【作者】 肖飞雁；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2008， 博士

【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．更多还原

【Abstract】 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.更多还原