Transformation Methods for Nonlinear Partial Differential Equations

by Edelen, Dominic G. B.; Wang, Jian-Hua

• ISBN13: 9789810209339
• ISBN10: 9810209339
• Format: Hardcover
• Copyright: 1992-05-01
• Publisher: World Scientific Pub Co Inc

Table of Contents

Forward
Contact Manifolds, Ideals and Partial Differential Equations	p. 1
The Space of Independent Variables	p. 1
Graph Space	p. 3
The First Order Contact Manifold	p. 5
Fundamental Ideals and Partial Differential Equations	p. 7
Contact Manifolds and Fundamental Ideals of Higher Order	p. 11
Isovectors, Lie Algebras, and Families of Solution Maps	p. 15
Explicit Characterization of Isovectors of Contact Ideals	p. 23
Symmetries that are Not Prolongations	p. 30
Balance Equations when R = N	p. 33
An Order Independent Method of Characteristics	p. 45
Introduction and Preview	p. 45
The Extension Operator on the Lie Manifold of a Vector Field	p. 51
Characteristic Solutions of PDE that are Determined by a Closed Ideal	p. 58
Problems with First Order Contact Forms	p. 62
Problems with Several Dependent Variables	p. 72
Higher Order Equations	p. 81
Comparison with Group Invariant (Similarity) Solutions	p. 86
Horizontal Ideals, Foliations, and Solutions	p. 91
Canonical Systems of Vector Fields	p. 92
Reformulation in terms of Horizontal and Vertical Ideals	p. 97
Closure Conditions and the Resulting Foliation Structures	p. 105
A Basic Existence Theorem for Systems of PDE	p. 116
Reduction by Isovectors of the Balance Ideal	p. 121
Examples with N = 1 and n = 2	p. 132
Inverse Problems	p. 145
Analysis of Second Order Systems of PDE in K[subscript 1]	p. 150
Primitive Integrals, Isovectors, and Resolutions	p. 161
Primitive Integrals and Constraints	p. 162
Isovectors of the Horizontal Ideal	p. 167
Vector Space Properties	p. 171
Lie Algebra Properties	p. 173
Transport Properties	p. 177
Resolutions	p. 180
Existence of Solutions to the System (2.3)	p. 184
Isovectors of Horizontal Ideals on K[subscript 2]	p. 188
Systems that Admit Specified First Integrals	p. 189
Extended Canonical Transformations	p. 211
Extended Canonical Transformations for First Order Contact Manifolds	p. 211
Examples with n = 2	p. 220
Extended Canonical Transformations of K[subscript 2]	p. 226
Group Properties	p. 228
An Extended Hamilton-Jacobi Method	p. 237
One Parameter Families of Transformations	p. 252
Auto-Balance Transformations and the Generation of Solutions	p. 269
Solutions of the Navier-Stokes Equations	p. 278
Appendix A. An Alternative Treatment of Extended Canonical Transformations	p. 291
A1. Extended Canonical Transformations	p. 291
A2. Group Properties	p. 295
Appendix B. Multiple Integral Variational Problems	p. 303
B1. Formulation of the Problem	p. 303
B2. An Existence Theorem for Euler-Lagrange Equations	p. 306
B3. Aspects Particular to Variational Problems	p. 312
References	p. 319
Index	p. 323
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