对称方法在偏微分方程中的应用

内容简介

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　　This book is a sequel to Symmetries and Integration Methods (2002), by George W. Bluman and Stephen C. Anco. It includes a significant update of the material in the last three chapters of Symmet’ries an,d Dzjjerential Equa-tions (1989; reprinted with corrections, 1996), by George W. Bluman and Sukeyuki Kumei. The emphasis in the present book is on how to find sys-tematically symmetries (local and nonlocal) and conservation laws (local and nonlocal) of a given PDE system and how to use systematically symmetries and conservation laws for related applications. In particular, for a given PDE system, it is shown how systematically (1) to find higher-order and nonlocal symmetries of the system; (2) to construct by direct methods its conserva- tion laws through finding sets of conservation law multipliers and formulas to obtain the fluxes of a conservation law from a known set of multipliers; (3) to determine whether it has a linearization by an invertible mapping and con- struct such a linearization when one exists from knowledge of its symmetries andlor conservation law multipliers, in the case wheii the given PDE system is nonlinear; (4) to use conservation laws to construct equivalent nonlocally related systems; (5) to use such nonlocally related systems to obtain nonlo- cal symmetries, nonlocal conservation laws and non-invertible mappings to linear systems; and (6) to construct specific solutions from reductions arising from its symmetries as well as from extensions of symmetry methods to find such reductions.　　This book is aimed at applied mathematicians; scientists and engineers interested in finding solutions of partial differential equations and is written in the style of the above-mentioned 1989 book by Bluman and Kumei. There are numerous examples involving various well-known physical and engineering PDE systems.

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目录

PrefaceIntroduction1 Local Transformations and Conservation Laws1.1 Introduction1.2 Local Transformations1.2.1 Point transformations1.2.2 Contact transformations1.2.3 Higher-order transformations1.2.4 One-parameter higher-order transformations1.2.5 Point symmetries1.2.6 Contact and higher-order symmetries1.2.7 Equivalence transformations and symmetry classification1.2.8 Recursion operators for local symmetries1.3 Conservation Laws1.3.1 Local conservation laws1.3.2 Equivalent conservation laws1.3.3 Multipliers for conservation laws.Euler operators1.3.4 The direct method for construction of conservation laws.Cauchy-Kovalevskaya form1.3.5 Examples1.3.6 Linearizing operators and adjoint equations1.3.7 Determination of fluxes of conservation laws from multipliers1.3.8 Self-adjoint PDE systems1.4 Noether’s Theorem1.4.1 Euler-Lagrange equations1.4.2 Noether’s formulation of Noether’s theorem1.4.3 Boyer’s formulation of Noether’s theorem1.4.4 Limitations of Noether’s theorem1.4.5 Examples1.5 Some Connections Between Symmetries and Conservation Laws1.5.1 Use of symmetries to find new conservation laws from known conservation laws1.5.2 Relationships among symmetries，solutions of adjoint equations，and conservation laws1.6 Discussion2 Construction of Mappings Relating Differential Equations2.1 Introduction2.2 Notations； Mappings of Infinitesimal Generators2.2.1 Theorems on invertible mappings2.3 Mapping of a Given PDE to a Specific Target PDE2.3.1 Construction of non-invertible mappings2.3.2 Construction of an invertible mapping by a point transformation2.4 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Symmetries2.4.1 Invertible mappings of nonlinear PDE systems（with at least two dependent variables）to linear PDE systems2.4.2 Invertible mappings of nonlinear PDE systems（with one dependent variable）to linear PDE systems2.5 Invertible Mappings of Linear PDEs to Linear PDEs with Constant Coefficients2.5.1 Examples of mapping variable coefficient linear PDEs to constant coefficient linear PDEs through invertible point transformations2.5.2 Example of finding the most general mapping of a given constant coefficient linear PDE to some constant coefficient linear PDE2.6 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Conservation Law Multipliers2.6.1 Computational steps2.6.2 Examples of linearizations of nonlinear PDEs through conservation law multipliers2.7 Discussion3 Nonlocally Related PDE Systems3.1 Introduction3.2 Nonlocally Related Potential Systems and Subsystems in Two Dimensions3.2.1 Potential systems3.2.2 Nonlocally related subsystems3.3 Trees of Nonlocally Related PDE Systems3.3.1 Basic procedure of tree construction3.3.2 A tree for a nonlinear diffusion equation3.3.3 A tree for planar gas dynamics（PGD）equations3.4 Nonlocal Conservation Laws3.4.1 Conservation laws arising from nonlocally related systems3.4.2 Nonlocal conservation laws for diffusion-convection equations3.4.3 Additional conservation laws of nonlinear telegraph equations3.5 Extended Tree Construction Procedure3.5.1 An extended tree construction procedure3.5.2 An extended tree for a nonlinear diffusion equation3.5.3 An extended tree for a nonlinear wave equation3.5.4 An extended tree for the planar gas dynamics equations3.6 Discussion4 Applications of Nonlocally Related PDE Systems4.1 Introduction4.2 Nonlocal Symmetries4.2.1 Nonlocal symmetries of a nonlinear diffusion equation4.2.2 NonlocAL symmetries of a nonlinear wave equation4.2.3 Classification of nonlocal symmetries of nonlinear telegraph equations arising from point symmetries of potential systems4.2.4 Nonlocal symmetries of nonlinear telegraph equations with power law nonlinearities4.2.5 Nonlocal symmetries of the planar gas dynamics equations4.3 Construction of Non-invertible Mappings Relating PDEs4.3.1 Non-invertible mappings of nonlinear PDE systems to linear PDE systems4.3.2 Non-invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficients.4.4 Discussion5 Further Applications of Symmetry Methods： Miscellaneous Extensions5.1 Introduction5.2 Applications of Symmetry Methods to the Construction of Solutions of PDEs5.2.1 The classical method5.2.2 The nonclassical method5.2.3 Invariant solutions arising from nonlocal symmetries that are local symmetries of nonlocally related systems5.2.4 Futrther extensions of symmetry methods for construction of solutions of PDEs connected with nonlocaUy related systems5.3 Nonlocally Related PDE Systems in Three or More Dimensions5.3.1 Divergence-type conservation laws and resulting potential systems5.3.2 Nonlocally related subsystems5.3.3 Tree construction，nonlocal conservation laws，and nonlocal symmetries5.3.4 Lower-degree conservation laws and related potential systems5.3.5 Examples of applications of nonlocally related systems in higher dimensions5.3.6 Symmetries and exact solutions of the three-dimensional MHD equilibrium equations5.4 Symbolic Software5.4.1 An example of symbolic computation of point symmetries5.4.2 An example of point symmetry classification5.4.3 An example of symbolic computation of conservation laws5.5 DiscussionReferencesTheorem，Corollary and Lemma IndexAuthor IndexSubject Index

封面

书名:对称方法在偏微分方程中的应用

作者:布鲁曼

页数:398

定价:¥79.0

出版社:世界图书出版公司

出版日期:2015-01-01

ISBN:9787510086267

PDF电子书大小:83MB 高清扫描完整版