抽象空间中非线性Volterra泛函微分方程的数值稳定性分析
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摘要
泛函微分方程广泛出现于生物学、物理学、经济及社会学、控制论及工程技术等诸多领域。其算法理论的研究对推动这些科技领域的发展无疑非常重要。近年来，泛函微分方程，特别是其特例——延迟微分方程的算法理论的研究得到了很大发展，获得了丰硕成果，这些成果可参见Barwell，Bellen，Torelli，Zennaro，Spijker，Watanabe，Roth，in’t Hout，Baker，Paul，Koto，Iserles以及李寿佛，匡蛟勋，刘明珠，黄乘明，张诚坚，田红炯，胡广大，甘四清等人的工作。但这些工作局限于讨论有限维内积空间中的初值问题，是以内积范数和单边Lipschitz条件为基础的。然而，科学与工程技术中还存在大量刚性问题，尽管问题本身是整体良态的，但当使用内积范数时，其最小单边Lipschitz常数却不可避免地取非常巨大的正值。因此，突破内积范数和单边Lipschitz常数的局限去建立泛函微分方程及其数值方法的理论是一项非常迫切而又富有意义的工作。
本文研究求解Banach空间中Volterra泛函微分方程的几类常用数值方法的非线性稳定性以及Hilbert空间中Volterra泛函微分方程及其数值方法的散逸性。所获主要结果如下：
(1) 提出了Banach空间中的Volterra泛函微分方程试验问题类D_(λ*)(α，β，μ_1，μ_2)和D_(λ*，δ)(α，β，μ_1，μ_2)，获得了理论解的一系列稳定性结果，并获得了D_0(α，β，μ_1，μ_2)类问题的基于对数矩阵范数的条件估计。这些结果是文献中已有的关于Banach空间中常微分方程初值问题类K(μ，λ~*)和K(μ，λ~*，δ)及泛函微分方程问题类D_0(α，β，μ_1，μ_2)的相应结果的推广。
(2) 获得了求解Banach空间中D_(0，0)(α，β，μ_1，μ_2)类非线性Volterra泛函微分方程初值问题的θ—方法的一系列数值稳定性结果。这些结果适用于延迟微分方程，积分微分方程及实际问题中遇到的各种类型的泛函微分方程，同时还包含当α+β＞0时的稳定性结果，因而比文献中已有的关于θ—方法的数值稳定性结果更为一般和深刻。同时还首次给出了Banach空间中非线性Volterra泛函微分方程θ—方法的数值渐近稳定性结果。
Many real-life phenomena in physics, biology, medicine, economics, control theory and engineering and so on can be modelled as initial value problem in Volterra functional differential equations (VFDEs). It is important for the development of these fields to study theory and application of numerical methods for VFDEs. In the last few decades, the theory of numerical methods for VFDEs, especially for the delay differential equations (DDEs), have been widely discussed by many authors such as Barwell, Bellen, Torelli, Zennaro,Spijker, Watanabe, Roth, in't Hout, Baker, Paul, Koto, Iser-les, Shoufu Li, Jiaoxun Kuang, Mingzhu Liu, Chengming Huang, Chengjian Zhang, Hongjiong Tian, Guangda Hu, Siqing Gan. However, the existing results for functional differential equations are mainly based upon the inner product and the corresponding norm in Euclidean space of finite dimension. But for some stiff problems, it may happen that the one-sided Lipschitz constant is very large. Therefore, it is urgent and meaningful to break through the restriction of the inner product and the corresponding norm.
The main results obtained in this paper are as follows:
(1) we introduce the test problem classes D_(λ*) (α, β, μ_1, μ_2) and D_(λ*,δ)(α,β, μ_1,μ_2) with respect to the initial value problems of nonlinear Volterra functional differential equations in Banach spaces. A series of stability results of the analytic solution are obtained and a condition estimate for the class D_0(α, β, μ_1, μ_2) which based on logarithmic matrix norm is also obtained. The above results extend the existing results for ordinary differential equa-tions(ODEs).
(2) A series of numerical stability results of θ—methods when applied to the class D_(0,0)(α, β, μ_1, μ_2) are obtained, which can be directly applied to the special problem of DDEs, integro-differential equations(IDEs) and VFDEs of other type which appear in practice, and are more general and deeper than the existing results for θ—methods in literature. A series of new asymptotic stability results are also obtained.
(3) A series of new stability criteria of the linear multistep methods and
本文研究求解Banach空间中Volterra泛函微分方程的几类常用数值方法的非线性稳定性以及Hilbert空间中Volterra泛函微分方程及其数值方法的散逸性。所获主要结果如下：
(1) 提出了Banach空间中的Volterra泛函微分方程试验问题类D_(λ*)(α，β，μ_1，μ_2)和D_(λ*，δ)(α，β，μ_1，μ_2)，获得了理论解的一系列稳定性结果，并获得了D_0(α，β，μ_1，μ_2)类问题的基于对数矩阵范数的条件估计。这些结果是文献中已有的关于Banach空间中常微分方程初值问题类K(μ，λ~*)和K(μ，λ~*，δ)及泛函微分方程问题类D_0(α，β，μ_1，μ_2)的相应结果的推广。
(2) 获得了求解Banach空间中D_(0，0)(α，β，μ_1，μ_2)类非线性Volterra泛函微分方程初值问题的θ—方法的一系列数值稳定性结果。这些结果适用于延迟微分方程，积分微分方程及实际问题中遇到的各种类型的泛函微分方程，同时还包含当α+β＞0时的稳定性结果，因而比文献中已有的关于θ—方法的数值稳定性结果更为一般和深刻。同时还首次给出了Banach空间中非线性Volterra泛函微分方程θ—方法的数值渐近稳定性结果。
Many real-life phenomena in physics, biology, medicine, economics, control theory and engineering and so on can be modelled as initial value problem in Volterra functional differential equations (VFDEs). It is important for the development of these fields to study theory and application of numerical methods for VFDEs. In the last few decades, the theory of numerical methods for VFDEs, especially for the delay differential equations (DDEs), have been widely discussed by many authors such as Barwell, Bellen, Torelli, Zennaro,Spijker, Watanabe, Roth, in't Hout, Baker, Paul, Koto, Iser-les, Shoufu Li, Jiaoxun Kuang, Mingzhu Liu, Chengming Huang, Chengjian Zhang, Hongjiong Tian, Guangda Hu, Siqing Gan. However, the existing results for functional differential equations are mainly based upon the inner product and the corresponding norm in Euclidean space of finite dimension. But for some stiff problems, it may happen that the one-sided Lipschitz constant is very large. Therefore, it is urgent and meaningful to break through the restriction of the inner product and the corresponding norm.
The main results obtained in this paper are as follows:
(1) we introduce the test problem classes D_(λ*) (α, β, μ_1, μ_2) and D_(λ*,δ)(α,β, μ_1,μ_2) with respect to the initial value problems of nonlinear Volterra functional differential equations in Banach spaces. A series of stability results of the analytic solution are obtained and a condition estimate for the class D_0(α, β, μ_1, μ_2) which based on logarithmic matrix norm is also obtained. The above results extend the existing results for ordinary differential equa-tions(ODEs).
(2) A series of numerical stability results of θ—methods when applied to the class D_(0,0)(α, β, μ_1, μ_2) are obtained, which can be directly applied to the special problem of DDEs, integro-differential equations(IDEs) and VFDEs of other type which appear in practice, and are more general and deeper than the existing results for θ—methods in literature. A series of new asymptotic stability results are also obtained.
(3) A series of new stability criteria of the linear multistep methods and
引文
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