对称方法在偏微分方程中的应用 英文版PDF|Epub|txt|kindle电子书版本下载

对称方法在偏微分方程中的应用 英文版PDF格式电子书版下载

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1 Local Transformations and Conservation Laws1

1.1 Introduction1

1.2 Local Transformations5

1.2.1 Point transformations6

1.2.2 Contact transformations8

1.2.3 Higher-order transformations10

1.2.4 One-parameter higher-order transformations10

1.2.5 Point symmetries16

1.2.6 Contact and higher-order symmetries20

1.2.7 Equivalence transformations and symmetry classification21

1.2.8 Recursion operators for local symmetries24

1.3 Conservation Laws38

1.3.1 Local conservation laws38

1.3.2 Equivalent conservation laws42

1.3.3 Multipliers for conservation laws.Euler operators43

1.3.4 The direct method for construction of conservation laws.Cauchy-Kovalevskaya form46

1.3.5 Examples50

1.3.6 Linearizing operators and adjoint equations53

1.3.7 Determination of fluxes of conservation laws from multipliers56

1.3.8 Self-adjoint PDE systems64

1.4 Noether's Theorem70

1.4.1 Euler-Lagrange equations71

1.4.2 Noether's formulation of Noether's theorem72

1.4.3 Boyer's formulation of Noether's theorem75

1.4.4 Limitations of Noether's theorem77

1.4.5 Examples79

1.5 Some Connections Between Symmetries and Conservation Laws89

1.5.1 Use of symmetries to find new conservation laws from known conservation laws90

1.5.2 Relationships among symmetries,solutions of adjoint equations,and conservation laws107

1.6 Discussion117

2 Construction of Mappings Relating Differential Equations121

2.1 Introduction121

2.2 Notations;Mappings of Infinitesimal Generators123

2.2.1 Theorems on invertible mappings127

2.3 Mapping of a Given PDE to a Specific Target PDE128

2.3.1 Construction of non-invertible mappings129

2.3.2 Construction of an invertible mapping by a point transformation133

2.4 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Symmetries139

2.4.1 Invertible mappings of nonlinear PDE systems (with at least two dependent variables)to linear PDE systems141

2.4.2 Invertible mappings of nonlinear PDE systems (with one dependent variable) to linear PDE systems146

2.5 Invertible Mappings of Linear PDEs to Linear PDEs with Constant Coefficients158

2.5.1 Examples of mapping variable coefficient linear PDEs to constant coefficient linear PDEs through invertible point transformations163

2.5.2 Example of finding the most general mapping of a given constant coefficient linear PDE to some constant coefficient linear PDE168

2.6 Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Conservation Law Multipliers173

2.6.1 Computational steps177

2.6.2 Examples of linearizations of nonlinear PDEs through conservation law multipliers179

2.7 Discussion184

3 Nonlocally Related PDE Systems187

3.1 Introduction187

3.2 Nonlocally Related Potential Systems and Subsystems in Two Dimensions191

3.2.1 Potential systems192

3.2.2 Nonlocally related subsystems193

3.3 Trees of Nonlocally Related PDE Systems199

3.3.1 Basic procedure of tree construction200

3.3.2 A tree for a nonlinear diffusion equation202

3.3.3 A tree for planar gas dynamics(PGD)equations204

3.4 Nonlocal Conservation Laws209

3.4.1 Conservation laws arising from nonlocally related systems210

3.4.2 Nonlocal conservation laws for diffusion-convection equations212

3.4.3 Additional conservation laws of nonlinear telegraph equations214

3.5 Extended Tree Construction Procedure222

3.5.1 An extended tree construction procedure223

3.5.2 An extended tree for a nonlinear diffusion equation225

3.5.3 An extended tree for a nonlinear wave equation228

3.5.4 An extended tree for the planar gas dynamics equations232

3.6 Discussion242

4 Applications of Nonlocally Related PDE Systems245

4.1 Introduction245

4.2 Nonlocal Symmetries248

4.2.1 Nonlocal symmetries of a nonlinear diffusion equation251

4.2.2 Nonlocal symmetries of a nonlinear wave equation256

4.2.3 Classification of nonlocal symmetries of nonlinear telegraph equations arising from point symmetries of potential systems270

4.2.4 Nonlocal symmetries of nonlinear telegraph equations with power law nonlinearities271

4.2.5 Nonlocal symmetries of the planar gas dynamics equations276

4.3 Construction of Non-invertible Mappings Relating PDEs283

4.3.1 Non-invertible mappings of nonlinear PDE systems to linear PDE systems284

4.3.2 Non-invertible mappings of linear PDEs with variable coefficients to linear PDEs with constant coefficients290

4.4 Discussion294

5 Further Applications of Symmetry Methods:Miscellaneous Extensions297

5.1 Introduction297

5.2 Applications of Symmetry Methods to the Construction of Solutions of PDEs301

5.2.1 The classical method302

5.2.2 The nonclassical method306

5.2.3 Invariant solutions arising from nonlocal symmetries that are local symmetries of nonlocally related systems314

5.2.4 Further extensions of symmetry methods for construction of solutions of PDEs connected with nonlocally related systems320

5.3 Nonlocally Related PDE Systems in Three or More Dimensions333

5.3.1 Divergence-type conservation laws and resulting potential systems334

5.3.2 Nonlocally related subsystems336

5.3.3 Tree construction,nonlocal conservation laws,and nonlocal symmetries337

5.3.4 Lower-degree conservation laws and related potential systems341

5.3.5 Examples of applications of nonlocally related systems in higher dimensions343

5.3.6 Symmetries and exact solutions of the three-dimensional MHD equilibrium equations350

5.4 Symbolic Software357

5.4.1 An example of symbolic computation of point symmetries357

5.4.2 An example of point symmetry classification359

5.4.3 An example of symbolic computation of conservation laws363

5.5 Discussion364

References369

Theorem,Corollary and Lemma Index383

Author Index385

Subject Index389