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Transformation Methods for Nonlinear Partial Differential Equations (Hardcover)

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Product Details: Transformation Methods for Nonlinear Partial Differential Equations

Author:	Dominic G. Edelen, Jian-Hua Wang, Dominic G. B Edelen
Format:	Hardcover
Publisher:	World Scientific Pub Co Inc (05/01/1992)
ISBN:	9810209339
ISBN13:	9789810209339
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Publisher Notes: Transformation Methods for Nonlinear Partial Differential Equations

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Ch. 1 Contact Manifolds, Ideals and Partial Differential Equations 1

1 The Space of Independent Variables 1

2 Graph Space 3

3 The First Order Contact Manifold 5

4 Fundamental Ideals and Partial Differential Equations 7

5 Contact Manifolds and Fundamental Ideals of Higher Order 11

6 Isovectors, Lie Algebras, and Families of Solution Maps 15

7 Explicit Characterization of Isovectors of Contact Ideals 23

8 Symmetries that are Not Prolongations 30

9 Balance Equations when R = N 33

Ch. 2 An Order Independent Method of Characteristics 45

1 Introduction and Preview 45

2 The Extension Operator on the Lie Manifold of a Vector Field 51

3 Characteristic Solutions of PDE that are Determined by a Closed Ideal 58

4 Problems with First Order Contact Forms 62

5 Problems with Several Dependent Variables 72

6 Higher Order Equations 81

7 Comparison with Group Invariant (Similarity) Solutions 86

Ch. 3 Horizontal Ideals, Foliations, and Solutions 91

1 Canonical Systems of Vector Fields 92

2 Reformulation in terms of Horizontal and Vertical Ideals 97

3 Closure Conditions and the Resulting Foliation Structures 105

4 A Basic Existence Theorem for Systems of PDE 116

5 Reduction by Isovectors of the Balance Ideal 121

6 Examples with N = 1 and n = 2 132

7 Inverse Problems 145

8 Analysis of Second Order Systems of PDE in K-subscript 1 150

Ch. 4 Primitive Integrals, Isovectors, and Resolutions 161

1 Primitive Integrals and Constraints 162

2 Isovectors of the Horizontal Ideal 167

3 Vector Space Properties 171

4 Lie Algebra Properties 173

5 Transport Properties 177

6 Resolutions 180

7 Existence of Solutions to the System (2.3) 184

8 Isovectors of Horizontal Ideals on K-subscript 2 188

9 Systems that Admit Specified First Integrals 189

Ch. 5 Extended Canonical Transformations 211

1 Extended Canonical Transformations for First Order Contact Manifolds 211

2 Examples with n = 2 220

3 Extended Canonical Transformations of K-subscript 2 226

4 Group Properties 228

5 An Extended Hamilton-Jacobi Method 237

6 One Parameter Families of Transformations 252

7 Auto-Balance Transformations and the Generation of Solutions 269

8 Solutions of the Navier-Stokes Equations 278

Appendix A. An Alternative Treatment of Extended Canonical Transformations 291

A1. Extended Canonical Transformations 291

A2. Group Properties 295

Appendix B. Multiple Integral Variational Problems 303

B1. Formulation of the Problem 303

B2. An Existence Theorem for Euler-Lagrange Equations 306

B3. Aspects Particular to Variational Problems 312

References 319

Index 323

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