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內容簡介

　　《非綫性物理科學：微分方程群性質理論講義》提供瞭確定和利用微分方程對稱性的李群方法簡明和清晰的介紹，並提供瞭在氣體動力學和其他非綫性模型中的大量應用，以及《非綫性物理科學：微分方程群性質理論講義》作者在這個經典領域的卓越貢獻。《非綫性物理科學：微分方程群性質理論講義》中還包含在其他現代書籍中不曾涉及的一些非常有剛的材料，例如：Ovsyannikow教授發展的部分不變解理論，該理論提供瞭求解非綫性微分方程和研究復雜數學模型強有力的工具。

作者簡介

　　L.V.Ovsyannikov，教授是20世紀60年代促進恢復微分方程群分析研究的領軍科學傢。他在不變解和部分不變解理論、微分方程群分類以及流體力學中的應用方麵作齣瞭基礎性的貢獻。在Ovsyannikow教授的影響下，李群分析目前已經發展成應用數學方麵相當活躍的領域。

內頁插圖

目錄

Editor's preface
Preface

1 One-parameter continuous transformation groups admitted by differential equations
1.1 One-parameter continuous transformation group
1.1.1 Definition
1.1.2 Canonical parameter
1.1.3 Examples
1.1.4 Auxiliary functions of groups
1.2 Infinitesimal operator of the group
1.2.1 Definition and examples
1.2.2 Transformation of functions
1.2.3 Change of coordinates
1.3 Invariants and invariant manifolds
1.3.1 Invariants
1.3.2 Invariant manifolds
1.3.3 Invariance of regularly defined manifolds
1.4 Theory of prolongation
1.4.1 Prolongation of the space
1.4.2 Prolonged group
1.4.3 First prolongation of the group operator
1.4.4 Second prolongation of the group operator
1.4.5 Properties of prolongations of operators
1.5 Groups admitted by differentialequations
1.5.1 Determining equations
1.5.2 First-order ordinary differential equations
1.5.3 Second-orderordinarydifferentialequations
1.5.4 Heat equation
1.5.5 Gasdynamic equations
1.6 Lie algebra of operators
1.6.1 Commutator. Definition of a Lie algebra
1.6.2 Properties of commutator
1.6.3 Lie algebra of admitted operators

2 Lie algebras and local Lie groups
2.1 Lie algebra
2.1.1 Definition and examples
2.1.2 Subalgebra and ideal
2.1.3 Structure of finite-dimensionalLie algebras
2.2 Adjoint algebra
2.2.1 Inner derivation
2.2.2Adjoint algebra
2.2.3 Inner automorphisms of a Lie algebra.
2.3 Local Lie group
2.3.1 Coordinates in a group
2.3.2 Subgroups
2.3.3 Canonical coordinates of the first kind
2.3.4 First fundamental theorem of Lie
2.3.5 Second fundamental theorem of Lie
2.3.6 Properties ofcanonicalcoordinate systems of the first kind
2.3.7 Third fundamental theorem of Lie
2.3.8 Lie algebra of a local Lie group
2.4 Subgroup, normal subgroup and factor group
2.4.1 Lemma on commutator
2.4.2 Subgroup
2.4.2 Subgroup
2.4.3 Normal subgroup
2.4.4 Factor grop
2.5 Inner automorphisms of a group and of its Lie algebra
2.5.1 Inner automorphism.
2.5.2 Lie algebra of GA and adjoint algebra of Lr
2.6 Local Lie group of transformations
2.6.1 Introduction
2.6.2 Lie's first theorem.
2.6.3 Lie's second theorem
2.6.4 Canonical coordinates of the second kind

3 Group invariant solutions of differential equations
3.1 Invariants of the group GNr
3.1.1 Invariance criterion
3.1.2 Functional independence
3.1.3 Linearly unconnected operators
3.1.4 Integration of jacobian systems
3.1.5 Computation ofinvariance
……

前言/序言

　　The theory of differential equations has two aspects of investigation，namely local and global，no matter whether the equations arise from applied problems of physics and mechanics or from abstract speculations （which is rather frequent in modern mathematics）.The local aspect is characterized by dealing with the inner structure of a family of solutions and its investigation in a neighborhood of a certain point.The global approach deals with solutions definedin some domain and having a given behavior on its boundary.
　　It would certainly be erroneous to oppose these directions to each other.However，it is no good to ignore the differences in approaches either.While the global approach necessitates the functional analytic apparatus，the local viewpoint allows one to get along with algebraic means only.A brilliant example of a profound local consideration is the famous Cauchy-Kovalevskaya theorem which is，in fact，an algebraic statement.Moreover，it is an easy matter to notice that the theory of boundary value problems also makes an essential application of various algebraic properties of the whole family of solutions. Therefore，the local aspect of the algebraic theory of differential equations is quite vital.
非綫性物理科學：微分方程群性質理論講義 [Lectures on the Theory of Group Properties of Differential Equations] 下載 mobi epub pdf txt 電子書

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