Stability Analysis Of Numerical Methods For Discrete And Distributed Delay Differential Equations

This doctoral dissertation is concerned with numerical stability of several classes of discrete and distributed delay differential equations, such as nonlinear neutral delay dif-ferential equations (NNDDEs), nonlinear neutral delay integro-differential equations (NN-DIDEs), stochastic delay integro-differential equations (SDIDEs) and stochastic Volterra integro-differential equations (SVIDEs). The main numerical schemes we consider include linear multistep methods and stochasticθ-methods. The asymptotic stability of linear multi-step methods for NNDEs and NNDIDEs, and the mean-square asymptotic stability of stochas-ticθ-methods for SDIDEs and SVIDEs are investigated respectively. The whole dissertation contains the following six parts:In Chapter 1, some application background of deterministic and stochastic delay differ-ential equations and the present state of the research of stability analysis of numerical methods for delay differential equations are briefly introduced. Also, the main works of this dissertation are listed.In Chapter 2, the asymptotic stability of A-stable linear multistep methods for nonlinear neutral delay differential equations is investigated. It is shown that any A-stale linear multistep methods with linear interpolation are GAS-stable.In Chapter 3, the analytical and numerical stability of nonlinear neutral delay integro-differential equations with variable delay are studied. First, some sufficient conditions for the analytical stability are derived. And then the asymptotic stability of A-stable linear multi-step methods for such equations is considered. It is shown that any A-stable linear multistep methods can preserve the asymptotic stability of the analytical solution with non-constrained meshes.In Chapter 4, the mean-square asymptotic stability of the stochasticθ-method for linear stochastic delay integro-differential equations is investigated. It is shown that the stochasticθ-methods can reproduce the mean square stability of the exact solution under appropriate conditions.In Chapter 5, we further investigate the mean-square asymptotic stability of the stochasticθ-method for nonlinear stochastic delay integro-differential equations. It is shown that under non-restrictive meshes, the stochasticθ-method is unconditional mean-square asymptotically stable ifθ[1/2,1]. Whenθ∈[0,]), the method is mean-square asymptotically stable with some mesh limitation.In Chapter 6, the stochasticθ-method is extended to solve nonlinear stochastic Volterra integro-differential equations. The mean-square convergence and asymptotic stability of the method are studied. First, we prove that the stochasticθ-method is convergent of order 1/2 in mean-square sense for such equations. Then, a sufficient condition for mean-square exponential stability of the true solution is given. Under this condition, it is shown that the stochasticθ-method is mean-square asymptotically stable for every stepsize if 1/2≤θ≤1, and when 0≤θ≤1/2, the stochastic 0-method is mean-square asymptotically stable for some small stepsizes.