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.
Author(s): Elemér E. Rosinger (Eds.)
Series: North-Holland Mathematics Studies 164
Publisher: Elsevier Science Ltd
Year: 1990
Language: English
Pages: III-XVI, 1-380
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
