Non-linear partial differential equations: an algebraic view of generalized solutions

Elemér E. Rosinger (Eds.)0444887008, 9780444887009, 9780080872759

A massive transition of interest from solving linear partial differential equations to solving nonlinear ones has taken place during the last two or three decades. The availability of better computers has often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomena have presented increasing difficulties in the mentioned order. In particular, the latter two phenomena necessarily lead to nonclassical or generalized solutions for nonlinear partial differential equations.

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Table of contents :
Content:
Edited by
Page III

Copyright page
Page IV

Dedication
Page V

Foreword
Pages VII-XVI
E.E. Rosinger

Chapter 1 Conflict Between Discontinuity, Multiplication and Differentiation
Pages 1-99

Chapter 2 Global Version of The Cauchy-Kovalevskaia Theorem on Analytic Nonlinear Partial Differential Equations
Pages 101-129

Chapter 3 Algebraic Characterization For The Solvability of Nonlinear Partial Differential Equations
Pages 131-171

Chapter 4 Generalized Solutions of Semilinear Wave Equations With Rough Initial Values
Pages 173-195

Chapter 5 Discontinuous, Shock, Weak and Generalized Solutions of Basic Nonlinear Partial Differential Equations
Pages 197-219

Chapter 6 Chains of Algebras of Generalized Functions
Pages 221-269

Chapter 7 Resolution of Singularities of Weak Solutions For Polynomial Nonlinear Partial Differential Equations
Pages 271-299

Chapter 8 The Particular Case of Colombeau’S Algebras
Pages 301-344

Appendix 1: The Natural Character of Colohbeau’s Differential Algebra
Pages 345-353

Appendix 2: Asymptotics Without a Topology
Pages 354-356

Appendix 3: Connections with Previous Attempts in Distribition Multiplication
Pages 357-360

Appendix 4 An Intuitive Illustration of the Struciure of Colombeau’s Algebras
Pages 361-366

Final Remarks
Pages 367-370

References
Pages 371-380