微分方程的定性理論

前言

Differential equations are mainly used to describe the changes of quanti-ties or behavior of certain systems in applications, such as those governedby Newton's laws in physics.When the differential equations under study are linear, the conventionalmethods, such as the Laplace transform method and the power series solu-tions, can be used to solve the differential equations analytically, that is, thesolutions can be written out using formulas.When the differential equations under study are nonlinear, analytical so-lutions cannot, in general, be found; that is, solutions cannot be writtenout using formulas. In those cases, one approach is to use numerical ap-proximations. In fact, the recent advances in computer technology makethe numerical approximation classes very popular because powerful softwareallows students to quickly approximate solutions of nonlinear differentialequations and visualize their properties.However, in most applications in biology, chemistry, and physics mod-eled by nonlinear differential equations where analytical solutions may beunavailable, people are interested in the questions related to the so-calledqualitative properties, such as: will the system have at least one solu-tion？ will the system have at most one solution？ can certain behavior ofthe system be controlled or stabilized？ or will the system exhibit some peri-odicity？ If these questions can be answered without solving the differentialequations, especially when analytical solutions are unavailable, we can stillget a very good understanding of the system. Therefore, besides learningsome numerical methods, it is also important and beneficial to learn howto analyze some qualitative properties.

內(nèi)容概要

　　書中主要講解了微分方程理論的基本方法，對(duì)微分方程的存在性、連續(xù)依賴性、穩(wěn)定性、周期解、自治微分系統(tǒng)、動(dòng)力系統(tǒng)等基本問題進(jìn)行詳細(xì)分析，并注重理論間的聯(lián)系?！段⒎址匠痰亩ㄐ岳碚摗坊A(chǔ)性強(qiáng)、應(yīng)用廣泛，是一本適合大學(xué)高年級(jí)選修課、研究生雙語教學(xué)以及讀者自學(xué)的英文教科書。

作者簡(jiǎn)介

劉和濤，教授，留美執(zhí)教數(shù)十年，曾在培生教育等國(guó)際著名、出版機(jī)構(gòu)出版過多種教材，為美國(guó)多所院校采用。本教材秉承了國(guó)外先進(jìn)教學(xué)理念，并針對(duì)國(guó)內(nèi)學(xué)生實(shí)際情況，尤其注、意了由淺入深的理論過渡，建立了完備的邏輯體系，語言地、道，是適合于雙語教學(xué)的優(yōu)秀教科書，亦適合學(xué)生自學(xué)。

書籍目錄

PrefaceChapter 1 A Brief Description1. Linear Differential Equations2. The Need for Qualitative Analysis3. Description and TerminologyChapter 2 Existence and Uniqueness1. Introduction2. Existence and Uniqueness3. Dependence on Initial Data and Parameters4. Maximal Interval of Existence5. Fixed Point MethodChapter 3 Linear Differential Equations1. Introduction2. General Nonhomogeneous Linear Equations3. Linear Equations with Constant Coefficients4. Periodic Coefficients and Floquet TheoryChapter 4 Autonomous Differential Equations in R21. Introduction2. Linear Autonomous Equations in R23. Perturbations on Linear Equations in R24. An Application: A Simple PendulumChapter 5 Stability1. Introduction2. Linear Differential Equations3. Perturbations on Linear Equations4. Liapunov's Method for Autonomous EquationsChapter 6 Periodic Solutions1. Introduction2. Linear Differential Equations3. Nonlinear Differential EquationsChapter 7 Dynamical Systems1. Introduction2. Poincare-Bendixson Theorem in R23. Limit Cycles4. An Application: Lotka-Volterra EquationChapter 8 Some New Equations1. Introduction2. Finite Delay Differential Equations3. Infinite Delay Differential Equations4. Integrodifferential Equations5. Impulsive Differential Equations6. Equations with Nonlocal Conditions7. Impulsive Equations with Nonlocal Conditions8. Abstract Differential EquationsAppendixReferencesIndex

章節(jié)摘錄

插圖：The study of linear differential equations is very important for the fol-lowing reasons. First, the study provides us with some basic knowledgefor understanding general nonlinear differential equations. Second, manynonlinear differential equations can be written as summations of linear dif-ferential equations and some small nonlinear perturbations. Thus, undercertain conditions, the qualitative properties of linear differential equationscan be used to infer essentially the same qualitative properties for nonlineardifferential equations.

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《微分方程的定性理論》是由中國(guó)科學(xué)技術(shù)大學(xué)出版社出版的。

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