几类非线性分数阶微分方程初值问题的数值求解及动力性质
|
摘要
近年来,随着分数阶微分方程在生物学、材料科学、化学动力学、电磁学、传输(扩散)、自动控制等许多科学领域中的日益广泛的应用,分数阶微分方程的相关理论逐步发展完善,分数阶微分方程的数值解自然也成为了非常重要的研究热点。
本文主要讨论几类非线性分数阶微分方程初值问题的几种近似解法,以及分数阶动力系统的特殊形状的混沌吸引子,主要内容和成果包括：
第一章主要介绍分数阶微积分、分数阶微分方程及其求解方法的研究背景和现状,并提出本文所要研究的问题。
第二章主要介绍分数阶微积分的基本定义和基本性质及几种常用的求解分数阶微分方程初值问题的数值方法。
第三章主要利用三次样条配置方法求解非线性分数阶微分方程初值问题,证明了相容性定理,得到了该方法求解与此类问题等价的分数阶积分方程初值问题的收敛性、稳定性结果。通过研究三次样条配置方法直接离散非线性分数阶微分方程初值问题所得到的数值格式与该方法离散与之等价的分数阶积分方程初值问题所得到的数值格式之间的关系,进一步证明了利用三次样条配置方法直接离散非线性分数阶微分方程初值问题所得到的数值格式是收敛和稳定的。还利用若干个数值算例验证了理论结果的正确性和方法的高效性。
第四章主要利用三次样条配置方法求解非线性分数阶多阶微分方程初值问题,推广了第三章所得到的结论,得到了相应的相容性、收敛性、稳定性结果,同样通过若干个数值算例验证了理论结果的正确性和方法的高效性。
第五章主要利用加入周期函数的方法,得到了一般的分数阶动力系统的平行条状以及矩形状混沌吸引子生成的一般方法,并通过分数阶Rosser系统、分数阶Lorenz系统以及分数阶Chua系统验证了方法的可靠性。
第六、七章利用变分迭代方法分别求解了一类一般的分数阶多阶微分方程初值问题及延迟积微分方程(包括多延迟微分方程)初值问题,并得到了相应的收敛性定理,通过若干个数值算例说明了变分迭代方法关于这些问题的有效性。
In recent years, many phenomena in material science, computational biology, chem-istry kinetics, control theory and other sciences have been described successfully by the mathematical models with fractional calculus, i.e. the theory of derivatives and integrals of non-integer order. Along with the development of the corresponding theories, the numerical solutions of fractional differential equations (FDEs) have been an important and hot topic in the world.
In this paper, we focus on the studies of some numerical methods and approximately analytical methods for solving the initial value problems (IVPs) of nonlinear fractional differential equations. And the chaotic attractors of fractional order dynamic systems are also discussed. The main contents and results are listed as follows.
In Chapter1, we introduce the background and current situations of the researches into fractional calculus, fractional differential equations and the methods for solving them. And the main research contents of this paper are proposed.
In Chapter2, we introduce the basic definitions and properties. Some numerical methods for solving IVPs of FDEs are listed too.
In Chapter3, we propose the cubic spline collocation method with two parameters for solving the IVPs of nonlinear FDEs. The theorem of the local truncation error of this method is given. And the results of the convergence and the stability of this cubic spline collocation method for the fractional order integral equations which is equivalent to the IVPs of nonlinear FDEs are obtained. We also obtain some results of the convergence and the stability of this method for the IVPs of nonlinear FDEs by using the relationship between the numerical solutions obtained respectively from the cubic spline method for the IVPs of nonlinear FDEs and the corresponding equivalent fractional order integral equations. Some illustrative examples successfully verify our theoretical results and show that this method is efficient.
In Chapter4, we propose the cubic spline collocation method with two parameters for the IVPs of nonlinear multi-order nonlinear fractional differential equations (M-FDEs). We extend the theoretical results obtained in Chapter3. And the corresponding results about the local truncation error, convergence and stability of the cubic spline method for the IVPs of nonlinear M-FDEs are obtained. In the same way, some illustrative examples also verify our theoretical results and show that this method is efficient.
In Chapter5, the generations of multi-stripe chaotic attractors of fractional order systems are considered. The original fractional order chaotir at tractors can be turned into a pattern with multiple "parallel" or " rectangular" stripes by employing certain simple periodic nonlinear functions. The relationships between the parameters in the periodic functions and the shapes of the generated attractors are analyzed. Theoretical investigations about the underlying mechanisms of the parallel stripes in the fractional order attractors are presented, with the fractional order Lorenz, Rossler and Chua systems as examples.
In Chapters6and7, the variational iteration method (VIM) is applied to obtain ap-proximately analytical solutions of M-FDEs and delay integro-differential equations (DIDEs) including multi-delay integro-differential equations (MDDEs). The corresponding theorems for convergence and error estimates of the VIM for solving M-FDEs and DIDEs are given respectively. The numerical results show that our theoretical analysis are right and the VIM is a powerful method for solving these equations.
本文主要讨论几类非线性分数阶微分方程初值问题的几种近似解法,以及分数阶动力系统的特殊形状的混沌吸引子,主要内容和成果包括：
第一章主要介绍分数阶微积分、分数阶微分方程及其求解方法的研究背景和现状,并提出本文所要研究的问题。
第二章主要介绍分数阶微积分的基本定义和基本性质及几种常用的求解分数阶微分方程初值问题的数值方法。
第三章主要利用三次样条配置方法求解非线性分数阶微分方程初值问题,证明了相容性定理,得到了该方法求解与此类问题等价的分数阶积分方程初值问题的收敛性、稳定性结果。通过研究三次样条配置方法直接离散非线性分数阶微分方程初值问题所得到的数值格式与该方法离散与之等价的分数阶积分方程初值问题所得到的数值格式之间的关系,进一步证明了利用三次样条配置方法直接离散非线性分数阶微分方程初值问题所得到的数值格式是收敛和稳定的。还利用若干个数值算例验证了理论结果的正确性和方法的高效性。
第四章主要利用三次样条配置方法求解非线性分数阶多阶微分方程初值问题,推广了第三章所得到的结论,得到了相应的相容性、收敛性、稳定性结果,同样通过若干个数值算例验证了理论结果的正确性和方法的高效性。
第五章主要利用加入周期函数的方法,得到了一般的分数阶动力系统的平行条状以及矩形状混沌吸引子生成的一般方法,并通过分数阶Rosser系统、分数阶Lorenz系统以及分数阶Chua系统验证了方法的可靠性。
第六、七章利用变分迭代方法分别求解了一类一般的分数阶多阶微分方程初值问题及延迟积微分方程(包括多延迟微分方程)初值问题,并得到了相应的收敛性定理,通过若干个数值算例说明了变分迭代方法关于这些问题的有效性。
In recent years, many phenomena in material science, computational biology, chem-istry kinetics, control theory and other sciences have been described successfully by the mathematical models with fractional calculus, i.e. the theory of derivatives and integrals of non-integer order. Along with the development of the corresponding theories, the numerical solutions of fractional differential equations (FDEs) have been an important and hot topic in the world.
In this paper, we focus on the studies of some numerical methods and approximately analytical methods for solving the initial value problems (IVPs) of nonlinear fractional differential equations. And the chaotic attractors of fractional order dynamic systems are also discussed. The main contents and results are listed as follows.
In Chapter1, we introduce the background and current situations of the researches into fractional calculus, fractional differential equations and the methods for solving them. And the main research contents of this paper are proposed.
In Chapter2, we introduce the basic definitions and properties. Some numerical methods for solving IVPs of FDEs are listed too.
In Chapter3, we propose the cubic spline collocation method with two parameters for solving the IVPs of nonlinear FDEs. The theorem of the local truncation error of this method is given. And the results of the convergence and the stability of this cubic spline collocation method for the fractional order integral equations which is equivalent to the IVPs of nonlinear FDEs are obtained. We also obtain some results of the convergence and the stability of this method for the IVPs of nonlinear FDEs by using the relationship between the numerical solutions obtained respectively from the cubic spline method for the IVPs of nonlinear FDEs and the corresponding equivalent fractional order integral equations. Some illustrative examples successfully verify our theoretical results and show that this method is efficient.
In Chapter4, we propose the cubic spline collocation method with two parameters for the IVPs of nonlinear multi-order nonlinear fractional differential equations (M-FDEs). We extend the theoretical results obtained in Chapter3. And the corresponding results about the local truncation error, convergence and stability of the cubic spline method for the IVPs of nonlinear M-FDEs are obtained. In the same way, some illustrative examples also verify our theoretical results and show that this method is efficient.
In Chapter5, the generations of multi-stripe chaotic attractors of fractional order systems are considered. The original fractional order chaotir at tractors can be turned into a pattern with multiple "parallel" or " rectangular" stripes by employing certain simple periodic nonlinear functions. The relationships between the parameters in the periodic functions and the shapes of the generated attractors are analyzed. Theoretical investigations about the underlying mechanisms of the parallel stripes in the fractional order attractors are presented, with the fractional order Lorenz, Rossler and Chua systems as examples.
In Chapters6and7, the variational iteration method (VIM) is applied to obtain ap-proximately analytical solutions of M-FDEs and delay integro-differential equations (DIDEs) including multi-delay integro-differential equations (MDDEs). The corresponding theorems for convergence and error estimates of the VIM for solving M-FDEs and DIDEs are given respectively. The numerical results show that our theoretical analysis are right and the VIM is a powerful method for solving these equations.
引文
[1]S. Abbasbandy. A new application of He's variational iteration method for quadratic Ric-cati differential equation by using Adomian's polynomials[Jl. Journal of Computational and Applied Mathematics,207(2007)59-63.
[2]M. A. Abdoua, A. A. Solimanb. Variational iteration method for solving Burger's and coupled Burger's equations[J]. J. Comput. Appl. Math.,181(2005) 245-251.
[3]W. G. Ajello, H. I. Freedman, J. Wu, A model of stage structured population growth with density dependent time delay[J]. SIAM J. Appl. Math.,52 (1992) 855-869.
[4]B. Batiha, M. S. M. Noorani, I. Hashim. Application of variational iteration method to heat and wave-like equations[J]. Phys. Lett. A,369(1-2)(2007)55-61.
[51 B. Batiha, M. S. M. Noorani, I. Hashim, E. S. Ismail. The multiple stage variational iteration method for class of nonlinear system of ODEs[J]. Phys. Scr.,76 (2007) 388-392.
[6]B. Baeumer, M. Kovacs and M. M. Meerschaert. Fractional reaction-diffusion equation for species growth and dispersal[J]. J. Math. Biolology,2007, http://www.maths.otago.ac.nz/ mcubed/JMBseed.pdf.
[71 A. Bcllen, M. Zermaro, Numerical methods for delay differential equations[M], Oxford: Clarendon Press.2003.
[8]L. Blank. Numerical treatment of differential equation of fraction order[R]. Numerical Analysis Report 287, Manchedter Center for Numerical Computational Mathematics(1996).
[91 H. Brunner, Collocation mehtods for Volterra integral and related functional diffrential equa-tions[M]. Combridge University Press,2004.
[10]H. Brunner, A. Pedas and G. Vainikko. A spline collocation method for linear Volterra integro-differential equations[J]. BIT.,41(5)(2001)891-900.
[11]D. A. Benson, S. W. Wheateraft and M. M. Meersehaert. Application of a fractional advection-despersion equation[J]. Water Resour. Res.,36(6)(2000)1403-1412.
[12]A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, CISM International Centre for Mechanical Sciences Series 378, Springer. Verlag, New York,1997.
[13]Yanzhao Cao, Terry et al, A hybrid collocation method for Voltcrra integral equations with weakly singular kernels[J]. SIAM J. Numer. Anal..41(1)(2003)364-381.
[14]G. Chen and X. Yu. Chaos control:Theory and applications[J]. Springer-Verlag, Berlin,2003.
[15]Q. Chen, Y. Hong, and G. Chen. Chaotic behaviors and toroidal/spherical attractors gener-ated by discontinuous dynamics[J]. Physica A,371(2006)293-302.
[16]Q. Chen, Y. Hong, G. Chen, and Q. Zhong. Chaotic attractors in striped rectangular shapes generated by a Rossler-like system[J]. Physics Letters A,348(2006)195-200.
[17]Q. Chen, Y. Zhang, and Y. Hong. Generation and control of striped attractors of Rossler systems with feedback[J]. Chaos, Solitons, Fractals,34(3)(2007)693-703.
[18]J. P. Coleman, S. C. Duxbury. Mixed collocation methods for y"=f(x,y)[J]. J. Comput. Appl. Math.,126(2000)47-75.
[19]Davod Khojasteh Salkuyeh. Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients[J]. Computers and Mathematics with Applica-tions,56(2008)2027-2033.
[20]J. Dixon and S. Mckee. Weakly singular discrete Grouwall inequalities[J]. Z. Angew Math. Mech.,66(11)(1986)535-544.
[21]K. Diethelm and G. Walz. Numerical solution of fractional order differential equation by extrapolation[J]. Numer. Algorithms,16(1997)231-253,
[22]K. Diethelm, N. J. Ford and Alan D. Freed. A predictor-corrector approach for the numerical solution of fractional differential equations [J]. Nonlinear Dynamics,29(2002) 3-22.
[23]Kai Diethelm and Neville J. Ford. Analysis of Fractional Differential Equations[M]. Springer, 2003.
[24]Kai Diethelm. The Analysis of Fractional Differential Equations[M]. Springer,2010.
[25]K. Diethelm, N. J. Ford and A. D. Reed. Detailed error analysis for a fractional Adams method [J]. Numerical Algorithms, Kluwer Academic Publishers,36(1)(2004)31-52.
[26]K. Diethelm. An algorithm for the numerical solution of differential equations of fractional order [J]. Electronic Transactions on Numerical Analysis,5(1997)1-6.
[27]K. Diethelm, N. J. Ford. Numerical solution of the Bagley-Torvik equation[J]. BIT, 42(2002)490-507.
[28]K. Diethelm, Y. Luchko. Numerical solution of linear multi-order differential equations of fractional order [J]. J. Comput. Anal. Appl.,6(2004)243-63.
[29]M. T. Darvishi, F. Khani, A. A. Soliman. The numerical simulation for stiff systems of ordinary differential equations[J]. Computers Math. Appl.,54(2007)1055-1063.
[30]David Elizarraraz and Luis Verde-Star. Fractional divided differences and the solution of differential equations of fractional order[J]. Advances in Applied Mathematics,24(2000)260-283.
[31]G. E. Draganescu. Application of a variational iteration method to linear and nonlinear vis-coelastic models with fractional derivatives[J]. J. Math. Phys.,47(8)(2006)082902.
[32]J. T. Edwards, N. J. Ford, A. C. Simpson. The numerical solution of linear multi-order frac-tional differential equations:systems of equations[J]. J. Comput. Appl. Math.,148(2002)401-418.
[33]A. Elwalil, S. Ozoguz. and M. Kennedy. Creation of a complex butterfuly attractor using a noval Lorenz-type system [J]. IEEE Trans. Circuits and Systems-1.49(2002)527-530.
[34]A. E. M. El-Mesiry, A. M. A. El-Sayed and H. A. A. El-Saka. Numerical methods for multi-term fractional (arbitrary) orders differential equations [J]. Appl. Math. Comput., 160(3)(2005)683-699.
[35]S. A. El-Wakil, M. A. Abdou, New applications of variational iteration method using Adomian polynomials [J]. Nonlinear Dyn,52(2008)41-49.
[36]H. M. El-Hawary, S. M. Mahmoud. On some 4-point spline collocation methods for solving ordianry initial value problems[J]. J. Comput. Math.,79(2002)931-944.
[37]H. M. El-Ha.wary, S. M. Mahmoud. On some 4-point spline collocation methods for solving second-order initial value problems [J]. Appl. Numer. Math.,38(2001)223-236.
[38]H. M. El-Hawary, S. M. Mahmoud. Spline collocation methods for solving delay-differential equations[J]. Appl. Math. Comput.,146(2003)359-372.
[39]樊华.泛函微分方程数值方法的B-理论在刚性微分方程数值分析中的应用[D].湘潭大学硕士学位论文,2005年.
[40]L. Fox, D. F. Mayers, J. A. Ockendon, A. B. Tayler. On a functional differential equation[J]. J. Inst. Math. Appl.8(1971)271-307.
[41]冯云.分数阶微分方程的混合配置及自适应配置算法[D].湘潭大学硕士学位论文,2010年.
[42]R. Gilmore and M. Ldfranc. Topology of chaos:Alice in stretch and squeezeland[M]. Wiley-Interscience, New York,2002.
[43]O. Heaviside. Electromagnetic Theory[M].3rd Ed., New York,1971.
[44]J. H. He. Approximate analytical solution for seepage flow with fractional deritives in porous media[J]. Comput. Methods Appl. Mech. Engrg.,167(1998)57-68.
[45]J. H. He, A new approach to linear partial differential equations[J], Commun. Nonlinear Sci. Numer. Simul.,2(4)(1997)230-235.
[46]J. H. He. Some applications of nonlinear fractional differential equations and their approxi-mations [J]. Bull. Sci. Technol.,15(12)(1999)86-90.
[47]J. H. He. Approximate solution of nonlinear differential equation with convolution product nonlinearities[J]. Comput. Methods Appl. Mech. Engrg..167(1998)69-73.
[481 J.H. He. Variational iteration method for autonomous ordinary differential systems[J]. Appl. Math. Comput.,114(2000)115-123.
[49]J. H. He. Variational iteration method for delay differential equations[J]. Commun. Nonlinear Sci. Numer. Simul.,2(4)(1997) 235-236.
[50]J. H. He, Variational iteration method -Some recent results and new interpretations[J]. J. Comput. Appl. Math.,207(2007) 3-17.
[51]Hilfer (Editor), Applications of Fractional Calculus in Physics[M], World Scientific, Singapore, 2000.
[52]C. D Hu, SA Meguid. Stability of Runge-Kutta methods for delay differential systems with multiple delays[J]. IMA Journal of Numerical Analysis.19(3)(1999)349-356.
[53]M. Inokuti, H. Sekine, T. Mura. General use of the Lagrange multiplier in nonlinear math-ematical physics, in:S. Nemat-Nasser (Ed.), Variational Methods in the Mechanics of Solids[M]. Pergamon Press. NewYork. (1978)156-162.
[54]A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and applications of fractional differential equations[M]. Elsevier,2006.
[55]F. Liu. I. Turner and V. Anh. An unstructured mesh finite volume method for modelling salt-water intrusion into coastal aquifers[J]. Korean J. Comput. and Appl. Math.,9(2)(2002)391-407.
[56]F. Liu, V. Anh, I. Turner. Numerical solution of the space fractional Fokker-Planck equa-tion[J]. Journal of Computational and Applied Mathematies,166(2004)209-219.
[57]F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage. Stability and convergence of the dif-ference methods for the space-time fractional advection-diffusion equation [J], Appllied Math-ematics and Computation,191(2007)12-20.
[58]R. Lin and F. Liu. Fractional high order methods for the nonlinear fractinal ordinary differ-ential equation[J]. Nonlinear Analysis,66(2007)856-869.
[59]D. Li, M. Z. Liu, Runge-Kutta methods for the multi-pantograph delay equation[J]. Appl. Math. Comput.163(2005)383-395.
[60]C. Li, G. Chen. Chaos and hyperchaos in the fractional-order Rossle equations[J]. Physica A, 341(2004)55-61.
[61]C. Li, G. Chen. Chaos in the fractional order Chen system and its control[J]. Chaos, Solitons and Fractals,22(2004)549-554.
[62]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.
[63]李敏.分数阶微分方程初值问题数值方法[D].中国科学院博士学位论文,2009年.
[64]李云飞.几类求解分数阶微分方程的Runge-Kutta方法[D].湘潭大学硕士学位论文,2010年.
[65]Changpin Li., An Chen, Junjie Yc. Numerical approaches to fractional calculus and fractional ordinary differential equation[J]. Journal of Computational Physics,230 (2011) 3352-3368.
[66]刘建军.求解分数阶微分方程问题的几类数值方法[D].湘潭大学硕士学位论文,2007年.
[67]刘祥.求解分数阶微分方程的配置方法[D].湘潭大学硕士学位论文,2009年.
[68]林玉闽.时间分数阶偏微分方程的解及其应用[D].厦门大学博士论文,2008.
[69]Jun-Feng Lu, Variational iteration method for solving two-point boundary value problems[J]. J. Comput, Appl. Math.207(2007) 92-95.
[70]Yu. Luchko, H. M. Srivastava. The exact solution of certain differential equations of fractional order by using operational calculus[J]. Computers Math. App.,29(8)(1995)73-85.
[71]Yurii Luchro, Rudolf Gorenflo. An operational method for solving fractional differential equa-tions with the Caputo derivatives[J]. Acta Mathematica Vietnamiea,24(2)(1999)207-233.
[72]C. Lubich. Runge-Kutta theory for Volterra and Abel integral equations of the second kind[J]. Mathematics of Computation,41(1983) 87-102.
[73]C. Lubich. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind[J]. Mathematics of Computation,45(1985)463-469.
[74]C. Lubich. Diseretized fractional calculus [J], SIAM Journal on Mathematical Analysis, 17(1986)704-719.
[75]B. B. Mandelbrot, The Fractal Geometry of Nature[M]. W H Freeman, New York,1982.
[76]R. Metzeler and J. Klafter. The random walk's guide to anomalous diffsion:a fractional dynamics approach [J]. Phys. Rep.,339(2000)1-77.
[77]R. Metzeler and J. Klafter. The restaurant at the end of the random walk recent developments in the descreption of anomalous transport by fractional dynamics[J]. J. Phys. A:Math. Gen., 37(2004)161-208.
[78]K. S. Miller and B. Ross. An introduction to the Fractional Calculus and Fractional Differ-ential Equations[M]. John Wiley and Sons, New York,1993.
[79]S. Momani and K. AI-Khaled. Numerical solutions for systems of fractional differential equa-tions by the decomposition method[J]. Appl. Math. Comp.,162(3)(2005)1351-1365.
[80]S. Momani. Analytic and appromate solutions of the spae- and time-fractional telegrah equations[J]. Applied Mathematics and Computation,170(2005)1126-1134.
[81]S. Momani, Z. Odibat. Analytic approach to linear fractional partial differential equations arising in fluid mechanics[J]. Phys. Lett. A.,355(2006)271-279.
[82]S. Momani, S. Abuasad. Application of He's variational iteration meathod to Helmhotz equa-tion[J]. Chaos, Solitons & Fractals,27(5)(2006)1119-1123.
[83]S. Momani, Z. Odibat. Numerical comparison of methods for solving linear differential equa-tions of fractional order [J]. Chaos, Solitons & Fractals,31(5)(2007)1248-1255.
[84]S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order[J]. J. Comput. Appl. Math..207(2007)96-110.
[85]S. Momani, Z. Odibat, A. Alawneh. Variational iteration method for solving the space-and time-fractional KdV equation[J]. Numer. Math. Partial Diff. Eqn.,24(1)(2007)262-271.
[86]R.Yulita Molliq, M. S. M. Noorani, I. Hashim. Variational iteration method for fractional heat-and wave-like equations[J]. Nonlinear Analysis:Real World Applications,10(2009)1854-1869.
[87]聂宁明.几类分数阶微分方程边值问题和初边值问题的数值方法[D].中国科学院博士论文,2010年.
[88]J. R. Ockendon, A. B. Tayler. The dynamics of a current collection system for an electric locomotive[J]. Proc. Roy. London Ser. A.322 (1971)447-468.
[89]Z. Odibat, S. Momani. Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives [J]. Phys. Lett. A.,369(5-6) (2007)349-358.
[90]Z. Odibat, S. Momani. Application of variational iteration method to nonlinear differential equations of fractional order [J]. Int. J. Nonlinear Sci. Numer. Simul., 1(7)(2006) 15-27.
[91]K. B. Oldham and J. Spanier. The Fractional Calculus[M]. Academic Press. New York,1974.
[921 E. Ott. Chaos in dynamical systems,2nd Edition [M]. Cambridge University Press, Cambridge, 2002.
[93]L. Pecora, T. Carroll, G. Johnson, D. Mar, and J. Heagy. Fundamentals of synchronization in chaotic systems:concepts and applications [J]. Chaos,7(1997)520-539.
[94]I. Podlubny. Fractional Differential Equations[M]. Academic Press, New York,1999.
[95]Arvet Pedas, Enn Tamme. Spline collocation methods for linear multi-term fractional differ-ential equations [J]. Journal of Computational and Applied Mathematics,236 (2011) 167-176.
[96]A. Pedas. E. Tamme, Spline collocation method for integro-differential equations with weakly singular kernels [J]. J. Comput. Appl. Math.,197 (2006) 253-269.
[97]A. Pedas, E. Tamme. On the convergence of spline collocation methods for solving fractional differential equations[J]. J. Comput. Appl. Math.,235 (2011) 3502-3514.
[98]D. Rothstein, E. Henry, J. Gollub. Persistent patterns in transcient chaotic mixing[J]. Nature, 401(1999)770-771.
[99]M. T. Rosenstein, J. J. Collins et al. A practical method for calculating largest Lyapunov exponents from small data sets[J]. Physica D:Nonlinear Phenomena,65(1993)117-134.
[100]S. Sallam. Stable quartic spline integration method for solving stiff ordiamy differential equa-tions [J]. Appl. Math. Comput.,116(2000)245-255.
[101]S. Sallam, A. A. Karaballi. A quartic C3-spline collocation method for solving second-order initial value problems[J]. J. Comput. Appl. Math.,75(1996)295-304.
[102]S. Sallam, M. Naim Anwar. One parameter quadratic C1-spline collocation method forsolving first order ordinary initial value problems[J]. Appl. Math. Model.,23(1999)153-160.
[103]S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives:Theory and Applications[M], Gordon and Breach Science publishers,1987.
[104]W. R. Schneider and W. Wyss, Fractional diffusion and wave equations[J]. J. Math. Phys.. 30(1989)134-144.
[105]M. F. Shlesinger, B. J. West and J. Klafter, Levy Dynamics of enhances diffusion:Appli-cation to turbulence[J]. Phys. Rew. Lett.,58(11)(1987)1100-1103.
[106]N. T. Shawagfeh. Analytical appromate solutions for nonlinear fractional differentia equ tions[J]. Appl. Math, and Comp.,13(2002)517-529.
[107]M. Siewe, F. M. Kakmeni, C. Tchawoua and P. Woafo. Bifurcations and chaos in the triple-well (?)6-Van der Pol oscillator driven by external and parametric excitations[J]. Physica A, 357(2005)383-396.
[108]A. A. Soliman. A numeric simulation an explicit solutions of KdV-Burgcrs'and Lax's seventh-order KdV equations[J]. Chaos, Solitons & Fractals,29(2)(2006)294-302.
[109]苏红.求解几类泛函微分方程的三次样条配置方法[D].湘潭大学硕士学位论文,2010.
[110]N. H. Swcilam, M. M. Khader and R. F. Al-Bar. Numerical studies for a multi-order fractional differential equation[J]. Physics Letters A,371(2007)26-33.
[111]Mehdi Tatari, Mehdi Dehghan. On the convergence of He's variational iteration method[J]. J. Comput. Appl. Math.207(2007) 121-128.
[112]Vedat Suat Erturk, Shaher Momani, Zaid Odibat, Application of generalized differential trans-form method to multi-order fractional differential equations[J]. Communications in Nonlinear Science and Numerical Simulation,13(2008)1642-1654.
[113]S. Westerlund and L. Eksam, Capacitor Theory, IEEE Trans, on Dielectrics and Electrical Insulation [J].1(5)(1994) 826-839.
[114]Wyss W. The fractional diffusion equation[J].J. Math. Phys,27(1986)2782-2785.
[115]Wan-Sheng Wang, Shou-Fu Li. On the one-leg θ-methods for solving nonlinear neutral func-tional differential equations[J]. Appl. Math. Comput.193 (2007) 285-301.
[116]Wan-Sheng Wang, Shou-Fu Li. Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations[J]. Front. Math. China.4(1)(2009)195-216.
[117]王海燕.求解分数阶微分方程的Runge-Kutta方法[D].湘潭大学硕士学位论文,2008年.
[118]M. Wcilbeer. Efficient numerical methods for fractional differential equations and their ana-lytical background [J]. Technical University of Braunsehwig,2005.
[119]J. Wu, Q. Chen, and Y. Hong. Striped attractor generation and synchronization analysis for coupled Rosslcr systems [J]. Chaos, Solitons, Fractals,39(1)(2009)322-331.
[120]J. Wu, Y. Hong, G. Chen. Occurrence and underlying mechanism of multi-stripe chaotic attractors[J]. Chaos. Solitons, Fractals,41(5)(2009)2250-2258.
[121]X. Wu, et al. Chaos in the fractional order unified system and its synchronization [J]. Journal of the Franklin Institute,345(4)(2008)392-401.
[122]杨光俊.分数阶积分方程[J].云南大学学报(自然科学版),19(3)(1997)328-334
[123]M. Yalcin, J. Suykens, and J. Vandevalle. Cellular neural networks, multi-scoll chaos and synchronization[M]. World Scientific, Singapore,2005.
[124]伊婕.变分迭代法关于Caputo分数阶常微分方程和中立型比例延迟微分方程的收敛性分析[D].湘潭大学硕士学位论文,2010年.
[125]Zhan-Hua Yu. Variational iteration method for solving the multi-pantograph delay equa tion[J]. Physics Letters A.372 (2008) 6475-6479.
[126]章红梅.含Riesz (-Feller)位势算子的分数阶偏微分方程的基本解与数值解[D].厦门大学博士论文,2007.
[127]张晓娟.多阶的分数阶常微分方程和分数阶扩散—波动方程[D].厦门大学硕士学位论文,2008.
[128]张晓娟,王婧.配置方法求多阶的分数阶常微分方程的数值解[J].华北水利水电学院学报,2010年第3期.
[129]张亚平.新迭代方法在几类分数阶微分方程中的应用[D].湘潭大学硕士学位论文,2009年4月.
[130]H. Zhu, S. Zhou, J. Zhang. Chaos and synchronization of the fractional-order Chua's system [J]. Chaos, Solitons and Fractals,39(4)(2009) 1595-1603.
[2]M. A. Abdoua, A. A. Solimanb. Variational iteration method for solving Burger's and coupled Burger's equations[J]. J. Comput. Appl. Math.,181(2005) 245-251.
[3]W. G. Ajello, H. I. Freedman, J. Wu, A model of stage structured population growth with density dependent time delay[J]. SIAM J. Appl. Math.,52 (1992) 855-869.
[4]B. Batiha, M. S. M. Noorani, I. Hashim. Application of variational iteration method to heat and wave-like equations[J]. Phys. Lett. A,369(1-2)(2007)55-61.
[51 B. Batiha, M. S. M. Noorani, I. Hashim, E. S. Ismail. The multiple stage variational iteration method for class of nonlinear system of ODEs[J]. Phys. Scr.,76 (2007) 388-392.
[6]B. Baeumer, M. Kovacs and M. M. Meerschaert. Fractional reaction-diffusion equation for species growth and dispersal[J]. J. Math. Biolology,2007, http://www.maths.otago.ac.nz/ mcubed/JMBseed.pdf.
[71 A. Bcllen, M. Zermaro, Numerical methods for delay differential equations[M], Oxford: Clarendon Press.2003.
[8]L. Blank. Numerical treatment of differential equation of fraction order[R]. Numerical Analysis Report 287, Manchedter Center for Numerical Computational Mathematics(1996).
[91 H. Brunner, Collocation mehtods for Volterra integral and related functional diffrential equa-tions[M]. Combridge University Press,2004.
[10]H. Brunner, A. Pedas and G. Vainikko. A spline collocation method for linear Volterra integro-differential equations[J]. BIT.,41(5)(2001)891-900.
[11]D. A. Benson, S. W. Wheateraft and M. M. Meersehaert. Application of a fractional advection-despersion equation[J]. Water Resour. Res.,36(6)(2000)1403-1412.
[12]A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, CISM International Centre for Mechanical Sciences Series 378, Springer. Verlag, New York,1997.
[13]Yanzhao Cao, Terry et al, A hybrid collocation method for Voltcrra integral equations with weakly singular kernels[J]. SIAM J. Numer. Anal..41(1)(2003)364-381.
[14]G. Chen and X. Yu. Chaos control:Theory and applications[J]. Springer-Verlag, Berlin,2003.
[15]Q. Chen, Y. Hong, and G. Chen. Chaotic behaviors and toroidal/spherical attractors gener-ated by discontinuous dynamics[J]. Physica A,371(2006)293-302.
[16]Q. Chen, Y. Hong, G. Chen, and Q. Zhong. Chaotic attractors in striped rectangular shapes generated by a Rossler-like system[J]. Physics Letters A,348(2006)195-200.
[17]Q. Chen, Y. Zhang, and Y. Hong. Generation and control of striped attractors of Rossler systems with feedback[J]. Chaos, Solitons, Fractals,34(3)(2007)693-703.
[18]J. P. Coleman, S. C. Duxbury. Mixed collocation methods for y"=f(x,y)[J]. J. Comput. Appl. Math.,126(2000)47-75.
[19]Davod Khojasteh Salkuyeh. Convergence of the variational iteration method for solving linear systems of ODEs with constant coefficients[J]. Computers and Mathematics with Applica-tions,56(2008)2027-2033.
[20]J. Dixon and S. Mckee. Weakly singular discrete Grouwall inequalities[J]. Z. Angew Math. Mech.,66(11)(1986)535-544.
[21]K. Diethelm and G. Walz. Numerical solution of fractional order differential equation by extrapolation[J]. Numer. Algorithms,16(1997)231-253,
[22]K. Diethelm, N. J. Ford and Alan D. Freed. A predictor-corrector approach for the numerical solution of fractional differential equations [J]. Nonlinear Dynamics,29(2002) 3-22.
[23]Kai Diethelm and Neville J. Ford. Analysis of Fractional Differential Equations[M]. Springer, 2003.
[24]Kai Diethelm. The Analysis of Fractional Differential Equations[M]. Springer,2010.
[25]K. Diethelm, N. J. Ford and A. D. Reed. Detailed error analysis for a fractional Adams method [J]. Numerical Algorithms, Kluwer Academic Publishers,36(1)(2004)31-52.
[26]K. Diethelm. An algorithm for the numerical solution of differential equations of fractional order [J]. Electronic Transactions on Numerical Analysis,5(1997)1-6.
[27]K. Diethelm, N. J. Ford. Numerical solution of the Bagley-Torvik equation[J]. BIT, 42(2002)490-507.
[28]K. Diethelm, Y. Luchko. Numerical solution of linear multi-order differential equations of fractional order [J]. J. Comput. Anal. Appl.,6(2004)243-63.
[29]M. T. Darvishi, F. Khani, A. A. Soliman. The numerical simulation for stiff systems of ordinary differential equations[J]. Computers Math. Appl.,54(2007)1055-1063.
[30]David Elizarraraz and Luis Verde-Star. Fractional divided differences and the solution of differential equations of fractional order[J]. Advances in Applied Mathematics,24(2000)260-283.
[31]G. E. Draganescu. Application of a variational iteration method to linear and nonlinear vis-coelastic models with fractional derivatives[J]. J. Math. Phys.,47(8)(2006)082902.
[32]J. T. Edwards, N. J. Ford, A. C. Simpson. The numerical solution of linear multi-order frac-tional differential equations:systems of equations[J]. J. Comput. Appl. Math.,148(2002)401-418.
[33]A. Elwalil, S. Ozoguz. and M. Kennedy. Creation of a complex butterfuly attractor using a noval Lorenz-type system [J]. IEEE Trans. Circuits and Systems-1.49(2002)527-530.
[34]A. E. M. El-Mesiry, A. M. A. El-Sayed and H. A. A. El-Saka. Numerical methods for multi-term fractional (arbitrary) orders differential equations [J]. Appl. Math. Comput., 160(3)(2005)683-699.
[35]S. A. El-Wakil, M. A. Abdou, New applications of variational iteration method using Adomian polynomials [J]. Nonlinear Dyn,52(2008)41-49.
[36]H. M. El-Hawary, S. M. Mahmoud. On some 4-point spline collocation methods for solving ordianry initial value problems[J]. J. Comput. Math.,79(2002)931-944.
[37]H. M. El-Ha.wary, S. M. Mahmoud. On some 4-point spline collocation methods for solving second-order initial value problems [J]. Appl. Numer. Math.,38(2001)223-236.
[38]H. M. El-Hawary, S. M. Mahmoud. Spline collocation methods for solving delay-differential equations[J]. Appl. Math. Comput.,146(2003)359-372.
[39]樊华.泛函微分方程数值方法的B-理论在刚性微分方程数值分析中的应用[D].湘潭大学硕士学位论文,2005年.
[40]L. Fox, D. F. Mayers, J. A. Ockendon, A. B. Tayler. On a functional differential equation[J]. J. Inst. Math. Appl.8(1971)271-307.
[41]冯云.分数阶微分方程的混合配置及自适应配置算法[D].湘潭大学硕士学位论文,2010年.
[42]R. Gilmore and M. Ldfranc. Topology of chaos:Alice in stretch and squeezeland[M]. Wiley-Interscience, New York,2002.
[43]O. Heaviside. Electromagnetic Theory[M].3rd Ed., New York,1971.
[44]J. H. He. Approximate analytical solution for seepage flow with fractional deritives in porous media[J]. Comput. Methods Appl. Mech. Engrg.,167(1998)57-68.
[45]J. H. He, A new approach to linear partial differential equations[J], Commun. Nonlinear Sci. Numer. Simul.,2(4)(1997)230-235.
[46]J. H. He. Some applications of nonlinear fractional differential equations and their approxi-mations [J]. Bull. Sci. Technol.,15(12)(1999)86-90.
[47]J. H. He. Approximate solution of nonlinear differential equation with convolution product nonlinearities[J]. Comput. Methods Appl. Mech. Engrg..167(1998)69-73.
[481 J.H. He. Variational iteration method for autonomous ordinary differential systems[J]. Appl. Math. Comput.,114(2000)115-123.
[49]J. H. He. Variational iteration method for delay differential equations[J]. Commun. Nonlinear Sci. Numer. Simul.,2(4)(1997) 235-236.
[50]J. H. He, Variational iteration method -Some recent results and new interpretations[J]. J. Comput. Appl. Math.,207(2007) 3-17.
[51]Hilfer (Editor), Applications of Fractional Calculus in Physics[M], World Scientific, Singapore, 2000.
[52]C. D Hu, SA Meguid. Stability of Runge-Kutta methods for delay differential systems with multiple delays[J]. IMA Journal of Numerical Analysis.19(3)(1999)349-356.
[53]M. Inokuti, H. Sekine, T. Mura. General use of the Lagrange multiplier in nonlinear math-ematical physics, in:S. Nemat-Nasser (Ed.), Variational Methods in the Mechanics of Solids[M]. Pergamon Press. NewYork. (1978)156-162.
[54]A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and applications of fractional differential equations[M]. Elsevier,2006.
[55]F. Liu. I. Turner and V. Anh. An unstructured mesh finite volume method for modelling salt-water intrusion into coastal aquifers[J]. Korean J. Comput. and Appl. Math.,9(2)(2002)391-407.
[56]F. Liu, V. Anh, I. Turner. Numerical solution of the space fractional Fokker-Planck equa-tion[J]. Journal of Computational and Applied Mathematies,166(2004)209-219.
[57]F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage. Stability and convergence of the dif-ference methods for the space-time fractional advection-diffusion equation [J], Appllied Math-ematics and Computation,191(2007)12-20.
[58]R. Lin and F. Liu. Fractional high order methods for the nonlinear fractinal ordinary differ-ential equation[J]. Nonlinear Analysis,66(2007)856-869.
[59]D. Li, M. Z. Liu, Runge-Kutta methods for the multi-pantograph delay equation[J]. Appl. Math. Comput.163(2005)383-395.
[60]C. Li, G. Chen. Chaos and hyperchaos in the fractional-order Rossle equations[J]. Physica A, 341(2004)55-61.
[61]C. Li, G. Chen. Chaos in the fractional order Chen system and its control[J]. Chaos, Solitons and Fractals,22(2004)549-554.
[62]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.
[63]李敏.分数阶微分方程初值问题数值方法[D].中国科学院博士学位论文,2009年.
[64]李云飞.几类求解分数阶微分方程的Runge-Kutta方法[D].湘潭大学硕士学位论文,2010年.
[65]Changpin Li., An Chen, Junjie Yc. Numerical approaches to fractional calculus and fractional ordinary differential equation[J]. Journal of Computational Physics,230 (2011) 3352-3368.
[66]刘建军.求解分数阶微分方程问题的几类数值方法[D].湘潭大学硕士学位论文,2007年.
[67]刘祥.求解分数阶微分方程的配置方法[D].湘潭大学硕士学位论文,2009年.
[68]林玉闽.时间分数阶偏微分方程的解及其应用[D].厦门大学博士论文,2008.
[69]Jun-Feng Lu, Variational iteration method for solving two-point boundary value problems[J]. J. Comput, Appl. Math.207(2007) 92-95.
[70]Yu. Luchko, H. M. Srivastava. The exact solution of certain differential equations of fractional order by using operational calculus[J]. Computers Math. App.,29(8)(1995)73-85.
[71]Yurii Luchro, Rudolf Gorenflo. An operational method for solving fractional differential equa-tions with the Caputo derivatives[J]. Acta Mathematica Vietnamiea,24(2)(1999)207-233.
[72]C. Lubich. Runge-Kutta theory for Volterra and Abel integral equations of the second kind[J]. Mathematics of Computation,41(1983) 87-102.
[73]C. Lubich. Fractional linear multistep methods for Abel-Volterra integral equations of the second kind[J]. Mathematics of Computation,45(1985)463-469.
[74]C. Lubich. Diseretized fractional calculus [J], SIAM Journal on Mathematical Analysis, 17(1986)704-719.
[75]B. B. Mandelbrot, The Fractal Geometry of Nature[M]. W H Freeman, New York,1982.
[76]R. Metzeler and J. Klafter. The random walk's guide to anomalous diffsion:a fractional dynamics approach [J]. Phys. Rep.,339(2000)1-77.
[77]R. Metzeler and J. Klafter. The restaurant at the end of the random walk recent developments in the descreption of anomalous transport by fractional dynamics[J]. J. Phys. A:Math. Gen., 37(2004)161-208.
[78]K. S. Miller and B. Ross. An introduction to the Fractional Calculus and Fractional Differ-ential Equations[M]. John Wiley and Sons, New York,1993.
[79]S. Momani and K. AI-Khaled. Numerical solutions for systems of fractional differential equa-tions by the decomposition method[J]. Appl. Math. Comp.,162(3)(2005)1351-1365.
[80]S. Momani. Analytic and appromate solutions of the spae- and time-fractional telegrah equations[J]. Applied Mathematics and Computation,170(2005)1126-1134.
[81]S. Momani, Z. Odibat. Analytic approach to linear fractional partial differential equations arising in fluid mechanics[J]. Phys. Lett. A.,355(2006)271-279.
[82]S. Momani, S. Abuasad. Application of He's variational iteration meathod to Helmhotz equa-tion[J]. Chaos, Solitons & Fractals,27(5)(2006)1119-1123.
[83]S. Momani, Z. Odibat. Numerical comparison of methods for solving linear differential equa-tions of fractional order [J]. Chaos, Solitons & Fractals,31(5)(2007)1248-1255.
[84]S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order[J]. J. Comput. Appl. Math..207(2007)96-110.
[85]S. Momani, Z. Odibat, A. Alawneh. Variational iteration method for solving the space-and time-fractional KdV equation[J]. Numer. Math. Partial Diff. Eqn.,24(1)(2007)262-271.
[86]R.Yulita Molliq, M. S. M. Noorani, I. Hashim. Variational iteration method for fractional heat-and wave-like equations[J]. Nonlinear Analysis:Real World Applications,10(2009)1854-1869.
[87]聂宁明.几类分数阶微分方程边值问题和初边值问题的数值方法[D].中国科学院博士论文,2010年.
[88]J. R. Ockendon, A. B. Tayler. The dynamics of a current collection system for an electric locomotive[J]. Proc. Roy. London Ser. A.322 (1971)447-468.
[89]Z. Odibat, S. Momani. Numerical solution of Fokker-Planck equation with space- and time-fractional derivatives [J]. Phys. Lett. A.,369(5-6) (2007)349-358.
[90]Z. Odibat, S. Momani. Application of variational iteration method to nonlinear differential equations of fractional order [J]. Int. J. Nonlinear Sci. Numer. Simul., 1(7)(2006) 15-27.
[91]K. B. Oldham and J. Spanier. The Fractional Calculus[M]. Academic Press. New York,1974.
[921 E. Ott. Chaos in dynamical systems,2nd Edition [M]. Cambridge University Press, Cambridge, 2002.
[93]L. Pecora, T. Carroll, G. Johnson, D. Mar, and J. Heagy. Fundamentals of synchronization in chaotic systems:concepts and applications [J]. Chaos,7(1997)520-539.
[94]I. Podlubny. Fractional Differential Equations[M]. Academic Press, New York,1999.
[95]Arvet Pedas, Enn Tamme. Spline collocation methods for linear multi-term fractional differ-ential equations [J]. Journal of Computational and Applied Mathematics,236 (2011) 167-176.
[96]A. Pedas. E. Tamme, Spline collocation method for integro-differential equations with weakly singular kernels [J]. J. Comput. Appl. Math.,197 (2006) 253-269.
[97]A. Pedas, E. Tamme. On the convergence of spline collocation methods for solving fractional differential equations[J]. J. Comput. Appl. Math.,235 (2011) 3502-3514.
[98]D. Rothstein, E. Henry, J. Gollub. Persistent patterns in transcient chaotic mixing[J]. Nature, 401(1999)770-771.
[99]M. T. Rosenstein, J. J. Collins et al. A practical method for calculating largest Lyapunov exponents from small data sets[J]. Physica D:Nonlinear Phenomena,65(1993)117-134.
[100]S. Sallam. Stable quartic spline integration method for solving stiff ordiamy differential equa-tions [J]. Appl. Math. Comput.,116(2000)245-255.
[101]S. Sallam, A. A. Karaballi. A quartic C3-spline collocation method for solving second-order initial value problems[J]. J. Comput. Appl. Math.,75(1996)295-304.
[102]S. Sallam, M. Naim Anwar. One parameter quadratic C1-spline collocation method forsolving first order ordinary initial value problems[J]. Appl. Math. Model.,23(1999)153-160.
[103]S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives:Theory and Applications[M], Gordon and Breach Science publishers,1987.
[104]W. R. Schneider and W. Wyss, Fractional diffusion and wave equations[J]. J. Math. Phys.. 30(1989)134-144.
[105]M. F. Shlesinger, B. J. West and J. Klafter, Levy Dynamics of enhances diffusion:Appli-cation to turbulence[J]. Phys. Rew. Lett.,58(11)(1987)1100-1103.
[106]N. T. Shawagfeh. Analytical appromate solutions for nonlinear fractional differentia equ tions[J]. Appl. Math, and Comp.,13(2002)517-529.
[107]M. Siewe, F. M. Kakmeni, C. Tchawoua and P. Woafo. Bifurcations and chaos in the triple-well (?)6-Van der Pol oscillator driven by external and parametric excitations[J]. Physica A, 357(2005)383-396.
[108]A. A. Soliman. A numeric simulation an explicit solutions of KdV-Burgcrs'and Lax's seventh-order KdV equations[J]. Chaos, Solitons & Fractals,29(2)(2006)294-302.
[109]苏红.求解几类泛函微分方程的三次样条配置方法[D].湘潭大学硕士学位论文,2010.
[110]N. H. Swcilam, M. M. Khader and R. F. Al-Bar. Numerical studies for a multi-order fractional differential equation[J]. Physics Letters A,371(2007)26-33.
[111]Mehdi Tatari, Mehdi Dehghan. On the convergence of He's variational iteration method[J]. J. Comput. Appl. Math.207(2007) 121-128.
[112]Vedat Suat Erturk, Shaher Momani, Zaid Odibat, Application of generalized differential trans-form method to multi-order fractional differential equations[J]. Communications in Nonlinear Science and Numerical Simulation,13(2008)1642-1654.
[113]S. Westerlund and L. Eksam, Capacitor Theory, IEEE Trans, on Dielectrics and Electrical Insulation [J].1(5)(1994) 826-839.
[114]Wyss W. The fractional diffusion equation[J].J. Math. Phys,27(1986)2782-2785.
[115]Wan-Sheng Wang, Shou-Fu Li. On the one-leg θ-methods for solving nonlinear neutral func-tional differential equations[J]. Appl. Math. Comput.193 (2007) 285-301.
[116]Wan-Sheng Wang, Shou-Fu Li. Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations[J]. Front. Math. China.4(1)(2009)195-216.
[117]王海燕.求解分数阶微分方程的Runge-Kutta方法[D].湘潭大学硕士学位论文,2008年.
[118]M. Wcilbeer. Efficient numerical methods for fractional differential equations and their ana-lytical background [J]. Technical University of Braunsehwig,2005.
[119]J. Wu, Q. Chen, and Y. Hong. Striped attractor generation and synchronization analysis for coupled Rosslcr systems [J]. Chaos, Solitons, Fractals,39(1)(2009)322-331.
[120]J. Wu, Y. Hong, G. Chen. Occurrence and underlying mechanism of multi-stripe chaotic attractors[J]. Chaos. Solitons, Fractals,41(5)(2009)2250-2258.
[121]X. Wu, et al. Chaos in the fractional order unified system and its synchronization [J]. Journal of the Franklin Institute,345(4)(2008)392-401.
[122]杨光俊.分数阶积分方程[J].云南大学学报(自然科学版),19(3)(1997)328-334
[123]M. Yalcin, J. Suykens, and J. Vandevalle. Cellular neural networks, multi-scoll chaos and synchronization[M]. World Scientific, Singapore,2005.
[124]伊婕.变分迭代法关于Caputo分数阶常微分方程和中立型比例延迟微分方程的收敛性分析[D].湘潭大学硕士学位论文,2010年.
[125]Zhan-Hua Yu. Variational iteration method for solving the multi-pantograph delay equa tion[J]. Physics Letters A.372 (2008) 6475-6479.
[126]章红梅.含Riesz (-Feller)位势算子的分数阶偏微分方程的基本解与数值解[D].厦门大学博士论文,2007.
[127]张晓娟.多阶的分数阶常微分方程和分数阶扩散—波动方程[D].厦门大学硕士学位论文,2008.
[128]张晓娟,王婧.配置方法求多阶的分数阶常微分方程的数值解[J].华北水利水电学院学报,2010年第3期.
[129]张亚平.新迭代方法在几类分数阶微分方程中的应用[D].湘潭大学硕士学位论文,2009年4月.
[130]H. Zhu, S. Zhou, J. Zhang. Chaos and synchronization of the fractional-order Chua's system [J]. Chaos, Solitons and Fractals,39(4)(2009) 1595-1603.