一类分数阶微分方程初值问题的数值方法
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摘要
近年来,分数阶微积分在科学工程领域的广泛应用引起了人们很大的兴趣。在电化学过程、色噪声、控制理论、流体力学、混沌、生物工程等领域,分数阶微积分是有用的数学工具。在使用分数阶导数的模型中,大部分情况下会导致出现一系列的分数阶微分方程。尽管有些方程的解析解可以求出来,但人们注意到,很多分数阶微分方程的解析解是由比较特殊的函数来表示,而要数值地表示这些特殊函数是很困难的,并且有些非线性方程是不可能求出其解析解的,于是,人们越来越关注分数阶微分方程的数值方法。
本文考虑了一类分数阶微分方程初值问题的数值求解Caputo导数的性质使得该初值问题可以等价地简化为Volterra积分方程,因此,求解Volterra积分方程的数值方法同样能够应用于分数阶微分方程的数值求解。在第三章,将求解普通积分方程的Adams技巧应用于分数阶微分方程模型,得到一个求解分数阶微分方程的显式算法,给出了误差估计,并通过数值实验说明该算法是有效的。在第四章,以[34]中的方法为基础,作了微小的改动,得到一个隐式的算法,并给出了误差分析。在第五章,以第三章和第四章的方法为基础,得到一个新的预校算法,给出了误差估计,并且通过数值实验证明该方法是有效的;实验说明,该算法在1＜α＜2时比[31]所提的预校算法具有更高的数值精度。在第三章和第五章中均对分数阶松驰—震荡方程作了计算,不但说明了文中算法的有效性,还通过图形显示出解从松驰性态逐渐过渡到震荡性态的情况。
In recent years, fractional integrals (FIs) and fractional derivatives (FDs) have drawn much attention due to its wide application in many science and engineering fields. They are very useful mathematic tools in electrochemical processes, colored noise, controllers theory, fluid mechanics, chaos, biology engineering etc. Modeling of systems using FDs lead in most cases, a set of fractional differntial equations (FDEs). Though some analytic solutions of FDEs can be resolved, many solutions of them are expressed by some special functions, and it is hard to express numerically. Generally, nonlinear FDEs can not be resolved analytically. Hence there has been a growing interest to develop numerical techniques.
Numerical methods for a class of initial problems of FDEs are considered in this paper. Properties of Caputo derivative allow one to reduce the FDEs into a Volterra-type integral equation, therefore, the numerical schemes for Volterra-type integral equations can also be applied to the numerical solution of FDEs. In the third chapter, we apply the Adams technics developed for Volterra-type integral equation to the FDEs, and then obtain an explicit numerical schemes, error analysis is also presented, and numerical examples verify the efficiency of the numerical method. In the fourth chapter, based on the approach presented in [34], with a little improvement, an implicit numerical scheme is obtained, and error analysis is also presented. In the fifth chapter, based on the schemes presented in the third chapter and fourth chapter, we obtain a new predictor-corrector method, then present the error analysis, and numerical examples verify the efficiency of the numerical method too, which show that the method can provide higher precision when 1 <α< 2 than the one in [31]. Both in the third and the fifth chapter, fractional Relaxation-Oscillation equation was numerically resolved, which doesn't only show the efficiency of the numerical method, and also explain the situation varying from relaxation to oscillation by the picture of the numerical solution.
本文考虑了一类分数阶微分方程初值问题的数值求解Caputo导数的性质使得该初值问题可以等价地简化为Volterra积分方程,因此,求解Volterra积分方程的数值方法同样能够应用于分数阶微分方程的数值求解。在第三章,将求解普通积分方程的Adams技巧应用于分数阶微分方程模型,得到一个求解分数阶微分方程的显式算法,给出了误差估计,并通过数值实验说明该算法是有效的。在第四章,以[34]中的方法为基础,作了微小的改动,得到一个隐式的算法,并给出了误差分析。在第五章,以第三章和第四章的方法为基础,得到一个新的预校算法,给出了误差估计,并且通过数值实验证明该方法是有效的;实验说明,该算法在1＜α＜2时比[31]所提的预校算法具有更高的数值精度。在第三章和第五章中均对分数阶松驰—震荡方程作了计算,不但说明了文中算法的有效性,还通过图形显示出解从松驰性态逐渐过渡到震荡性态的情况。
In recent years, fractional integrals (FIs) and fractional derivatives (FDs) have drawn much attention due to its wide application in many science and engineering fields. They are very useful mathematic tools in electrochemical processes, colored noise, controllers theory, fluid mechanics, chaos, biology engineering etc. Modeling of systems using FDs lead in most cases, a set of fractional differntial equations (FDEs). Though some analytic solutions of FDEs can be resolved, many solutions of them are expressed by some special functions, and it is hard to express numerically. Generally, nonlinear FDEs can not be resolved analytically. Hence there has been a growing interest to develop numerical techniques.
Numerical methods for a class of initial problems of FDEs are considered in this paper. Properties of Caputo derivative allow one to reduce the FDEs into a Volterra-type integral equation, therefore, the numerical schemes for Volterra-type integral equations can also be applied to the numerical solution of FDEs. In the third chapter, we apply the Adams technics developed for Volterra-type integral equation to the FDEs, and then obtain an explicit numerical schemes, error analysis is also presented, and numerical examples verify the efficiency of the numerical method. In the fourth chapter, based on the approach presented in [34], with a little improvement, an implicit numerical scheme is obtained, and error analysis is also presented. In the fifth chapter, based on the schemes presented in the third chapter and fourth chapter, we obtain a new predictor-corrector method, then present the error analysis, and numerical examples verify the efficiency of the numerical method too, which show that the method can provide higher precision when 1 <α< 2 than the one in [31]. Both in the third and the fifth chapter, fractional Relaxation-Oscillation equation was numerically resolved, which doesn't only show the efficiency of the numerical method, and also explain the situation varying from relaxation to oscillation by the picture of the numerical solution.
引文
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[2]H.H.Sun,B.Onaral,Y.Tsao,Application of positive reality principle to metal electrode linear polarization phenomena[J],IEEE Trans.Biomed.Eng.,BME-31(10)(1984) 664-674.
[3]B.Mandelbrot,Some noises with 1/f spetrum,a bridge between direct current and white noise[J],IEEE T.Inform.Theory,13(2)(1967)289-298.
[4]I.Pudlubny,Fractional Differential Equations[M],Academic Press,San Diego,1999.
[5]L.Debnath,Fractional integral and fractional differential equations in fluid mechanics[J],An International Journal for Theory and Applicatons,6(2)(2003).
[6]T.T.Hartley,C.F.Lorenzo,H.K.Qammar,Chaoes in a fractional order chua system[J],IEEE Trans.Circuits Systems,I 42(8)(1995) 485-490.
[7]R.L.Magin,Fractional calculus in bioengineering[J],Crit.Rev.Biomed.Eng.,32(1)(2004) 1-104.
[8]R.L.Magin,Fractional calculus in bioenginering-part 2[J],Crit.Rev.Biomed.Eng.,32(2)(2004) 105-193.
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[10]R.L.Bagley,P.J.Torvik,A theoretical basis for the application of fractional calculus to viscoelasticity,J.Rheology[J]27(3)(1983)201-210.
[11]R.L.Bagley,P.J.Torvik,Fractional calculus — a different approach to the analysis of viscoelastically damped structures[J],AIAA J.21(5)(1983) 741-748.
[12]宋道云,江体乾.带分数导数的Jefrreys模型的建立及其应用.见:现代科学与进步.清华大学出版社,1997,639-642.
[13] K. B. Oldham, J. Spanier, The fractional calulus[M], Academic Press, New York,1974.
[14] S. G. Samko, A. A. Kilbas, O. I. Marichev, fractional Integrals and Derivatives-Theory and Applications[M], Gordon and Breach Science Publishers, Longhorne,PA, 1993.
[15] R. Gorenflo, F. Mainardi, Fractional caclulus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractioanl Vaculus in Continuum Mechanics, Spinger, Wein, New York, 1997, pp.223-276.
[16] K. S. Miller, B. Ross, An interoduction to the fractional calculus and fractional differential equations[M], Wiley, New York, 1993.
[17] L. Gaul, P. Klein, S. Kempfle, Impulse response function of an oscillator with fractional derivative in damping description[J], Mech. Res. Comm., 16 (5) (1989)4447-4472.
[18] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional opera-tors[J], Mech. Systems Signal Process., 5 (2) (1991) 8-88.
[19] S. Momani and R. Qaralleh, Numerical approximations and Pade approximants for a fractional population growth model[J], Applied Mathematical M , odelling, 31(2007) 1907-1914.
[20] S. Momani and Z. Odibat, Numerical comparison of metheods for solving linear differential equations of fractional order[J], Chaos, Solitions and Fractials, 31 (2007)1248-1255.
[21] S. B. Yuste, Weighted average finite difference methods for fractional diffusion equa-tions[J], Journal of Computational Physics, 216 (2006) 264-274.
[22] Mark Meerschaert, Hans-Peter Scheffler, Charles Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation[J], Journal of Computational Physics, 211 (2006) 249-261.
[23] R. Gorenflo, Fractional calculus: some numerical methods, in: A. Carpinteri,F. Maomardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics,Springer, Wein, New York, 1997, pp. 277-290.
[24]J.Padovan,Computational algorithms and finite element,formulation involving fractional operators[J],Comput.Mech.,2(1987) 271-287.
[25]P.Ruge,N.Wagner,Time-domain solutions for vibration systems with feding memory,European Conference of Computational Mechanics,Munchen,Germany,August 31-September 3,1999.
[26]S.Shen,F.Liu,Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends[J],ANZIAM J.,46(E) pp.C871-C887,2005
[27]S.Shen and F.Liu,A computational effective methods/br fractional order Bagley-Torvik equation[J].厦门大学学报(自科版),45(3)(2004)306-311.
[28]Yang C.and Liu F.,A computationally effective predictor-corrector method for simulating fractional-order dynamical control system[J],ANZIAM J.,47,(2006)168-184.
[29]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[J].厦门大学学报(自科版):45(3)(2005)761-765.
[30]沈淑君,刘发旺,解分数阶Bagley-Torvik方程的一种计算有效的数值方法[J],厦门大学学报,43(3),(2004)306-311.
[31]K.Diethehn,N.J.Ford,A.D.Freed,A predictor-corrector approach for the numerical solution of fractional differential equations[J],Nonlinear Dynamics,29(1-4)(2002)3-22.
[32]K.Diethelm,N.J.Ford,A.D.Freed,Y.Luchko,Algorithms for the fractional calculus:a selection of numerical methods]J],Comput.Methods Appl.Mech.Eng.,194(2005) 743-773.
[33]P.Kumar,O.P.Agrawal,A cubic scheme for numerical solution of fractional differential equations,in:Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference,ENOC-2005,pp.1479-1487.
[34]P.Kumar,O.P.Agrawal,An approximate method for numerical solution of fractional differential equations[J],Signal Processing,86(2006) 2602-2610.
[35]K.Diethelm and G.Walz,Numerical solution of fractional order differdential equations by extrapolation[J],Numer.Algorithms,16(1997) 231-253.
[36]K.Diethelm,N.J.Ford and A.D.Freed,Detailed error analysis for a fractional Adams method,Berichte der Mathematischen Institut der TU Braunschweig 02/02,Braunschweig,submitted for publication.
[37]Neville J.Ford and A.Charles Simpson,The numerical solution of fractional differential equations:Speed versus accuracy,Numerical Algorithms[J],26(2001) 333-346.
[38]W.H.Deng,Short memory principle and a predictor corrector approach for fractional differential equations[J],Journal of Computational and Applied Mathematics,Received 28 April 2006.
[39]K.Diethelm,N.J.Ford,Analysis of fractional differential equations[J],J.Math.Anal.Appl.,265(2002) 229-248.
[40]王仁宏,数值逼近[M],高等教育出版社,1999年6月.
[41]C.Lubich,Discretized fractional calculus[J],SIAM J.Math.Anal.,17(3)(1986)pp.704-716.
[42]R.Lin,F.Liu,A high order approximation of fractional order ordinary differential equation initial value problem[J],厦门大学学报(自科版),43(1)(2004)25-30.
[43]K.Diethelm,An algorithm for the numerical solution of differential equations of fractional order[J],Electronic Transactions on Numerical Analysis,Volume 5.pp.1-6,March 1997.
[44]F.W.Liu,Fractional differential equations and its application.Australia,2006.
[451 N.J.Ford,C.A.Simpson,The approximate solution of fractional differential equations of order greater than 1,Internet download,URL(http://www.chester.ac.uk/maths/nevillepub.html) May 2003.
[46]K.Diethelm,Efficient solution of multi-term fractional differential equation using P(EC)~m E methods[J],Computing,71(2003)305-319.
[47]K.Diethelm,N.J.Ford,Multi-order fractional differential equations and their numerical solution[J],Appl.Math.Comput.,154(3)(2004)621-640
[48]K.Diethelm,N.J.Ford,Analysis of fractional differential equations[J],J.Math.Anal.Appl.,265(2002)229-228
[49]Luciano Galeone,Roberto Garrappa,On Multistep Methods for Differential Equations of Fractional Order[J],Mediterr.a.math.,3(2006),565-580.
[50]X.N.Cao,K.Burrage,F.Abdullah,Variable stepsize implementation for fractional differential equations.
[51]何吉欢,非线性分数微分方程的应用及近似求解方法[J],Bulletin of science and technology,No.2.1999
[2]H.H.Sun,B.Onaral,Y.Tsao,Application of positive reality principle to metal electrode linear polarization phenomena[J],IEEE Trans.Biomed.Eng.,BME-31(10)(1984) 664-674.
[3]B.Mandelbrot,Some noises with 1/f spetrum,a bridge between direct current and white noise[J],IEEE T.Inform.Theory,13(2)(1967)289-298.
[4]I.Pudlubny,Fractional Differential Equations[M],Academic Press,San Diego,1999.
[5]L.Debnath,Fractional integral and fractional differential equations in fluid mechanics[J],An International Journal for Theory and Applicatons,6(2)(2003).
[6]T.T.Hartley,C.F.Lorenzo,H.K.Qammar,Chaoes in a fractional order chua system[J],IEEE Trans.Circuits Systems,I 42(8)(1995) 485-490.
[7]R.L.Magin,Fractional calculus in bioengineering[J],Crit.Rev.Biomed.Eng.,32(1)(2004) 1-104.
[8]R.L.Magin,Fractional calculus in bioenginering-part 2[J],Crit.Rev.Biomed.Eng.,32(2)(2004) 105-193.
[9]R.L.Magin,Fractional calculus in bioengineering-part 3[J],Crit.Rev.Biomed.Eng.,32(2)(2004) 194-377.
[10]R.L.Bagley,P.J.Torvik,A theoretical basis for the application of fractional calculus to viscoelasticity,J.Rheology[J]27(3)(1983)201-210.
[11]R.L.Bagley,P.J.Torvik,Fractional calculus — a different approach to the analysis of viscoelastically damped structures[J],AIAA J.21(5)(1983) 741-748.
[12]宋道云,江体乾.带分数导数的Jefrreys模型的建立及其应用.见:现代科学与进步.清华大学出版社,1997,639-642.
[13] K. B. Oldham, J. Spanier, The fractional calulus[M], Academic Press, New York,1974.
[14] S. G. Samko, A. A. Kilbas, O. I. Marichev, fractional Integrals and Derivatives-Theory and Applications[M], Gordon and Breach Science Publishers, Longhorne,PA, 1993.
[15] R. Gorenflo, F. Mainardi, Fractional caclulus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractioanl Vaculus in Continuum Mechanics, Spinger, Wein, New York, 1997, pp.223-276.
[16] K. S. Miller, B. Ross, An interoduction to the fractional calculus and fractional differential equations[M], Wiley, New York, 1993.
[17] L. Gaul, P. Klein, S. Kempfle, Impulse response function of an oscillator with fractional derivative in damping description[J], Mech. Res. Comm., 16 (5) (1989)4447-4472.
[18] L. Gaul, P. Klein, S. Kempfle, Damping description involving fractional opera-tors[J], Mech. Systems Signal Process., 5 (2) (1991) 8-88.
[19] S. Momani and R. Qaralleh, Numerical approximations and Pade approximants for a fractional population growth model[J], Applied Mathematical M , odelling, 31(2007) 1907-1914.
[20] S. Momani and Z. Odibat, Numerical comparison of metheods for solving linear differential equations of fractional order[J], Chaos, Solitions and Fractials, 31 (2007)1248-1255.
[21] S. B. Yuste, Weighted average finite difference methods for fractional diffusion equa-tions[J], Journal of Computational Physics, 216 (2006) 264-274.
[22] Mark Meerschaert, Hans-Peter Scheffler, Charles Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation[J], Journal of Computational Physics, 211 (2006) 249-261.
[23] R. Gorenflo, Fractional calculus: some numerical methods, in: A. Carpinteri,F. Maomardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics,Springer, Wein, New York, 1997, pp. 277-290.
[24]J.Padovan,Computational algorithms and finite element,formulation involving fractional operators[J],Comput.Mech.,2(1987) 271-287.
[25]P.Ruge,N.Wagner,Time-domain solutions for vibration systems with feding memory,European Conference of Computational Mechanics,Munchen,Germany,August 31-September 3,1999.
[26]S.Shen,F.Liu,Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends[J],ANZIAM J.,46(E) pp.C871-C887,2005
[27]S.Shen and F.Liu,A computational effective methods/br fractional order Bagley-Torvik equation[J].厦门大学学报(自科版),45(3)(2004)306-311.
[28]Yang C.and Liu F.,A computationally effective predictor-corrector method for simulating fractional-order dynamical control system[J],ANZIAM J.,47,(2006)168-184.
[29]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[J].厦门大学学报(自科版):45(3)(2005)761-765.
[30]沈淑君,刘发旺,解分数阶Bagley-Torvik方程的一种计算有效的数值方法[J],厦门大学学报,43(3),(2004)306-311.
[31]K.Diethehn,N.J.Ford,A.D.Freed,A predictor-corrector approach for the numerical solution of fractional differential equations[J],Nonlinear Dynamics,29(1-4)(2002)3-22.
[32]K.Diethelm,N.J.Ford,A.D.Freed,Y.Luchko,Algorithms for the fractional calculus:a selection of numerical methods]J],Comput.Methods Appl.Mech.Eng.,194(2005) 743-773.
[33]P.Kumar,O.P.Agrawal,A cubic scheme for numerical solution of fractional differential equations,in:Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference,ENOC-2005,pp.1479-1487.
[34]P.Kumar,O.P.Agrawal,An approximate method for numerical solution of fractional differential equations[J],Signal Processing,86(2006) 2602-2610.
[35]K.Diethelm and G.Walz,Numerical solution of fractional order differdential equations by extrapolation[J],Numer.Algorithms,16(1997) 231-253.
[36]K.Diethelm,N.J.Ford and A.D.Freed,Detailed error analysis for a fractional Adams method,Berichte der Mathematischen Institut der TU Braunschweig 02/02,Braunschweig,submitted for publication.
[37]Neville J.Ford and A.Charles Simpson,The numerical solution of fractional differential equations:Speed versus accuracy,Numerical Algorithms[J],26(2001) 333-346.
[38]W.H.Deng,Short memory principle and a predictor corrector approach for fractional differential equations[J],Journal of Computational and Applied Mathematics,Received 28 April 2006.
[39]K.Diethelm,N.J.Ford,Analysis of fractional differential equations[J],J.Math.Anal.Appl.,265(2002) 229-248.
[40]王仁宏,数值逼近[M],高等教育出版社,1999年6月.
[41]C.Lubich,Discretized fractional calculus[J],SIAM J.Math.Anal.,17(3)(1986)pp.704-716.
[42]R.Lin,F.Liu,A high order approximation of fractional order ordinary differential equation initial value problem[J],厦门大学学报(自科版),43(1)(2004)25-30.
[43]K.Diethelm,An algorithm for the numerical solution of differential equations of fractional order[J],Electronic Transactions on Numerical Analysis,Volume 5.pp.1-6,March 1997.
[44]F.W.Liu,Fractional differential equations and its application.Australia,2006.
[451 N.J.Ford,C.A.Simpson,The approximate solution of fractional differential equations of order greater than 1,Internet download,URL(http://www.chester.ac.uk/maths/nevillepub.html) May 2003.
[46]K.Diethelm,Efficient solution of multi-term fractional differential equation using P(EC)~m E methods[J],Computing,71(2003)305-319.
[47]K.Diethelm,N.J.Ford,Multi-order fractional differential equations and their numerical solution[J],Appl.Math.Comput.,154(3)(2004)621-640
[48]K.Diethelm,N.J.Ford,Analysis of fractional differential equations[J],J.Math.Anal.Appl.,265(2002)229-228
[49]Luciano Galeone,Roberto Garrappa,On Multistep Methods for Differential Equations of Fractional Order[J],Mediterr.a.math.,3(2006),565-580.
[50]X.N.Cao,K.Burrage,F.Abdullah,Variable stepsize implementation for fractional differential equations.
[51]何吉欢,非线性分数微分方程的应用及近似求解方法[J],Bulletin of science and technology,No.2.1999