离散与分布式延迟微分方程数值方法稳定性分析
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摘要
本博士论文主要考虑几类离散和分布式延迟微分方程的数值稳定性,诸如非线性中立型延迟微分方程,非线性中立型延迟积分微分方程,随机延迟积分微分方程以及随机Volterra积分微分方程。主要考虑的数值格式是线性多步方法和随机θ-方法。我们分别研究了线性多步方法在求解非线性中立型延迟微分方程和非线性中立型延迟积分微分方程时的渐近稳定性,以及随机θ-方法求解随机延迟积分微分方程和随机Volterra积分微分方程时的均方渐近稳定性。整个论文如下包括6个部分：
第一章,我们简要介绍确定性和随机延迟微分方程的一些应用背景,以及延迟微分方程数仇方法稳定性分析的研究现状。同时,我们给出本文的工作概要。
第二章研究A-稳定的线性多步方法在求解非线性中立型延迟微分方程时的渐近稳定性,证明了带有线性插值的任何A-稳定的线性多步方法是GAS-稳定的。
第三章研究非线性中立型变延迟积分微分方程解析解和数值解的稳定性,给出了方程解析解稳定的充分条件；接着考虑A-稳定的线性多步方法求解此类方程时的渐近稳定性,证明了在非约束网格下,线性多步方法能够保持解析解的渐近稳定性。
第四章研究随机θ-方法求解线性随机延迟积分微分方程时的均方渐近稳定性,证明了在适当条件下,随机θ-方法能够保持方程真解的均方渐近稳定性。
第五章进一步研究随机θ-方法求解非线性随机延迟积分微分方程的均方渐近稳定性,证明了在约束网格条件下,当θ∈[1/2,1]时,随机θ-方法是无条件均方渐近稳定的。当θ∈[0,1/2)时,在适当步长限制条件-下,方法是均方渐近稳定的。
第六章,我们将随机θ-方法扩展应用到求解非线性随机Volterra积分微分方程。首先,我们证明随机θ-方法在求解此类方程时具有1/2阶均方收敛性。接着,我们给出了非线性随机Volterra积分微分方程真解均方指数稳定的充分条件。在此条件-下,我们证明了随机θ-方法当1/2≤θ≤1时对任意步长是均方渐近稳定的；当0≤θ<1/2时,随机θ-方法对适当小的步长是均方渐近稳定的。
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.
第一章,我们简要介绍确定性和随机延迟微分方程的一些应用背景,以及延迟微分方程数仇方法稳定性分析的研究现状。同时,我们给出本文的工作概要。
第二章研究A-稳定的线性多步方法在求解非线性中立型延迟微分方程时的渐近稳定性,证明了带有线性插值的任何A-稳定的线性多步方法是GAS-稳定的。
第三章研究非线性中立型变延迟积分微分方程解析解和数值解的稳定性,给出了方程解析解稳定的充分条件；接着考虑A-稳定的线性多步方法求解此类方程时的渐近稳定性,证明了在非约束网格下,线性多步方法能够保持解析解的渐近稳定性。
第四章研究随机θ-方法求解线性随机延迟积分微分方程时的均方渐近稳定性,证明了在适当条件下,随机θ-方法能够保持方程真解的均方渐近稳定性。
第五章进一步研究随机θ-方法求解非线性随机延迟积分微分方程的均方渐近稳定性,证明了在约束网格条件下,当θ∈[1/2,1]时,随机θ-方法是无条件均方渐近稳定的。当θ∈[0,1/2)时,在适当步长限制条件-下,方法是均方渐近稳定的。
第六章,我们将随机θ-方法扩展应用到求解非线性随机Volterra积分微分方程。首先,我们证明随机θ-方法在求解此类方程时具有1/2阶均方收敛性。接着,我们给出了非线性随机Volterra积分微分方程真解均方指数稳定的充分条件。在此条件-下,我们证明了随机θ-方法当1/2≤θ≤1时对任意步长是均方渐近稳定的；当0≤θ<1/2时,随机θ-方法对适当小的步长是均方渐近稳定的。
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.
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