Non-Linear Partial Differential Equations

An Algebraic View of Generalized Solutions

By

  • E.E. Rosinger, Department of Mathematics, University of Pretoria, South Africa

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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Book information

  • Published: November 1990
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-88700-9

Reviews

The author of this volume proposes a systematic and wide-ranging nonlinear theory of GF's, based on differential algebras, applicable to a very broad class of NLPDE's. He also provides an excellent historical survey and derives some new results.
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Table of Contents

1. Conflict Between Discontinuity, Multiplication and Differentiation. 2. Global Version of the Cauchy Kovalevskaia Theorem on Analytic Nonlinear Partial Differential Equations. 3. Algebraic Characterization for the Solvability of Nonlinear Partial Differential Equations. 4. Generalized Solutions of Semilinear Wave Equations with Rough Initial Values. 5. Discontinuous, Shock, Weak and Generalized Solutions of Basic Nonlinear Partial Differential Equations. 6. Chains of Algebras of Generalized Functions. 7. Resolution of Singularities of Weak Solutions for Polynomial Nonlinear Partial Differential Equations. 8. The Particular Case of Colombeau's Algebras. Final Remarks. References.