Solution of Continuous Nonlinear PDEs through Order Completion 1st Edition by M B Oberguggenberger, E E Rosinger ISBN 9780080872926

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ISBN 13: 9780080872926

Author: M.B. Oberguggenberger, E.E. Rosinger

This work inaugurates a new and general 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. However, the nonlinear PDEs dealt with need not satisfy any kind of monotonicity properties. Moreover, the solution method is completely type independent. In other words, it does not assume anything about the nonlinear PDEs, except for the continuity of their left hand term, which includes the unkown function. Furthermore the right hand term of such nonlinear PDEs can in fact be given any discontinuous and measurable function.

Table of contents:

PART I: GENERAL EXISTENCE OF SOLUTIONS THEORY
Section 1. Introduction
Section 2. Approximation of Solutions of Continuous Nonlinear PDEs
Section 3. Spaces of Generalized Functions
Section 4. Extending T(x,D) to the Order Completion of Spaces of Smooth Functions
Section 5. Existence of Generalized Solutions
Section 6. A Few First Examples
Section 7. Generalized Solutions as Measurable Functions
7.1. And the Measurable Functions
7.2. The Flabby Sheaf Structure of uˆm(n)*
7.3. Further Examples and Counterexamples
PART II: APPLICATIONS TO SPECIFIC CLASSES OF LINEAR AND NONLINEAR PDEs
Section 8. The Cauchy Problem for Nonlinear First Order Systems
Section 9. An Abstract Existence Result
Section 10. PDEs with Sufficiently Many Smooth Solutions
Section 11. Nonlinear Systems with Measures as Initial Data
11.1. Extension of an Existence Result
11.2. The Meaning of Generalized Solutions
Section 12. Solution of PDEs and the Completion of Uniform Spaces
Section 13. Partial Orders Compatible with a Nonlinear Partial Differential Operator
Section 14. Miscellaneous Results
14.1. Preliminaries
14.2. Simultaneous Solvability of a Family of Nonlinear PDEs
14.3. Strengthened Approximation and Existence Results with Conditions on Derivatives
PART III: GROUP INVARIANCE OF GLOBAL GENERALIZED SOLUTIONS OF NONLINEAR PDEs
Section 15. Introduction
Section 16. Group Invariance of Global Generalized Solutions of Nonlinear PDEs Obtained Through the
16.0. Preliminaries
16.1. Hilbert’s Fifth Problem
16.2. Review of the Global Existence Result
16.3. Group Transformations for Functions
16.4. Group Transformations for Algebras of Generalized Functions
16.5. Group Invariance for Global Generalized Solutions
16.6. Computing Symmetry Groups for Global Generalized Solutions
16.7. Application to Delta Waves of Semilinear Hyperbolic PDEs
16.8. Application to Shock Waves
16.9. The Case of Local Groups of Transformations
16.10. Local Group Invariance
Section 17. Group Invariance of Generalized Solutions Obtained Through the Algebraic Method : An Alt
17.0. Preliminaries
17.1. A Brief Review of the Structure of Colombeau’s Generalized Functions
17.2. Group Transformations on G(LRn)
17.3. Group Invariance of Global Generalized Solutions
17.4.Symmetry Groups for Generalized Solutions
Section 18. Group Invariance of Global Generalized Solutions Obtained Through the Order Completion M
18.1. Introduction
18.2. Group Transformations of Spaces of Generalized Functions
18.3. Group Transformations of Dedekind Order Completions
18.4. Group Invariance of Global Generalized Solutions
18.5. Computing Symmetry Groups for Global Generalized Solutions
18.6. Application to Delta Waves, Klein-Gordon Equations and Shock Waves

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Tags: M B Oberguggenberger, E E Rosinger, Solution of Continuous, Nonlinear PDEs through, Order Completion