几类随机延迟微分代数系统的数值分析
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摘要
随机延迟微分代数系统(SDDAS)是既考虑了延迟现象又考虑了非确定因素的微分代数系统的扩展．因此，它能更真实的反映和模拟应用中出现的实际问题．由于绝大部分的随机延迟微分代数系统无法显式的求出理论解，从而对其数值方法的研究更加的重要和迫切．稳定性和收敛性是衡量方法优劣的重要指标，故而对数值方法的收敛性和稳定性的研究是数值分析中的重要研究课题．
然而，由于这类系统既有延迟项又有随机项，还兼有代数约束条件，对其进行数值分析十分困难，目前较少有相关文献论及此类问题．鉴于此，本文就SDDAS的数值算法做了一些尝试性的工作．由于较少关于SDDAS的解析解的理论研究结果，在第二章，我们对SDDAS解的存在性进行了讨论，在一定条件下证明了SDDAS初值问题存在唯一解．根据后续研究的需要，本章还给出了关于SDDAS解的一个均方矩估计定理．
在第三章，我们主要研究了一类随机常延迟微分代数系统初值问题．将Euler-Maruyama方法应用于此类问题，方法为1／2阶相容，并且1／2阶收敛的．
在第四章，我们将关于常延迟问题的研究推广到了随机变延迟微分代数系统(SVDDAS)，并证明了扩展后的θ-方法是1／2阶相容且1／2阶收敛的．
当代数约束条件消失后，随机延迟微分代数方程(SDDAEs)退化为随机延迟微分方程(SDDEs)．本文的第五，六章主要针对一类带无界延迟的非线性随机微分方程做了若干数值分析．在第五章，我们首先考察了该类方程解析解的均方渐近稳定性，其次，我们将θ-方法改造后用于求解非线性随机Pantograph方程并证明所得数值解是均方渐近稳定的．
在第六章里，我们进一步讨论了求解非线性随机Pantograph方程的Milstein方法的均方渐近稳定性，证明用Milstein方法求解此类问题．当步长满足一定条件时，其数值解均方渐近稳定．
当扩散项g≡0时，随机延迟微分代数系统退化为确定性延迟微分代数系统．在第七、八章，本文就变延迟微分代数系统(VDDAS)的数值算法展开了研究．第七章将常微分方程的B-收敛和延迟微分方程的D-收敛的概念推广到了变时滞微分代数系统问题类K_(α，β，γ)~((A))，给出了D_A-收敛的定义，讨论了该问题类的D_A-收敛性，证明如果G-稳定的单支方法(ρ，σ)对于常微分方程初值问题在经典意义下是p阶相容的且β_κ／α_κ＞0，那么具有线性插值过程的该方法是p阶D_A-收敛的，这里p=1或2．
第八章将求解常微分方程的Runge-Kutta方法改造后用于求解变延迟微分代数系统，并且证明如果代数稳定且对角稳定的Runge-Kutta方法(A，b，c)对于常微分方程初值问题在经典意义下是p阶相容的，那么具有Lagrange插值过程的该方法是M阶D_A-收敛的，M=min{p，u+q+1}，u+q为Lagrange插值多项式的次数．
Stochastic delay-differential-algebraic system(SDDAS) is the extension of differentialalgebraic system, and takes into account of both delay factors and randomfactors. It can reflect and imitate the problems arising in the application more factually.Because most of the stochastic delay-differential-algebraic system can't get anexplicit analytical solution, it gets more important and impendent to investigate itsnumerical methods. Stability and convergence are basilic indexes to weigh the validityof a method, so the research of stability and convergence of numerical methods isthe main task of the numerical analysis.
However, besides algebraic constrain, there still are delay term and random termin this class of system. All of these makes the numerical analysis be difficult. Thereis few literature which discusses this problem coming into my eyes. In view of this,some work on the numerical algorithm of SDDAS is done here.
In the second chapter, the existence of the solution of the SDDAS problemsis discussed, and it is also proved that the initial value problems of SDDAS whichsatisfy certain conditions have unique solution. According to the need of succeedingresearch, a mean-square estimation theory about the solution of SDDAS is proposedin this chapter.
In the third chapter, initial value problem of stochastic differential algebraicsystem with constant delay are studied. In this process, Euler-Maruyama method isapplied to this class of problems, and it is proved that when some conditions satisfied.the method is consistent with order 1/2 and convergent of order 1/2.
In the chapter 4, the research of the constant delay problems is extended to thestochastic variable delay differential algebraic systems(SVDDAS), and it is provedthat the extendedθ-method is consistent with order 1/2 and convergent of order 1/2.
When algebraic constrain disappears, stochastic delay differential algebraic equations(SDDAEs) are degraded into stochastic delay differential equations(SDDEs).The fifth chapter and the sixth chapter of this paper are focus on the numerical analysis of the nonlinear stochastic differential equations with infinite delay. In thechaper 5, the mean-square asymptotical stability of the solution of this class of equationsis investigated first,then theθ-method is represented and used to solve nonlinearstochastic pantograph differential equations. It is proved that the method is meansquareasymptotically stable.
In the chapter 6, mean-square asymptotic stability of the Milstein method whichis used to numerically solve the nonlinear stochastic pantograph differential equationsis further discussed. And it is proved that when stepsize satisfies certain conditions,the Milstein method is mean-square asymptotically stable.
When diffusion term turns into 0, stochastic delay differential algebraic systemis degraded into a deterministic delay differential algebraic system. In the chapter7 and 8, the numerical algorithm for the variable delay differential algebraic system(VDDAS) is studied. In the chapter 7, the concepts of B-convergence in theordinary differential equations and D-convergence in delay differential equations areextended to the problem class of variable delay differential algebraic system. Thedefinition of D_A-Convergence is given. It is also proved that if the One-Leg methodswhich is G-convergence are consistent with order p andβ_k/α_k＞0 the extended One-Legmethod with linear interpolation procedure is D_A-Convergent of order p. Here pis 1 or 2.
In chapter 8. we extend the Runge-Kutta methods to variable delay differentialalgebraic system. It is proved that if the Runge-Kutta method which is algebraicallystable and diagonally stable is consistent with order p , the extended Runge-Kuttamethod with Lagrange interpolation procedure is D_A-convergence with order M. HereM=min{p, u + q + 1 }, and u + q is the degree of Lagrange interpolation polynomial.
然而，由于这类系统既有延迟项又有随机项，还兼有代数约束条件，对其进行数值分析十分困难，目前较少有相关文献论及此类问题．鉴于此，本文就SDDAS的数值算法做了一些尝试性的工作．由于较少关于SDDAS的解析解的理论研究结果，在第二章，我们对SDDAS解的存在性进行了讨论，在一定条件下证明了SDDAS初值问题存在唯一解．根据后续研究的需要，本章还给出了关于SDDAS解的一个均方矩估计定理．
在第三章，我们主要研究了一类随机常延迟微分代数系统初值问题．将Euler-Maruyama方法应用于此类问题，方法为1／2阶相容，并且1／2阶收敛的．
在第四章，我们将关于常延迟问题的研究推广到了随机变延迟微分代数系统(SVDDAS)，并证明了扩展后的θ-方法是1／2阶相容且1／2阶收敛的．
当代数约束条件消失后，随机延迟微分代数方程(SDDAEs)退化为随机延迟微分方程(SDDEs)．本文的第五，六章主要针对一类带无界延迟的非线性随机微分方程做了若干数值分析．在第五章，我们首先考察了该类方程解析解的均方渐近稳定性，其次，我们将θ-方法改造后用于求解非线性随机Pantograph方程并证明所得数值解是均方渐近稳定的．
在第六章里，我们进一步讨论了求解非线性随机Pantograph方程的Milstein方法的均方渐近稳定性，证明用Milstein方法求解此类问题．当步长满足一定条件时，其数值解均方渐近稳定．
当扩散项g≡0时，随机延迟微分代数系统退化为确定性延迟微分代数系统．在第七、八章，本文就变延迟微分代数系统(VDDAS)的数值算法展开了研究．第七章将常微分方程的B-收敛和延迟微分方程的D-收敛的概念推广到了变时滞微分代数系统问题类K_(α，β，γ)~((A))，给出了D_A-收敛的定义，讨论了该问题类的D_A-收敛性，证明如果G-稳定的单支方法(ρ，σ)对于常微分方程初值问题在经典意义下是p阶相容的且β_κ／α_κ＞0，那么具有线性插值过程的该方法是p阶D_A-收敛的，这里p=1或2．
第八章将求解常微分方程的Runge-Kutta方法改造后用于求解变延迟微分代数系统，并且证明如果代数稳定且对角稳定的Runge-Kutta方法(A，b，c)对于常微分方程初值问题在经典意义下是p阶相容的，那么具有Lagrange插值过程的该方法是M阶D_A-收敛的，M=min{p，u+q+1}，u+q为Lagrange插值多项式的次数．
Stochastic delay-differential-algebraic system(SDDAS) is the extension of differentialalgebraic system, and takes into account of both delay factors and randomfactors. It can reflect and imitate the problems arising in the application more factually.Because most of the stochastic delay-differential-algebraic system can't get anexplicit analytical solution, it gets more important and impendent to investigate itsnumerical methods. Stability and convergence are basilic indexes to weigh the validityof a method, so the research of stability and convergence of numerical methods isthe main task of the numerical analysis.
However, besides algebraic constrain, there still are delay term and random termin this class of system. All of these makes the numerical analysis be difficult. Thereis few literature which discusses this problem coming into my eyes. In view of this,some work on the numerical algorithm of SDDAS is done here.
In the second chapter, the existence of the solution of the SDDAS problemsis discussed, and it is also proved that the initial value problems of SDDAS whichsatisfy certain conditions have unique solution. According to the need of succeedingresearch, a mean-square estimation theory about the solution of SDDAS is proposedin this chapter.
In the third chapter, initial value problem of stochastic differential algebraicsystem with constant delay are studied. In this process, Euler-Maruyama method isapplied to this class of problems, and it is proved that when some conditions satisfied.the method is consistent with order 1/2 and convergent of order 1/2.
In the chapter 4, the research of the constant delay problems is extended to thestochastic variable delay differential algebraic systems(SVDDAS), and it is provedthat the extendedθ-method is consistent with order 1/2 and convergent of order 1/2.
When algebraic constrain disappears, stochastic delay differential algebraic equations(SDDAEs) are degraded into stochastic delay differential equations(SDDEs).The fifth chapter and the sixth chapter of this paper are focus on the numerical analysis of the nonlinear stochastic differential equations with infinite delay. In thechaper 5, the mean-square asymptotical stability of the solution of this class of equationsis investigated first,then theθ-method is represented and used to solve nonlinearstochastic pantograph differential equations. It is proved that the method is meansquareasymptotically stable.
In the chapter 6, mean-square asymptotic stability of the Milstein method whichis used to numerically solve the nonlinear stochastic pantograph differential equationsis further discussed. And it is proved that when stepsize satisfies certain conditions,the Milstein method is mean-square asymptotically stable.
When diffusion term turns into 0, stochastic delay differential algebraic systemis degraded into a deterministic delay differential algebraic system. In the chapter7 and 8, the numerical algorithm for the variable delay differential algebraic system(VDDAS) is studied. In the chapter 7, the concepts of B-convergence in theordinary differential equations and D-convergence in delay differential equations areextended to the problem class of variable delay differential algebraic system. Thedefinition of D_A-Convergence is given. It is also proved that if the One-Leg methodswhich is G-convergence are consistent with order p andβ_k/α_k＞0 the extended One-Legmethod with linear interpolation procedure is D_A-Convergent of order p. Here pis 1 or 2.
In chapter 8. we extend the Runge-Kutta methods to variable delay differentialalgebraic system. It is proved that if the Runge-Kutta method which is algebraicallystable and diagonally stable is consistent with order p , the extended Runge-Kuttamethod with Lagrange interpolation procedure is D_A-convergence with order M. HereM=min{p, u + q + 1 }, and u + q is the degree of Lagrange interpolation polynomial.
引文
[1] Barwell V K. On the asymptotic behavior of the solution of a differential difference equations. Util. Math., 1974, 6: 189-194
[2] Barwell V K. Special stability problems for functional differential equations. BIT, 1975, 15: 130-135
[3] Watanabe D S, Roth M G. The stability of difference formulas for delay differential equations. SIAM J. Numer. Anal., 1985, 22:132-145
[4] Zennaro M. P-stability properties of Runge-Kutta methods for delay differential equations. Numer. Math., 1986, 49: 305-318
[5] Liu M Z, Spijker M N. The stability of the θ-methods in the numerical solution of delay differential equations. IMA J. Numer. Anal., 1990, 10: 31-48
[6] In't Hout K J. A new intepolation procedure for adapting Runge-Kutta methods to delay differential equations. BIT, 1992, 32:634-649
[7] In't Hout K J. The stability of theta-methods for systems of delay differential equations. Ann. Numer. Math. 1994, 1: 323-334
[8] In't Hout K J. Stability analysis of Runge-Kutta methods for systems of delay differential equations. IMA J. Numer. Anal., 1997, 17: 17-27
[9] Koto T. Astability properties of A-stable natural Runge-Kutta methods for systems of delay differential equations. BIT, 1994, 34: 262-267
[10] 李冬松，刘明珠．比例延迟微分方程数值解的渐近稳定性．1994，31(1)：57-59
[11] Tian H J, Kuang J X. The stability of linear multistep methods for systems of delay differential equations. Numer. Math.(English Series), 1995, 4: 10-16
[12] Tian H J, Kuang J X. The stability analysis of the θ -methods for delay differential equations with many delays. SIAM J. Numer. Anal., 1996, 33(3): 883-889
[13] Buhmann M D, Iserles A. On the dynamics of a discretized neutral equation. IMA J.Numer. Anal., 1992, 12: 339-363
[14] Buhmann M D, Iserles A. Stability of the discretized pantograph differential equation. Math. Comp., 1993, 60: 575-589
[15] Liu Y. On the 0 -methods for delay differential equations with infinity lag. J. Comput. Appl. Math., 1996, 71: 177-190
[16] Liu Y. Numerical investigation of the pantograph equation. Appl. Numer. Math., 1997,24: 309-317
[17] Iserles A. Exact and discretized stability of pantograph equation. Appl. Numer. Math., 1997, 24: 295-308
[18] Bellen A, Guglielmi N, Torelli L. Asymptotic properties of 0 -methods for the pantograph equation. Appl. Numer. Math., 1997, 24: 279-293
[19] Bellen A. Contractivity of continuous Runge-Kutta methods for delay differential equations. Appl. Numer. Math., 1997, 24: 219-232
[20] Hu G D, Hu G D, Meguid S A. Stability of Runge-Kutta methods for delay differential systems with multiple delays. IMA J. Numer. Anal., 1999, 19(3): 349-356
[21] Torelli L. Stability of numerical methods for delay differential equations. J. Comput. Appl. Math., 1989, 25: 15-26
[22] Torelli L. A sufficient condition for GRN-stability for delay differential equations. Numer. Math., 1991, 59: 311-320
[23] Bellen A, Zennaro M. strong contractively properties of numerical methods for ordinary and delay differential equations. Appl. Numer. Math.. 1992. 9: 321-346
[24] Zhang C J, Zhou S Z. Nonlinear stability and D-convergence of runge-kutta methods for delay differential equations. J. Comput. Appl. Math., 1997, 85: 225-237
[25] Huang C M, Li S F, Fu H Y et.al. Stability and error analysis of one-leg methods for nonlinear delay differential equations. J. Comput. Appl. Math., 1999, 103: 263-279
[26] Huang C M, Fu H Y, Li S F et.al. Stability analysis of Runge-Kutta methods for non-linear delay differential equations. BIT, 1999, 39: 270-280
[27] Huang C M, Li S F, Fu H Y et.al. Nonlinear stability of general linear methods for delay differential equations. BIT, 2002, 42: 380-392
[28] Huang C M, Chen G G, Li S F et.al. D-convergence of general linear methods for stiff delay differential equations. Comput. Math. Appl., 2001, 41:627-639
[29] Huang C M, Li S F, Fu H Y et.al. D-convergence of one-leg methods for stiff delay differential equations. J. Comput. Math., 2001, 19:601-606
[30] Zennaro M. Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations. Numer. Math., 1997, 77: 549-563
[31] Bellen A, Zennaro M. Numerical methods for delay differential equations and volterra integral equations with variable delay. BIT, 1997, 37: 37-42
[32] Brunner H. On the discretization of differential and Volterra integral equations with variable delay. BIT, 1997, 37:1-12
[33] 王文强，李寿佛．非线性刚性变延迟微分方程单支方法的数值稳定性．计算数学，2002，4：417-430
[34] 王文强，李寿佛．非线性变延迟微分方程隐式Euler方法的数值稳定性．应用数学，2004，1：22-25
[35] 王文强，李寿佛．非线性刚性变延迟微分方程单支方法D-收敛性．计算数学，2004，26(2)：247-256
[36] Zhang C J, Sun G. The discrete dynamics of nonlinear infinite-delay-differential equations. Appl. Math. Lett., 2002, 15(5): 521-526
[37] Zhang C J, Sun G. Nonlinear stability of runge-kutta methods applied to infinite-delay-differential equations. Math. Comput. Model., 2004, 39:495-503
[38] Huang C M, Vandwalle S. Discretized stability and error growth of the nonautomous pantograph equation. SIAM J. Numer. Anal., 2005, 42(5): 2020-2042
[39] 匡蛟勋．泛涵微分方程的数值处理．北京：科学出版社，1999．
[40] Bellen A, Zennaro M. Numerical methods for delay differential equations. Oxford: Clarendon Press, 2003.
[41] Mao X. Stochastic differential equations and applications. New York: Horwood, 1997.
[42] Milstein G N. Numerical integration of stochastic differential equations. Dordrecht: Kluwer Academic, 1995.
[43] Kloeden P E, Platen E. Numerical solution of stochastic differential equations. Berlin: Springer-Verlag, 1992.
[44] Burrage P M. Runge-Kutta methods for stochastic differential equations: PHD thesis. Queensland: The University of Queensland, 1999.
[45] Tian T H. Implicit numerical methods for stiff stochastic differential equations and numerical simulations of stochastic models: PHD thesis. Queensland: The University of Queensland, 2001.
[46] Baker C T H, Buckwar E. Intruduction to the numerical analysis of stochastic delay differential equations. J. Comput. Appl. Math, 2000, 125: 297-307
[47] Baker C T H, Buckwar E. Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math., 2000, 3:315-335
[48] Baker C T H, Buckwar E. Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math., 2005, 184:404-427
[49] Buckwar E. One-step approximations for stochastic functional differential equations. Appl. Numer. Math., 2006,56:667-681
[50] K(u|¨)chler U, Platen E. Strong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simul., 2000, 54:189-205
[51] K(u|¨)chler U, Platen E. Weak discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simul., 2002, 59:497-507
[52] Mao X R, Sabanis S. Numerical solutions of stochastic differential delay equations under local Lipschitz condition. J. Comp. Appl. Math., 2003, 151:215-227
[53] Cao W, Liu M, Fan Z. MS-stability of the Euler-Maruyama method for stochastic differential dalay equations. Appl. Math. Comput., 2004, 159:127-135
[54] Liu M, Cao W, Fan Z. Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math., 2004, 170: 255-268
[55] Wang Z Y, Zhang C J. An analysis of stability of Milstein method for stochastic differential equations with delay. Comput. Math. Appl., 2006, 51:1445-1452
[56]王志勇，张诚坚．随机延迟微分方程的Milstein方法的非线性均方稳定性．应用数学，2008，21(1)：201-206
[57]王文强．几类非线性随机延迟微分方程数值方法的稳定性和收敛性：博士学位论文．湘潭：湘潭大学，2007．
[58]王文强，黄山，李寿佛．非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性．计算数学，2007，29(2)：217-224
[59]王文强，李寿佛，黄山．非线性随机延迟微分方程Euler-Maruyama方法的收敛性．系统仿真学报，2007，19(17)：3910-3913
[60] Tian T H. Burrage K, Burrage P M et.al. Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math., 2007, 205:696-707
[61] Norbert H, Thomas M G. A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag. J. Comp. Appl. Math., 2006, 197: 89-121
[62] Mao X R. Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math., 2007, 200:297-316
[63] 范振成．几类随机延迟微分方程解析解及其数值方法的收敛性与稳定性：博士学位论文．哈尔滨：哈尔滨工业大学，2006．
[64] 范振成，刘明珠．线性随机比例方程的渐近均方稳定性．应用数学，2007，20(3)：519-523
[65] Fan Z C, Liu M Z, Cao W R. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. J. Math. Anal. Appl., 2007, 325: 1142-1159
[66] Baker C T H, Buckwar E. Continuous θ-methods for the Stochastic Pantograph Equation. Electron. Trans. Numer. Anal., 2000, 11: 131-151.
[67] Gear C W. The simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory, 1971, 18: 89-95
[68] Petzold L R. Differential-algebraic equations are not ODEs. SIAM J. Sci. Statist. Comput., 1982, 3: 367-384.
[69] Brenan K E, Campbell S L, Petzold L. Numerical solution of initial-value problems in differential-algebraic equations. New York: North-Holland, 1989.
[70] Gear C W. Differential-algebraic equations, indices and integral algebraic equations. SIAM J. Numer. Anal., 1990, 27: 984-996
[71] Campbell S L, Gear C W. The index of general nonlinear DAEs. Numer. Math., 1995, 72:173-196[72] Hairer E, Wanner G. Solving ordinary differential equations Ⅱ: stiff and differential-algebraic problems(second edition). Berlin: Springer-Verlag, 1996.
[73] Rabier P, Rheinboldt W. A general existence and uniqueness theory for implicit differential-algebraic equations. Diff. Int. Equations, 1991, 4(3): 563-582
[74] Rabier P J, Rheinboldt W C. On the computation of impasse points of quasilinear differential algebraic equations. Math. Comp., 1994, 205(62): 133-154
[75] Rentrop P, Roche M, Steinebach G. The application of Rosenbrock- Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 1989, 55: 545-563
[76] Ascher U M. On symmetric schemes and differential-algebraic equations. SIAM J. Sci. Stat. Comput. 1989, 10: 937-949
[77] Ascher U M, Petzold L R. Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal. 1991, 28: 1097-1120
[78] Ascher U M, Spiter R J. Collocation software for boundary value differential-algebraic equations, SIAM J. Sci. Comput. 1994, 15: 938-952
[79] Arnold M, Strehmel K, Weiner R. Half-explicit runge-kutta methods for semi-explicit differential-algebraic equations of index 1. Numer. Math., 1993, 64:409-431
[80] 罗新龙，刘德贵．计算微分代数系统的实时仿真算法．数值计算与计算机应用．2001．2：97-105
[81] 罗新龙，宋晓秋，刘德贵．实时控制计算微分代数系统的代数约束算法．系统工程与电子技术，1998，12：66-71
[82] Feeher W L, Tolsma J E, Barton P I. Efficicnt sensitivity analysis of large-scale differential algebraie systems. Appl. Numer. Math., 1997, 25:41-54
[83] Maly T, Petzold L R. Numerical methods and software for sensitivity analysis of differential-algebraic systems. Appl. Numer. Math., 1996, 20: 57-79
[84] 费景高．微分代数控制问题的实时计算．自动化学报，1993：19(4)：385-392
[85] 费景高．微分代数系统的实时控制计算．计算机工程与设计，1996，12：3-10
[86] Fei J G. A class of parallel Runge-Kutta methods for differential-algebraic systems of index 2. J. Sys. Eng. Elect., 1999, 10(3): 64-75
[87] Fei J G. A class of parallel Runge-Kutta methods for differential-algebraic systems of index 2. J. Sys. Eng. Elect., 2000, 22(4): 59-63
[88] Sun W, Jiang Y L. Runge-Kutta methods of dynamic iteration for index-2 differential-algebraic equations. Appl. Math. Comput., 2005, 169(2): 1346-1363
[89] 孙卫，樊晓光，周友运等．指标-2微分-代数系统的连续扩展RUulge-Kutta迭代方法．高等学校计算数学学报，2006，28(4)：329-336
[90] Tischendorf C. Solution of index-2 differential algebraic equations and its application in circuit simulation: PHD Thesis. Berlin: Humboldt University, 1996.
[91] Higueras I, M(a|¨)rz R. Tischendorf C. Stability preserving integration of index- 2 DAEs. Appl. Numer. Math., 2003, 45:201-229
[92] Celik E. On the numerical solution of differential-algebraic equation with index-2. Appl. Math. Comput., 2004, 156:541-550
[93] Guzel N, Bayram M. Numerical solution of differential-aigebraic equations with index-2. Appl. Math. Comput., 2006, 174(2): 1279-1289
[94] Guzel N, Bayram M. On the numerical solution of differential-algebraic equations with index-3. Appl. Math. Comput., 2006, 175(2): 1320-1331
[95] Campbell S L. Singular linear systems of differential equations with delays. Appl. Anal.,1980, 2:129-136
[96] Ascher U M, Petzold L R. The numerical solution of delay-differential-algebraic equations of retarded and neutral type. SIAM J. Numer. Anal., 1995, 32:1635-1657
[97] Hauber R. Numerical treatment of retarded differential-algebraic equations by collocation methods. Adv. Comput. Math., 1997, 7:573-592
[98] Zhu W, Petzold L R. Asymptotic stability of linear delay differential-algebraic equations and numerical methods. Appl. Numer. Math., 1997, 24:247-264
[99] Zhu W, Petzold L R. Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type. Appl. Numer. Math., 1998, 27:309-325
[100] Liu Y. Runge-Kutta-collocation methods for systems of functional-differential and functional equations. Adv. Comput. Math., 1999, 11:315-329
[101] Baker C T H, Paul C A H, Tian H. Differential algebraic equations with after-effect. J. Comput. Appl. Math., 2002, 140: 63-80
[102] Huang C M, Chang Q S. Stability analysis of numerical methods for systems of functional-differential and functional equations. Comput. Math. Appl., 2002, 44: 717-729
[103] Zhang C J, Liao X X. Nonlinear stability of(p, σ)-methods for stiff delay-differential-algebraic systems. J. Cont. The. Appl., 2001, 18(6): 827-832
[104]甘四清．奇异摄动问题与时滞奇异摄动问题数值分析：博士学位论文．北京：中国科学院，2001．
[105]甘四清，孙耿．时滞奇异摄动问题单支方法的收敛性．中国科学(A辑)，2001，31(4)：314-323
[106] Gan S Q, Sun G, Zheng W M. Errors of linear multistep methods and Runge-Kutta methods for singular perturbation problems with delays. Comput. Math. Appl., 2002, 44: 1157-1173
[107] Schein O, Denk G. Numerical solution of stochastic differential-algebraic equations with applications to trunsient noise simulation of microelectronic circuis. J. Comput. Appl. Math., 1998, 100(1): 77-92
[108] Christain P. Analysis and numerical integration of stochastic differential-algebraic equations and application in electronic circuit simulation: PHD thesis. Munich: Technische Universit(a|¨)t M(u|¨)nchen, 1999.
[109] Gerdin M, Sjoberg J. Nonlinear stochastic differential-algebraic equations with application to particle filtering. IEEE Conf. Deci. Contr., 2006, 45: 6630-6635
[110] Winkler R. Stochastic dfferential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math., 2003, 157(2): 477-505
[111] Alabert A, Ferrante M. Linear stochastic differential-algebraic equations with constant coefficients. Elect. Com. Prob., 2006, 11:316-335
[112] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 2001, 43(3): 525-546
[113] 李寿佛．刚性微分方程算法理论．长沙：湖南科学技术出版社，1997．
[114] Dahlquist G. A special stability problem for linear multistep methods. BIT, 1963, 3: 27-43
[115] Bahar A, Mao X. Stochastic delay Lotka-Volterra model. J. Math. Anal. Appl., 2004, 292: 364-380
[116] Baker C T H, Ford J M, Ford N J. Bifurcations in approximate solutions of stochastic delay differential equations. Int. J. Bifurcat. Chaos Appl. Sci. Eng., 2004, 14(9): 2999-3021
[117] Mao X. Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl., 2005, 304: 296-320
[118] Mao X. Almost sure exponential stability for delay stochastic differential equations with respect to semimartingales. Stoch. Anal. Appl., 1991, 9(2): 177-194
[119] Mao X. Existence and uniqueness of the solutions of delay stochastic integral equations. Stoch. Anal. Appl., 1989, 7(1): 59-74
[120] Winkler R. Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math., 2004, 163(2): 435-463
[121] In't Hout K J. The stability of θ -methods for systems of delay differential equations. Ann. Numer. Math., 1994, 1(3): 323-334
[122] Kuang J X, Xiang J X, Tian H J. The asymptotic stability of one-parameter methods for neutral differential equations. BIT, 1994, 34(3): 400-408
[123] Higham D J. A-stability and stochastic mean-square stability. BIT, 2000, 40:404-409
[124] Higham D J. Mean-Square and asymptotical stability of numerical methods for stochastic ordinary differential equations. SIAM J. Appl. Math., 2000, 38:753-769
[125] Baker C T H, Buckwar E. On Halanay-type analysis of exponential stability for the theta-Maruyama method for stochastic delay differential equations. Stoch. Dyn., 2005, 5: 201-209
[126] K(u|¨)chler U, Platen E. Stong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simulat., 2000, 54:189-205
[127] Guglielmi N, Hairer E. Implementing radau ⅡA methods for stiff delay differential equations. Computing, 2001, 67: 1-12
[128]李宏智，李建国．块隐式单步方法求解一类延迟微分代数方程．华中科技大学学报(自)，2003，10：111-113
[129]甘四清，孙耿．单支方法的收敛性．应用数学，2001，14：30-33
[130] Guglielmi N. On the Newton iteration in the application of collocation methods to implicit delay equations. Appl. Numer. Math., 2005, 53: 281-297
[2] Barwell V K. Special stability problems for functional differential equations. BIT, 1975, 15: 130-135
[3] Watanabe D S, Roth M G. The stability of difference formulas for delay differential equations. SIAM J. Numer. Anal., 1985, 22:132-145
[4] Zennaro M. P-stability properties of Runge-Kutta methods for delay differential equations. Numer. Math., 1986, 49: 305-318
[5] Liu M Z, Spijker M N. The stability of the θ-methods in the numerical solution of delay differential equations. IMA J. Numer. Anal., 1990, 10: 31-48
[6] In't Hout K J. A new intepolation procedure for adapting Runge-Kutta methods to delay differential equations. BIT, 1992, 32:634-649
[7] In't Hout K J. The stability of theta-methods for systems of delay differential equations. Ann. Numer. Math. 1994, 1: 323-334
[8] In't Hout K J. Stability analysis of Runge-Kutta methods for systems of delay differential equations. IMA J. Numer. Anal., 1997, 17: 17-27
[9] Koto T. Astability properties of A-stable natural Runge-Kutta methods for systems of delay differential equations. BIT, 1994, 34: 262-267
[10] 李冬松，刘明珠．比例延迟微分方程数值解的渐近稳定性．1994，31(1)：57-59
[11] Tian H J, Kuang J X. The stability of linear multistep methods for systems of delay differential equations. Numer. Math.(English Series), 1995, 4: 10-16
[12] Tian H J, Kuang J X. The stability analysis of the θ -methods for delay differential equations with many delays. SIAM J. Numer. Anal., 1996, 33(3): 883-889
[13] Buhmann M D, Iserles A. On the dynamics of a discretized neutral equation. IMA J.Numer. Anal., 1992, 12: 339-363
[14] Buhmann M D, Iserles A. Stability of the discretized pantograph differential equation. Math. Comp., 1993, 60: 575-589
[15] Liu Y. On the 0 -methods for delay differential equations with infinity lag. J. Comput. Appl. Math., 1996, 71: 177-190
[16] Liu Y. Numerical investigation of the pantograph equation. Appl. Numer. Math., 1997,24: 309-317
[17] Iserles A. Exact and discretized stability of pantograph equation. Appl. Numer. Math., 1997, 24: 295-308
[18] Bellen A, Guglielmi N, Torelli L. Asymptotic properties of 0 -methods for the pantograph equation. Appl. Numer. Math., 1997, 24: 279-293
[19] Bellen A. Contractivity of continuous Runge-Kutta methods for delay differential equations. Appl. Numer. Math., 1997, 24: 219-232
[20] Hu G D, Hu G D, Meguid S A. Stability of Runge-Kutta methods for delay differential systems with multiple delays. IMA J. Numer. Anal., 1999, 19(3): 349-356
[21] Torelli L. Stability of numerical methods for delay differential equations. J. Comput. Appl. Math., 1989, 25: 15-26
[22] Torelli L. A sufficient condition for GRN-stability for delay differential equations. Numer. Math., 1991, 59: 311-320
[23] Bellen A, Zennaro M. strong contractively properties of numerical methods for ordinary and delay differential equations. Appl. Numer. Math.. 1992. 9: 321-346
[24] Zhang C J, Zhou S Z. Nonlinear stability and D-convergence of runge-kutta methods for delay differential equations. J. Comput. Appl. Math., 1997, 85: 225-237
[25] Huang C M, Li S F, Fu H Y et.al. Stability and error analysis of one-leg methods for nonlinear delay differential equations. J. Comput. Appl. Math., 1999, 103: 263-279
[26] Huang C M, Fu H Y, Li S F et.al. Stability analysis of Runge-Kutta methods for non-linear delay differential equations. BIT, 1999, 39: 270-280
[27] Huang C M, Li S F, Fu H Y et.al. Nonlinear stability of general linear methods for delay differential equations. BIT, 2002, 42: 380-392
[28] Huang C M, Chen G G, Li S F et.al. D-convergence of general linear methods for stiff delay differential equations. Comput. Math. Appl., 2001, 41:627-639
[29] Huang C M, Li S F, Fu H Y et.al. D-convergence of one-leg methods for stiff delay differential equations. J. Comput. Math., 2001, 19:601-606
[30] Zennaro M. Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations. Numer. Math., 1997, 77: 549-563
[31] Bellen A, Zennaro M. Numerical methods for delay differential equations and volterra integral equations with variable delay. BIT, 1997, 37: 37-42
[32] Brunner H. On the discretization of differential and Volterra integral equations with variable delay. BIT, 1997, 37:1-12
[33] 王文强，李寿佛．非线性刚性变延迟微分方程单支方法的数值稳定性．计算数学，2002，4：417-430
[34] 王文强，李寿佛．非线性变延迟微分方程隐式Euler方法的数值稳定性．应用数学，2004，1：22-25
[35] 王文强，李寿佛．非线性刚性变延迟微分方程单支方法D-收敛性．计算数学，2004，26(2)：247-256
[36] Zhang C J, Sun G. The discrete dynamics of nonlinear infinite-delay-differential equations. Appl. Math. Lett., 2002, 15(5): 521-526
[37] Zhang C J, Sun G. Nonlinear stability of runge-kutta methods applied to infinite-delay-differential equations. Math. Comput. Model., 2004, 39:495-503
[38] Huang C M, Vandwalle S. Discretized stability and error growth of the nonautomous pantograph equation. SIAM J. Numer. Anal., 2005, 42(5): 2020-2042
[39] 匡蛟勋．泛涵微分方程的数值处理．北京：科学出版社，1999．
[40] Bellen A, Zennaro M. Numerical methods for delay differential equations. Oxford: Clarendon Press, 2003.
[41] Mao X. Stochastic differential equations and applications. New York: Horwood, 1997.
[42] Milstein G N. Numerical integration of stochastic differential equations. Dordrecht: Kluwer Academic, 1995.
[43] Kloeden P E, Platen E. Numerical solution of stochastic differential equations. Berlin: Springer-Verlag, 1992.
[44] Burrage P M. Runge-Kutta methods for stochastic differential equations: PHD thesis. Queensland: The University of Queensland, 1999.
[45] Tian T H. Implicit numerical methods for stiff stochastic differential equations and numerical simulations of stochastic models: PHD thesis. Queensland: The University of Queensland, 2001.
[46] Baker C T H, Buckwar E. Intruduction to the numerical analysis of stochastic delay differential equations. J. Comput. Appl. Math, 2000, 125: 297-307
[47] Baker C T H, Buckwar E. Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math., 2000, 3:315-335
[48] Baker C T H, Buckwar E. Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations. J. Comput. Appl. Math., 2005, 184:404-427
[49] Buckwar E. One-step approximations for stochastic functional differential equations. Appl. Numer. Math., 2006,56:667-681
[50] K(u|¨)chler U, Platen E. Strong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simul., 2000, 54:189-205
[51] K(u|¨)chler U, Platen E. Weak discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simul., 2002, 59:497-507
[52] Mao X R, Sabanis S. Numerical solutions of stochastic differential delay equations under local Lipschitz condition. J. Comp. Appl. Math., 2003, 151:215-227
[53] Cao W, Liu M, Fan Z. MS-stability of the Euler-Maruyama method for stochastic differential dalay equations. Appl. Math. Comput., 2004, 159:127-135
[54] Liu M, Cao W, Fan Z. Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation. J. Comput. Appl. Math., 2004, 170: 255-268
[55] Wang Z Y, Zhang C J. An analysis of stability of Milstein method for stochastic differential equations with delay. Comput. Math. Appl., 2006, 51:1445-1452
[56]王志勇，张诚坚．随机延迟微分方程的Milstein方法的非线性均方稳定性．应用数学，2008，21(1)：201-206
[57]王文强．几类非线性随机延迟微分方程数值方法的稳定性和收敛性：博士学位论文．湘潭：湘潭大学，2007．
[58]王文强，黄山，李寿佛．非线性随机延迟微分方程Euler-Maruyama方法的均方稳定性．计算数学，2007，29(2)：217-224
[59]王文强，李寿佛，黄山．非线性随机延迟微分方程Euler-Maruyama方法的收敛性．系统仿真学报，2007，19(17)：3910-3913
[60] Tian T H. Burrage K, Burrage P M et.al. Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math., 2007, 205:696-707
[61] Norbert H, Thomas M G. A modified Milstein scheme for approximation of stochastic delay differential equations with constant time lag. J. Comp. Appl. Math., 2006, 197: 89-121
[62] Mao X R. Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations. J. Comput. Appl. Math., 2007, 200:297-316
[63] 范振成．几类随机延迟微分方程解析解及其数值方法的收敛性与稳定性：博士学位论文．哈尔滨：哈尔滨工业大学，2006．
[64] 范振成，刘明珠．线性随机比例方程的渐近均方稳定性．应用数学，2007，20(3)：519-523
[65] Fan Z C, Liu M Z, Cao W R. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. J. Math. Anal. Appl., 2007, 325: 1142-1159
[66] Baker C T H, Buckwar E. Continuous θ-methods for the Stochastic Pantograph Equation. Electron. Trans. Numer. Anal., 2000, 11: 131-151.
[67] Gear C W. The simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circuit Theory, 1971, 18: 89-95
[68] Petzold L R. Differential-algebraic equations are not ODEs. SIAM J. Sci. Statist. Comput., 1982, 3: 367-384.
[69] Brenan K E, Campbell S L, Petzold L. Numerical solution of initial-value problems in differential-algebraic equations. New York: North-Holland, 1989.
[70] Gear C W. Differential-algebraic equations, indices and integral algebraic equations. SIAM J. Numer. Anal., 1990, 27: 984-996
[71] Campbell S L, Gear C W. The index of general nonlinear DAEs. Numer. Math., 1995, 72:173-196[72] Hairer E, Wanner G. Solving ordinary differential equations Ⅱ: stiff and differential-algebraic problems(second edition). Berlin: Springer-Verlag, 1996.
[73] Rabier P, Rheinboldt W. A general existence and uniqueness theory for implicit differential-algebraic equations. Diff. Int. Equations, 1991, 4(3): 563-582
[74] Rabier P J, Rheinboldt W C. On the computation of impasse points of quasilinear differential algebraic equations. Math. Comp., 1994, 205(62): 133-154
[75] Rentrop P, Roche M, Steinebach G. The application of Rosenbrock- Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 1989, 55: 545-563
[76] Ascher U M. On symmetric schemes and differential-algebraic equations. SIAM J. Sci. Stat. Comput. 1989, 10: 937-949
[77] Ascher U M, Petzold L R. Projected implicit Runge-Kutta methods for differential-algebraic equations. SIAM J. Numer. Anal. 1991, 28: 1097-1120
[78] Ascher U M, Spiter R J. Collocation software for boundary value differential-algebraic equations, SIAM J. Sci. Comput. 1994, 15: 938-952
[79] Arnold M, Strehmel K, Weiner R. Half-explicit runge-kutta methods for semi-explicit differential-algebraic equations of index 1. Numer. Math., 1993, 64:409-431
[80] 罗新龙，刘德贵．计算微分代数系统的实时仿真算法．数值计算与计算机应用．2001．2：97-105
[81] 罗新龙，宋晓秋，刘德贵．实时控制计算微分代数系统的代数约束算法．系统工程与电子技术，1998，12：66-71
[82] Feeher W L, Tolsma J E, Barton P I. Efficicnt sensitivity analysis of large-scale differential algebraie systems. Appl. Numer. Math., 1997, 25:41-54
[83] Maly T, Petzold L R. Numerical methods and software for sensitivity analysis of differential-algebraic systems. Appl. Numer. Math., 1996, 20: 57-79
[84] 费景高．微分代数控制问题的实时计算．自动化学报，1993：19(4)：385-392
[85] 费景高．微分代数系统的实时控制计算．计算机工程与设计，1996，12：3-10
[86] Fei J G. A class of parallel Runge-Kutta methods for differential-algebraic systems of index 2. J. Sys. Eng. Elect., 1999, 10(3): 64-75
[87] Fei J G. A class of parallel Runge-Kutta methods for differential-algebraic systems of index 2. J. Sys. Eng. Elect., 2000, 22(4): 59-63
[88] Sun W, Jiang Y L. Runge-Kutta methods of dynamic iteration for index-2 differential-algebraic equations. Appl. Math. Comput., 2005, 169(2): 1346-1363
[89] 孙卫，樊晓光，周友运等．指标-2微分-代数系统的连续扩展RUulge-Kutta迭代方法．高等学校计算数学学报，2006，28(4)：329-336
[90] Tischendorf C. Solution of index-2 differential algebraic equations and its application in circuit simulation: PHD Thesis. Berlin: Humboldt University, 1996.
[91] Higueras I, M(a|¨)rz R. Tischendorf C. Stability preserving integration of index- 2 DAEs. Appl. Numer. Math., 2003, 45:201-229
[92] Celik E. On the numerical solution of differential-algebraic equation with index-2. Appl. Math. Comput., 2004, 156:541-550
[93] Guzel N, Bayram M. Numerical solution of differential-aigebraic equations with index-2. Appl. Math. Comput., 2006, 174(2): 1279-1289
[94] Guzel N, Bayram M. On the numerical solution of differential-algebraic equations with index-3. Appl. Math. Comput., 2006, 175(2): 1320-1331
[95] Campbell S L. Singular linear systems of differential equations with delays. Appl. Anal.,1980, 2:129-136
[96] Ascher U M, Petzold L R. The numerical solution of delay-differential-algebraic equations of retarded and neutral type. SIAM J. Numer. Anal., 1995, 32:1635-1657
[97] Hauber R. Numerical treatment of retarded differential-algebraic equations by collocation methods. Adv. Comput. Math., 1997, 7:573-592
[98] Zhu W, Petzold L R. Asymptotic stability of linear delay differential-algebraic equations and numerical methods. Appl. Numer. Math., 1997, 24:247-264
[99] Zhu W, Petzold L R. Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type. Appl. Numer. Math., 1998, 27:309-325
[100] Liu Y. Runge-Kutta-collocation methods for systems of functional-differential and functional equations. Adv. Comput. Math., 1999, 11:315-329
[101] Baker C T H, Paul C A H, Tian H. Differential algebraic equations with after-effect. J. Comput. Appl. Math., 2002, 140: 63-80
[102] Huang C M, Chang Q S. Stability analysis of numerical methods for systems of functional-differential and functional equations. Comput. Math. Appl., 2002, 44: 717-729
[103] Zhang C J, Liao X X. Nonlinear stability of(p, σ)-methods for stiff delay-differential-algebraic systems. J. Cont. The. Appl., 2001, 18(6): 827-832
[104]甘四清．奇异摄动问题与时滞奇异摄动问题数值分析：博士学位论文．北京：中国科学院，2001．
[105]甘四清，孙耿．时滞奇异摄动问题单支方法的收敛性．中国科学(A辑)，2001，31(4)：314-323
[106] Gan S Q, Sun G, Zheng W M. Errors of linear multistep methods and Runge-Kutta methods for singular perturbation problems with delays. Comput. Math. Appl., 2002, 44: 1157-1173
[107] Schein O, Denk G. Numerical solution of stochastic differential-algebraic equations with applications to trunsient noise simulation of microelectronic circuis. J. Comput. Appl. Math., 1998, 100(1): 77-92
[108] Christain P. Analysis and numerical integration of stochastic differential-algebraic equations and application in electronic circuit simulation: PHD thesis. Munich: Technische Universit(a|¨)t M(u|¨)nchen, 1999.
[109] Gerdin M, Sjoberg J. Nonlinear stochastic differential-algebraic equations with application to particle filtering. IEEE Conf. Deci. Contr., 2006, 45: 6630-6635
[110] Winkler R. Stochastic dfferential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math., 2003, 157(2): 477-505
[111] Alabert A, Ferrante M. Linear stochastic differential-algebraic equations with constant coefficients. Elect. Com. Prob., 2006, 11:316-335
[112] Higham D J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev., 2001, 43(3): 525-546
[113] 李寿佛．刚性微分方程算法理论．长沙：湖南科学技术出版社，1997．
[114] Dahlquist G. A special stability problem for linear multistep methods. BIT, 1963, 3: 27-43
[115] Bahar A, Mao X. Stochastic delay Lotka-Volterra model. J. Math. Anal. Appl., 2004, 292: 364-380
[116] Baker C T H, Ford J M, Ford N J. Bifurcations in approximate solutions of stochastic delay differential equations. Int. J. Bifurcat. Chaos Appl. Sci. Eng., 2004, 14(9): 2999-3021
[117] Mao X. Stochastic differential delay equations of population dynamics. J. Math. Anal. Appl., 2005, 304: 296-320
[118] Mao X. Almost sure exponential stability for delay stochastic differential equations with respect to semimartingales. Stoch. Anal. Appl., 1991, 9(2): 177-194
[119] Mao X. Existence and uniqueness of the solutions of delay stochastic integral equations. Stoch. Anal. Appl., 1989, 7(1): 59-74
[120] Winkler R. Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math., 2004, 163(2): 435-463
[121] In't Hout K J. The stability of θ -methods for systems of delay differential equations. Ann. Numer. Math., 1994, 1(3): 323-334
[122] Kuang J X, Xiang J X, Tian H J. The asymptotic stability of one-parameter methods for neutral differential equations. BIT, 1994, 34(3): 400-408
[123] Higham D J. A-stability and stochastic mean-square stability. BIT, 2000, 40:404-409
[124] Higham D J. Mean-Square and asymptotical stability of numerical methods for stochastic ordinary differential equations. SIAM J. Appl. Math., 2000, 38:753-769
[125] Baker C T H, Buckwar E. On Halanay-type analysis of exponential stability for the theta-Maruyama method for stochastic delay differential equations. Stoch. Dyn., 2005, 5: 201-209
[126] K(u|¨)chler U, Platen E. Stong discrete time approximation of stochastic differential equations with time delay. Math. Comput. Simulat., 2000, 54:189-205
[127] Guglielmi N, Hairer E. Implementing radau ⅡA methods for stiff delay differential equations. Computing, 2001, 67: 1-12
[128]李宏智，李建国．块隐式单步方法求解一类延迟微分代数方程．华中科技大学学报(自)，2003，10：111-113
[129]甘四清，孙耿．单支方法的收敛性．应用数学，2001，14：30-33
[130] Guglielmi N. On the Newton iteration in the application of collocation methods to implicit delay equations. Appl. Numer. Math., 2005, 53: 281-297
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