师范学院学报. 2007, 25(4):1~3
15 A. Y. Rossikhinu, M. V. Shitikova. Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids. Appl. Mech. Rev. 1997, 50(1):16~67
16陈春华,卢旋珠.时间分数阶偏微分方程的基本解.莆田学院学报. 2005, 12(5):11~13
17 J. L. Lavoie, T. J. Osler. Tremblay. Fractional Derivatives and Special Functions. SIAM Rev. 1976, 18(3):240~268
18 M. Riesz, L’integrale dé. Riemann-Liouville ei léProblem déCauchy. Acta.Math. 1949, 81:1~223
19 K. B. Oldham, J. Spanier. The Fractional Calculus. Academic Press. New York, 1974:1~335
20 D. A. Benson, M. M. Meerschert. Application of a Fractional Advection-Despersion Equation. Water Resour. Res. 2000, 36(6):1403~1412
21 W. Wyss. The Fractional Black-Scholes Equation. Fractional Calculus Appl. Anal. 2000, 3(4):51~61
22 D. Vries. Mechanics of the Labyrinth Organs. Acta Otplaryngol. 1987, 38:262~273
23 E. F. Elshehawey, M. El-Shahed. On the Solution of Endolymph Equation Using Fractional Calculus. Applied Mathematics and Computation. 2001,124(5):337~341
24 B. Ross. Fractional Calculus and Its Applications. Springer-Verlag. Berlin, 1975:112~132
25 P. L. Butzer, U. Wesphal. An Introduction to Fractional Calculus. Applications of Fractional Calculus in Physics. World Scientific. Singapore, 2000, (6):1~85
26 R. Hilfer. Fractional Time Evolution. Applications of Fractional Calculus in Physics. World Scientific. Singapore, 2000:34~55
27 B. B. Mandelbrot.The Fratal Geometry of Nature. W H Freeman. New York, 1982:322~350
28 B. B. Mandelbrot, J. W. Ness. Fractional Brownian Motion Fractional Noises and Applications. SLAM Rev. 1968, (10):422~437
29 B. B. Mandelbrot. Fractals Form and Chance and Dimension. W H Freeman. San Francisco, 1977:210~234
30 B. B. Mandelbrot. Is Nature Fractals? Science. 1998, 279(5):5352
31 B. B. Mandelbrot. Some Mathematical Questions Arising in Fractal Geometry. Development of Mathematics 1950-2000. Birkhauser Verlag. Berlin, 2000, (4):795~811
32 Y. A. Rossikhin, M. V.Shitikova. A New Method for Solving Dynamic Problems of Fractional Dericative Viscoelasticity. Int J. Eng Sci. 2000, 39(5):149~176
33 Y. A. Rossikhin, M. V. Shitikova. Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids. Appl. Mech. Rev. 1997, 50(1):15~67
34 T. F. Nonnenmacher, R. Metzler. Applicatons of Fractional Calculus Techniquesto Problems in Biophysics. Applications of Fractional Calculus in Physics. World Scientific Publishing Copte Ltd. Singapore, 2000, (3):377~427
35徐明瑜.药物动力学的现状与展望.生物力学研究和应用.华南理工大学出版社, 1990:33~37
36 Z. Mor a??vek, J. Fiala. Fractal Dynamics in the Growth of Root. Chaos Solitons and Fractals. 2006, (19):31~34
37 M. Y. Xu. An Analytical Solution for the Model of Drug Distribution and Absorption in Small Intestine. ACTA Mech Sinica. 1990, 6(4):316~323
38 T. Sun, P. Meaakin, T. Jossang. Minimum Energy Dissipation and Fractal Structures of Vascular Systems Fractals. 1995, 3(1):123~153
39 B. J. Best. Fractal Physilolgy and Chaos in Medicine. World Scientific. New York, 1993:1~288
40 W. G. Gl o??ckle. T. F. Nonenmacher. A Fractonal Calculus Approach to Self-similar Protein Dynamics. Biophys J. 1995, 68(5):46~53
41 I. M. Tolic-Norrelykke, E. L. Munteanu, G. Thon. Anomalous Diffusion in Living Yeast Cells. Phys Rev Lett. 2006, 93(7):781~790
42 L. V. Yakushevich. Nonlinear Dynamics of DNA. Nonlinear Science at the Dawn of the 21st century. Springer-Verlag. Berlin Heideberg, 2000:373-391
43 C. K. Peng, S. Buldyrev, A. L. Goldberg. Long-range Correlations in Nucleotide Sequences. Nature. 1992, 356(4):168-171
44 H. E. Stanley, S. Buldyrev, A. L. Goldberg. Statistical Mechanics in Biology: How Ubiquitous are Long-range Correlations. Physica A. 1994:205~214
45 P. Allegrini, M. Buacci, P. Grigolin. Fractional Brownian Motion as a Nonstationary Process: An Alernative Paradigm for DNA Sequences. Phys Rev E. 1998, 57(4):1~10
46 D. R. Bickel, B. J. West. Multiplicative and Fractal Processes in DNA Evlution. Fractals. 1998, 6:211~217
47 Z. G. Yu, V. Anh, K.S. Lau. Measure Representation and Multifractal Analysis of Complete Genomes. Phys Rev F. 2005, 64(3): 190~193
48 L. V. Yakushevivh. Nonlinear Physics of DNA. John Wiley. New York, 1998
49 T. Vicsek. Fractal Growth Phenomena.2nd. World Scientific. Singapore, 1992
50 I. Podlubny. Fractional Differential Equation. Academic Press .NewYork, 1999
51 C.Lubich. Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the Second Kind. Math .Comp. 1985, 45(12):463~469
52 K. Diethelm, N. J. Ford. Algorithms for the Fractional Calculus: A Selection of Numerical Methods. Comput. Methods Appl. Mech. Engrg. 2005,194(6):743~7 73
53 J. M. Ford, N. J. Ford. Pitfalls in Fast Numerical Solvers for Fractionl Differential Equations. J.Comput. Appl. Math. 2006, 186(2):482~503
54 S. B. Yuste, L.Acedo. An Explicit Finite Difference of a New Vonneumann-Type Stability Analysis for Fractioanl Diffusion Equations. SIAM. J. Numer. Anal. 2005,42(5):1862~1874
55戴华.矩阵论.科学出版社, 2003:277~278
2 A. Y. Rossikhin, M. V. Shitikova. Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids. Appl. Mech. Rev. 1997, 50(1):16~67
3 D. A. Benson, S. W. Wheatcraft. Application of a Fractional Advection Despersion Equation. Water Resour. Res. 2000, 36(6):1403~1412
4 W. Wyss. The Fractional Black-Scholes Equation. Fractional Calculus Appl. Anal. 2000, (3):51~61
5 I. Podlubny. Fractional Differential Equations. Academic Press, 1999:183~192
6 R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific. Singaopore, 2000:125~127
7 R. E. Scalas, F. Mainardi. Fractional Calculus and Continuous-Time Finance. Physica A. 2000, (24):376~384
8 W. Wyss. The Fractional Black-Scholes Equations. Fractional Calculus Appl. Anal. 2000, 3:51~61
9 F. Huang, F. Liu. The Time Fractional Diffusion Equation and Advection-Dispersion Equation. ANZIAM J. 2005, 46(E):317~330
10 N. Su, G. Sander, F. Liu. Similarity Solution of Fokker-Planck Equation with Time and Scale-Dependent Dispersivity for Solute Ttransport in Fractal Porous Media. Applied Mathematical Modelling. 2005, 2:852~870
11 F. Huang, F. Liu. The Space-Time Fractional Diffusion Equation with Caputo Derivatives. J. Appl. Math. Comp. 2005, (19):233~245
12 F. Huang, F. Liu. The Fundamental Solution of The Space-Time Fractional aDvection Equation. J. Appl. Math. Comp. 2005, 18:339~350
13 R. Gorenflo, F. Maniardi, D. Moretti. Descrete Random Walk Models for Space-Time Fractional Duffusion. Chemical Physics. 2002, 284:521~541
14陈世平.用变量分离法求解分数阶微分方程.泉州师范学院学报. 2007, 25(4):1~3
15 A. Y. Rossikhinu, M. V. Shitikova. Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids. Appl. Mech. Rev. 1997, 50(1):16~67
16陈春华,卢旋珠.时间分数阶偏微分方程的基本解.莆田学院学报. 2005, 12(5):11~13
17 J. L. Lavoie, T. J. Osler. Tremblay. Fractional Derivatives and Special Functions. SIAM Rev. 1976, 18(3):240~268
18 M. Riesz, L’integrale dé. Riemann-Liouville ei léProblem déCauchy. Acta.Math. 1949, 81:1~223
19 K. B. Oldham, J. Spanier. The Fractional Calculus. Academic Press. New York, 1974:1~335
20 D. A. Benson, M. M. Meerschert. Application of a Fractional Advection-Despersion Equation. Water Resour. Res. 2000, 36(6):1403~1412
21 W. Wyss. The Fractional Black-Scholes Equation. Fractional Calculus Appl. Anal. 2000, 3(4):51~61
22 D. Vries. Mechanics of the Labyrinth Organs. Acta Otplaryngol. 1987, 38:262~273
23 E. F. Elshehawey, M. El-Shahed. On the Solution of Endolymph Equation Using Fractional Calculus. Applied Mathematics and Computation. 2001,124(5):337~341
24 B. Ross. Fractional Calculus and Its Applications. Springer-Verlag. Berlin, 1975:112~132
25 P. L. Butzer, U. Wesphal. An Introduction to Fractional Calculus. Applications of Fractional Calculus in Physics. World Scientific. Singapore, 2000, (6):1~85
26 R. Hilfer. Fractional Time Evolution. Applications of Fractional Calculus in Physics. World Scientific. Singapore, 2000:34~55
27 B. B. Mandelbrot.The Fratal Geometry of Nature. W H Freeman. New York, 1982:322~350
28 B. B. Mandelbrot, J. W. Ness. Fractional Brownian Motion Fractional Noises and Applications. SLAM Rev. 1968, (10):422~437
29 B. B. Mandelbrot. Fractals Form and Chance and Dimension. W H Freeman. San Francisco, 1977:210~234
30 B. B. Mandelbrot. Is Nature Fractals? Science. 1998, 279(5):5352
31 B. B. Mandelbrot. Some Mathematical Questions Arising in Fractal Geometry. Development of Mathematics 1950-2000. Birkhauser Verlag. Berlin, 2000, (4):795~811
32 Y. A. Rossikhin, M. V.Shitikova. A New Method for Solving Dynamic Problems of Fractional Dericative Viscoelasticity. Int J. Eng Sci. 2000, 39(5):149~176
33 Y. A. Rossikhin, M. V. Shitikova. Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids. Appl. Mech. Rev. 1997, 50(1):15~67
34 T. F. Nonnenmacher, R. Metzler. Applicatons of Fractional Calculus Techniquesto Problems in Biophysics. Applications of Fractional Calculus in Physics. World Scientific Publishing Copte Ltd. Singapore, 2000, (3):377~427
35徐明瑜.药物动力学的现状与展望.生物力学研究和应用.华南理工大学出版社, 1990:33~37
36 Z. Mor a??vek, J. Fiala. Fractal Dynamics in the Growth of Root. Chaos Solitons and Fractals. 2006, (19):31~34
37 M. Y. Xu. An Analytical Solution for the Model of Drug Distribution and Absorption in Small Intestine. ACTA Mech Sinica. 1990, 6(4):316~323
38 T. Sun, P. Meaakin, T. Jossang. Minimum Energy Dissipation and Fractal Structures of Vascular Systems Fractals. 1995, 3(1):123~153
39 B. J. Best. Fractal Physilolgy and Chaos in Medicine. World Scientific. New York, 1993:1~288
40 W. G. Gl o??ckle. T. F. Nonenmacher. A Fractonal Calculus Approach to Self-similar Protein Dynamics. Biophys J. 1995, 68(5):46~53
41 I. M. Tolic-Norrelykke, E. L. Munteanu, G. Thon. Anomalous Diffusion in Living Yeast Cells. Phys Rev Lett. 2006, 93(7):781~790
42 L. V. Yakushevich. Nonlinear Dynamics of DNA. Nonlinear Science at the Dawn of the 21st century. Springer-Verlag. Berlin Heideberg, 2000:373-391
43 C. K. Peng, S. Buldyrev, A. L. Goldberg. Long-range Correlations in Nucleotide Sequences. Nature. 1992, 356(4):168-171
44 H. E. Stanley, S. Buldyrev, A. L. Goldberg. Statistical Mechanics in Biology: How Ubiquitous are Long-range Correlations. Physica A. 1994:205~214
45 P. Allegrini, M. Buacci, P. Grigolin. Fractional Brownian Motion as a Nonstationary Process: An Alernative Paradigm for DNA Sequences. Phys Rev E. 1998, 57(4):1~10
46 D. R. Bickel, B. J. West. Multiplicative and Fractal Processes in DNA Evlution. Fractals. 1998, 6:211~217
47 Z. G. Yu, V. Anh, K.S. Lau. Measure Representation and Multifractal Analysis of Complete Genomes. Phys Rev F. 2005, 64(3): 190~193
48 L. V. Yakushevivh. Nonlinear Physics of DNA. John Wiley. New York, 1998
49 T. Vicsek. Fractal Growth Phenomena.2nd. World Scientific. Singapore, 1992
50 I. Podlubny. Fractional Differential Equation. Academic Press .NewYork, 1999
51 C.Lubich. Fractional Linear Multistep Methods for Abel-Volterra Integral Equations of the Second Kind. Math .Comp. 1985, 45(12):463~469
52 K. Diethelm, N. J. Ford. Algorithms for the Fractional Calculus: A Selection of Numerical Methods. Comput. Methods Appl. Mech. Engrg. 2005,194(6):743~7 73
53 J. M. Ford, N. J. Ford. Pitfalls in Fast Numerical Solvers for Fractionl Differential Equations. J.Comput. Appl. Math. 2006, 186(2):482~503
54 S. B. Yuste, L.Acedo. An Explicit Finite Difference of a New Vonneumann-Type Stability Analysis for Fractioanl Diffusion Equations. SIAM. J. Numer. Anal. 2005,42(5):1862~1874
55戴华.矩阵论.科学出版社, 2003:277~278