UID:
almahu_9947367746802882
Format:
1 online resource (429 p.)
ISBN:
1-281-79307-8
,
9786611793074
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0-08-087257-3
Series Statement:
North-Holland mathematics studies ; 146
Content:
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Note:
Description based upon print version of record.
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Front Cover; Generalized Solutions of Nonlinear Partial Differential Equations; Copyright Page; Foreword; TABLE OF CONTENTS; PART 1: REVIEW OF SOME ESSENTIALS IN WEAK SOLUTIONS AND DISTRIBUTIONS; CHAPTER 1. CLASSICAL VERSUS DISTRIBUTION SOLUTIONS; 1. Integral Versus Partial Differential Equations; 2. The Necessity of Generalized Solutions; 3. The Concept of Generalized Solution; 4. Why Use Distributions?; 5. Multiplication, Localization, Regularization, Convolution and Change of Variables; CHAPTER 2. IMPOSSIBILITY AND DEGENERACY RESULTS IN DISTRIBUTIONS
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1. Difficulties with Distribution Multiplication2. The Schwartz Impossibility Result; 3. Extensions; 4. Degeneracy Results; CHAPTER 3. LIMITATIONS OF THE LINEAR DISTRIBUTION THEORY; 1. The Lewy Inexistence Result; 2. Stability Paradoxes Concerning Weak Solutions of Nonlinear Partial Differential Equations; PART 2: COLOMBEAU'S NONLINEAR THEORY OF GENERALIZED FUNCTIONS; CHAPTER 1. THE DIFFERENTIAL ALGEBRA G AS AN EXTENTION OF THE D' DISTRIBUTIONS; 1. Smooth Approximations and Representations; 2. Properties of the Differential Algebra G(RN); 3. Generalized Complex Numbers
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4. Point Values of Generalized Functions5. Integrals of Generalized Functions; 6. Coupled Calculus in G(RN); 7. Recapitulation; 8. General Nonlinear Operations on G(RN); 9. Generalized Functions on Arbitrary Domains; 10. Variants of Colombeau's Differential Algebras G; 11. Changes of Variables in Generalized Functions; Appendix 1. The Natural Character of Colombeau's Differential Algebra; Appendix 2. Asymptotics Without a Topology; Appendix 3. Properties of the Index Set O; Appendix 4. The Sheaf Structure of Generalized Functions
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Appendix 5. Connections With Previous Attempts in Distribution MultiplicationAppendix 6. An Intuitive Illustration of the Structure of Colombeau's Algebra G; CHAPTER 2. GENERALIZED SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS; 1. Preliminary Remarks; 2. Shock Wave Equations and Solutions . .; 3. Generalized Solutions of Nonlinear Wave Equations in Quantum Field Interaction; 4. Restrictions of Generalized Functions to Subspaces; 5. Generalized Solutions of Nonlinear Parabolic Equations; 6. Generalized Functions on Arbitrary Subsets
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CHAPTER 3. GENERALIZED SOLUTIONS FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS1. Preliminary Remarks; 2. Smooth Partial Derivatives; 3. Solutions for Distributionally Unsolvable Equations; 4. Linear Scattering; PART 3: A NECESSARY STRUCTURE FOR NONLINEAR THEORIES OF GENERALIZED FUNCTIONS; CHAPTER 1. STABILITY, GENERALITY AND EXACTNESS OF GENERALIZED SOLUTIONS; 1. Preliminary Remarks; 2. Some Basics of the Customary Sequential Approach in Solving Nonlinear Partial Diffe- rential Equations; 3. A Natural Framework for Generalized Solutions of Nonlinear Partial Differential Equations
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4. The Neutrix Condition
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English
Additional Edition:
ISBN 0-444-70310-1
Language:
English
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