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Solution of Continuous Nonlinear PDEs Through Order Completion

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This work examines a solution method for arbitrary continuous nonlinear PDEs. The solution method is based on Dedekind order completion of usual spaces of smooth functions defined on domains in Euclidean spaces.

Author(s): Michael B. Oberguggenberger and Elemér E. Rosinger (Eds.)
Series: North-Holland Mathematics Studies 181
Publisher: Elsevier, Academic Press
Year: 1994

Language: English
Pages: ii-xii, 1-432

Content:
Editor
Page ii

Edited by
Page iii

Copyright page
Page iv

Dedication
Page v

Foreword
Pages vii-xii

Part I. General Existence of Solutions Theory
Page 1

1. Introduction
Pages 3-10

2. Approximation of Solutions of Continuous Nonlinear PDEs
Pages 11-23

3. Spaces of Generalized Functions
Pages 24-30

4. Extending T(x, D) to the Order Completion of Spaces of Smooth Functions
Pages 31-37

5. Existence of Generalized Solutions
Pages 38-64

6. A Few First Examples
Pages 65-73

7. Generalized Solutions as Measurable Functions
Pages 74-158

Part II. Applications to Specific Classes of Nonlinear and Linear PDEs
Page 159

8. The Cauchy Problem for Nonlinear First Order Systems
Pages 161-183

9. An Abstract Existence Result
Pages 184-194

10. PDEs with Sufficiently Many Smooth Solutions
Pages 195-209

11. Nonlinear Systems with Measures as Initial Data
Pages 210-236

12. Solution of PDEs and the Completion of Uniform Spaces
Pages 237-262

13. Partial Orders Compatible with a Nonlinear Partial Differential Operator
Pages 263-277

14. Miscellaneous Results
Pages 278-293

Part III. Group Invariance of Global Generalized Solutions of Nonlinear PDEs
Page 295

15. Introduction
Pages 297-298

16. Group Invariance of Global Generalized Solutions of Nonlinear PDEs Obtained Through the Algebraic Method
Pages 299-353

17. Group Invariance of Generalized Solutions Obtained Through the Algebraic Method : An Alternative Approach
Pages 354-364

18. Group Invariance of Global Generalized Solutions Obtained Through the Order Completion Method
Pages 365-387

Appendix. MacNeille Order Completion Through Dedekind Cuts and Related Results
Pages 390-420

References
Pages 421-428

Index
Pages 429-432