The Numerical Analysis Of Several Classes Of Stochastic Delay-differential-algebraic Systems

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.