微分方程中的小波方法
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摘要
自本世纪30年代以来，在物理、工程、化学、生物、经济等众多领域中产
生的大量数学模型可以用带有极限环的平面自治系统来描述，极限环的问题已
变得愈来愈重要，并引起了许多理论数学和应用数学工作者的注意。工程中的
许多问题常常可归结于常微分方程的求解，由于自动控制理论的众多问题需要
用微分方程来描述，这为小波方法的使用提供了可能，小波分析及其应用的日
益活跃也引起了建筑工程及抗震设计等方面工程技术人员的关注，传统的各种
分析与计算方法在具有其各自优点的同时，难免有其各自的缺点。本文对小波
变换及小波理论进行研究的基础上，着重研究小波在微分方程中应用的一些基
本问题和基本方法，主要提出并解决了如下的问题：
1．提出了微分方程解的小波逼近的理论框架。对一类有紧支撑的正交小波
的正则度进行了分析，同时对这类小波及相关尺度函数的正则性指数做出估
计，提出了一种新的估计方法，得出了一个优于I.Daubechies的估计。在区间
样条小波插值的最佳逼近性的基础上，对其误差进行了精确估计；提出了有限
区间上的自由结点多尺度分析及小波；针对求解小波逼近函数给出了一种基于
逐项插值的快速小波变换的算法。
2．由于小波函数既有有限元基函数的特征，又有频域方法的优点，加之对
高频信号的聚焦能力，非常适合求解微分方程边值问题。我们利用小波具有良
好的局部化特点来研究线性边值问题的奇异性，提出了改进的小波配点法。
3．我们采用小波逼近的方法，结合Lyapunov稳定性理论，提出了一种崭
新的不需要进行系统辨识的在线的连续自适应控制算法。对于一类未知非线性
系统，我们提出的算法能够保证闭环的稳定性，跟踪误差可以收敛于零的一个
邻域内。
4．如何将小波分析应用于建筑物体系，在地震作用下的输出信号分析是一
个崭新的课题。我们利用小波变换对建筑物系统通过叠加的方法，对任一信号
激励下的响应进行预测，并直接给出预测信号的小波分析。利用小波分别对单
质点弹性体系（如水塔、单层房屋等）和多质点弹性体系（如厂房、烟囱等）
进行小波变换，给出了建筑物体系在脉冲响应下的输出信号变换与该体系的输
入信号的关系，为全面分析建筑物在地震等脉冲响应下的输出信号提供了良好
的理论基础。
5．确定微分动力系统的极限环的位置无论在理论上还是在应用上都具有重
要的意义。我们首先总结出求极限环方程的三种方法，其中的两种方法都是我
们在研究平面微分动力系统的极限环时最早独立提出来的，然后借助谐波小波
对极限环的位置进行研究，设计了寻求极限环方程的算法，开辟了用小波研究
极限环理论的新途径。
From the 1930's, very many mathematical models from physics, engineering, chemistry,
biology, economics. ect., were displayed as plane autonomous systems with limit cycles.
The problem of limit cycles has become more and more important and has attracted the
attention of many pure and applied mathematicians. A lot of problems in science can be
eventually presented in the form of ordinary differential equation (ODE). It is possible for
wavelet methods to solve plenty of questions, which are described by ODE, in autonomous
control theory, Architectural engineers begin to pay close attention to wavelet analysis and
its application. While the methods now we using to analysis and compute ODE have their
own advantages, they also have some shortcomings. In this dissertation, based on the
research of wavelet transformation and theory, the following problems are discussed:
1. The theory frame of wavelet approximation for the solution of ODE is put forward.
The regularity of a class of compactly supported orthogonal wavelet is analyzed. A new
method for the estimation of exponent regularity of the kind wavelet and related scale
function is proposed. Based on the optimal approximation of spline interval wavelet, the
interpolation error estimation is given. An approach to the study of multiresolution analysis
and wavelet on finite interval with free knots is presented. A sort of algorithm of fast
wavelet transformation is obtained to find Out wavelet approximation function.
2. The wavelet function is very suitable for solving the boundary value problem of ODE,
not only for it characteristic of finite basis function, but has advantage of frequency method.
We research the singularity of linear boundary value problem by making use of localizable
distinguishing feature of wavelet. The improved wavelet-collocation method is given.
3. Combining with Lyapunov stability theory and taking advantage of wavelet
approximation method, we propose a new algorithm of continuous adaptive control without
system recognition on lines. To a kind of unknown nonlinear system, our algorithm can
insurance the stability of closed loop. The trace error can be convergence into a
neighborhood of zero.
4. How to manage the output signal analysis of architectural systems under the
earthquake with wavelet analysis is a new subject. The systems of simple elastic particle
(water tower, house, ect.) and multiple elastic particles(factory building, chimney. ect.)
are analyzed with wavelet transformations, the relationship between the wavelet
transformation of output signal with the pulse respond of architectural system and input
signal of that system is put forward. The ideal theory basis is offered to analysis output
signal of architectural under the pulse respond.
5. It is of significance to find out the limit cycles of plane autonomous systems both in
theory and in application. First, we sum up three methods to look for the equation of limit
cycle, two of them are our earliest results of researching limit cycles. Then, we study the
place of limit cycle by means of harmonic wavelet . A sort of algorithm to compute the
equation of limit cycle is designed. A new path leading to the limit cycle theory is hewed
out.
生的大量数学模型可以用带有极限环的平面自治系统来描述，极限环的问题已
变得愈来愈重要，并引起了许多理论数学和应用数学工作者的注意。工程中的
许多问题常常可归结于常微分方程的求解，由于自动控制理论的众多问题需要
用微分方程来描述，这为小波方法的使用提供了可能，小波分析及其应用的日
益活跃也引起了建筑工程及抗震设计等方面工程技术人员的关注，传统的各种
分析与计算方法在具有其各自优点的同时，难免有其各自的缺点。本文对小波
变换及小波理论进行研究的基础上，着重研究小波在微分方程中应用的一些基
本问题和基本方法，主要提出并解决了如下的问题：
1．提出了微分方程解的小波逼近的理论框架。对一类有紧支撑的正交小波
的正则度进行了分析，同时对这类小波及相关尺度函数的正则性指数做出估
计，提出了一种新的估计方法，得出了一个优于I.Daubechies的估计。在区间
样条小波插值的最佳逼近性的基础上，对其误差进行了精确估计；提出了有限
区间上的自由结点多尺度分析及小波；针对求解小波逼近函数给出了一种基于
逐项插值的快速小波变换的算法。
2．由于小波函数既有有限元基函数的特征，又有频域方法的优点，加之对
高频信号的聚焦能力，非常适合求解微分方程边值问题。我们利用小波具有良
好的局部化特点来研究线性边值问题的奇异性，提出了改进的小波配点法。
3．我们采用小波逼近的方法，结合Lyapunov稳定性理论，提出了一种崭
新的不需要进行系统辨识的在线的连续自适应控制算法。对于一类未知非线性
系统，我们提出的算法能够保证闭环的稳定性，跟踪误差可以收敛于零的一个
邻域内。
4．如何将小波分析应用于建筑物体系，在地震作用下的输出信号分析是一
个崭新的课题。我们利用小波变换对建筑物系统通过叠加的方法，对任一信号
激励下的响应进行预测，并直接给出预测信号的小波分析。利用小波分别对单
质点弹性体系（如水塔、单层房屋等）和多质点弹性体系（如厂房、烟囱等）
进行小波变换，给出了建筑物体系在脉冲响应下的输出信号变换与该体系的输
入信号的关系，为全面分析建筑物在地震等脉冲响应下的输出信号提供了良好
的理论基础。
5．确定微分动力系统的极限环的位置无论在理论上还是在应用上都具有重
要的意义。我们首先总结出求极限环方程的三种方法，其中的两种方法都是我
们在研究平面微分动力系统的极限环时最早独立提出来的，然后借助谐波小波
对极限环的位置进行研究，设计了寻求极限环方程的算法，开辟了用小波研究
极限环理论的新途径。
From the 1930's, very many mathematical models from physics, engineering, chemistry,
biology, economics. ect., were displayed as plane autonomous systems with limit cycles.
The problem of limit cycles has become more and more important and has attracted the
attention of many pure and applied mathematicians. A lot of problems in science can be
eventually presented in the form of ordinary differential equation (ODE). It is possible for
wavelet methods to solve plenty of questions, which are described by ODE, in autonomous
control theory, Architectural engineers begin to pay close attention to wavelet analysis and
its application. While the methods now we using to analysis and compute ODE have their
own advantages, they also have some shortcomings. In this dissertation, based on the
research of wavelet transformation and theory, the following problems are discussed:
1. The theory frame of wavelet approximation for the solution of ODE is put forward.
The regularity of a class of compactly supported orthogonal wavelet is analyzed. A new
method for the estimation of exponent regularity of the kind wavelet and related scale
function is proposed. Based on the optimal approximation of spline interval wavelet, the
interpolation error estimation is given. An approach to the study of multiresolution analysis
and wavelet on finite interval with free knots is presented. A sort of algorithm of fast
wavelet transformation is obtained to find Out wavelet approximation function.
2. The wavelet function is very suitable for solving the boundary value problem of ODE,
not only for it characteristic of finite basis function, but has advantage of frequency method.
We research the singularity of linear boundary value problem by making use of localizable
distinguishing feature of wavelet. The improved wavelet-collocation method is given.
3. Combining with Lyapunov stability theory and taking advantage of wavelet
approximation method, we propose a new algorithm of continuous adaptive control without
system recognition on lines. To a kind of unknown nonlinear system, our algorithm can
insurance the stability of closed loop. The trace error can be convergence into a
neighborhood of zero.
4. How to manage the output signal analysis of architectural systems under the
earthquake with wavelet analysis is a new subject. The systems of simple elastic particle
(water tower, house, ect.) and multiple elastic particles(factory building, chimney. ect.)
are analyzed with wavelet transformations, the relationship between the wavelet
transformation of output signal with the pulse respond of architectural system and input
signal of that system is put forward. The ideal theory basis is offered to analysis output
signal of architectural under the pulse respond.
5. It is of significance to find out the limit cycles of plane autonomous systems both in
theory and in application. First, we sum up three methods to look for the equation of limit
cycle, two of them are our earliest results of researching limit cycles. Then, we study the
place of limit cycle by means of harmonic wavelet . A sort of algorithm to compute the
equation of limit cycle is designed. A new path leading to the limit cycle theory is hewed
out.
引文
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