Nonlinear Theory Of Generalized Functions

Author : Michael Oberguggenberger,Michael Grosser,Michael Kunzinger,Gunther Hormann
Publisher : CRC Press
Page : 396 pages
File Size : 54,7 Mb
Release : 1999-03-16
Category : Mathematics
ISBN : 0849306493

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Nonlinear Theory of Generalized Functions by Michael Oberguggenberger,Michael Grosser,Michael Kunzinger,Gunther Hormann Pdf

Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis. The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods.

Author : Hebe de Azevedo Biagioni
Publisher : Springer
Page : 226 pages
File Size : 42,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540469810

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A Nonlinear Theory of Generalized Functions by Hebe de Azevedo Biagioni Pdf

This book provides a simple introduction to a nonlinear theory of generalized functions introduced by J.F. Colombeau, which gives a meaning to any multiplication of distributions. This theory extends from pure mathematics (it presents a faithful generalization of the classical theory of C? functions and provides a synthesis of most existing multiplications of distributions) to physics (it permits the resolution of ambiguities that appear in products of distributions), passing through the theory of partial differential equations both from the theoretical viewpoint (it furnishes a concept of weak solution of pde's leading to existence-uniqueness results in many cases where no distributional solution exists) and the numerical viewpoint (it introduces new and efficient methods developed recently in elastoplasticity, hydrodynamics and acoustics). This text presents basic concepts and results which until now were only published in article form. It is in- tended for mathematicians but, since the theory and applications are not dissociated it may also be useful for physicists and engineers. The needed prerequisites for its reading are essentially reduced to the classical notions of differential calculus and the theory of integration over n-dimensional euclidean spaces.

Author : Hebe de Azevedo Biagioni
Publisher : Springer
Page : 218 pages
File Size : 52,7 Mb
Release : 1990-03-21
Category : Mathematics
ISBN : 3540524088

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A Nonlinear Theory of Generalized Functions by Hebe de Azevedo Biagioni Pdf

This book provides a simple introduction to a nonlinear theory of generalized functions introduced by J.F. Colombeau, which gives a meaning to any multiplication of distributions. This theory extends from pure mathematics (it presents a faithful generalization of the classical theory of C? functions and provides a synthesis of most existing multiplications of distributions) to physics (it permits the resolution of ambiguities that appear in products of distributions), passing through the theory of partial differential equations both from the theoretical viewpoint (it furnishes a concept of weak solution of pde's leading to existence-uniqueness results in many cases where no distributional solution exists) and the numerical viewpoint (it introduces new and efficient methods developed recently in elastoplasticity, hydrodynamics and acoustics). This text presents basic concepts and results which until now were only published in article form. It is in- tended for mathematicians but, since the theory and applications are not dissociated it may also be useful for physicists and engineers. The needed prerequisites for its reading are essentially reduced to the classical notions of differential calculus and the theory of integration over n-dimensional euclidean spaces.

Author : Hebe Azevedo Biagioni
Publisher : Unknown
Page : 228 pages
File Size : 48,6 Mb
Release : 1988
Category : Nonlinear theories
ISBN : UOM:39015029903252

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Introduction to a Nonlinear Theory of Generalized Functions by Hebe Azevedo Biagioni Pdf

Author : M. Oberguggenberger,M. Grosser,M. Kunzinger
Publisher : Unknown
Page : 304 pages
File Size : 44,7 Mb
Release : 1998-11-01
Category : Electronic
ISBN : 0582356822

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Nonlinear Theory of Generalized Functions by M. Oberguggenberger,M. Grosser,M. Kunzinger Pdf

Author : E.E. Rosinger
Publisher : Elsevier
Page : 406 pages
File Size : 45,6 Mb
Release : 1987-11-01
Category : Mathematics
ISBN : 0080872573

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Generalized Solutions of Nonlinear Partial Differential Equations by E.E. Rosinger Pdf

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 concerning existence, uniqueness regularity, etc., of generalized solutions for nonlinear partial differential equations can be reduced to elementary calculus in Euclidean spaces, combined with elementary algebra in quotient rings of families of smooth functions on Euclidean spaces, all of that joined by certain asymptotic interpretations. In this way, one avoids the complexities and difficulties of the customary functional analytic methods which would involve sophisticated topologies on various function spaces. The result is a rather elementary yet powerful and far-reaching method which can, among others, give generalized solutions to linear and nonlinear partial differential equations previously unsolved or even unsolvable within distributions or hyperfunctions. Part 1 of the volume discusses the basic limitations of the linear theory of distributions when dealing with linear or nonlinear partial differential equations, particularly the impossibility and degeneracy results. Part 2 examines the way Colombeau constructs a nonlinear theory of generalized functions and then succeeds in proving quite impressive existence, uniqueness, regularity, etc., results concerning generalized solutions of large classes of linear and nonlinear partial differential equations. Finally, Part 3 is a short presentation of the nonlinear theory of Rosinger, showing its connections with Colombeau's theory, which it contains as a particular case.

Author : Michael Oberguggenberger,Michael Grosser,Michael Kunzinger,Gunther Hormann
Publisher : Routledge
Page : 400 pages
File Size : 46,7 Mb
Release : 2022-02-28
Category : Mathematics
ISBN : 9781351428033

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Nonlinear Theory of Generalized Functions by Michael Oberguggenberger,Michael Grosser,Michael Kunzinger,Gunther Hormann Pdf

Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis. The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods.

Author : Ram P. Kanwal
Publisher : Springer Science & Business Media
Page : 490 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9780817681746

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Generalized Functions by Ram P. Kanwal Pdf

Provides a more cohesive and sharply focused treatment of fundamental concepts and theoretical background material, with particular attention given to better delineating connections to varying applications Exposition driven by additional examples and exercises

Author : J.F. Colombeau
Publisher : Elsevier
Page : 280 pages
File Size : 46,9 Mb
Release : 2011-08-18
Category : Mathematics
ISBN : 0080872247

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Elementary Introduction to New Generalized Functions by J.F. Colombeau Pdf

The author's previous book `New Generalized Functions and Multiplication of Distributions' (North-Holland, 1984) introduced `new generalized functions' in order to explain heuristic computations of Physics and to give a meaning to any finite product of distributions. The aim here is to present these functions in a more direct and elementary way. In Part I, the reader is assumed to be familiar only with the concepts of open and compact subsets of R&eegr;, of C∞ functions of several real variables and with some rudiments of integration theory. Part II defines tempered generalized functions, i.e. generalized functions which are, in some sense, increasing at infinity no faster than a polynomial (as well as all their partial derivatives). Part III shows that, in this setting, the partial differential equations have new solutions. The results obtained show that this setting is perfectly adapted to the study of nonlinear partial differential equations, and indicate some new perspectives in this field.

Author : M Nedeljkov,S Pilipovic,D Scarpalezos
Publisher : CRC Press
Page : 172 pages
File Size : 52,6 Mb
Release : 1998-05-20
Category : Mathematics
ISBN : 0582356830

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Linear Theory of Colombeau Generalized Functions by M Nedeljkov,S Pilipovic,D Scarpalezos Pdf

Results from the now-classical distribution theory involving convolution and Fourier transformation are extended to cater for Colombeau's generalized functions. Indications are given how these particular generalized functions can be used to investigate linear equations and pseudo differential operators. Furthermore, applications are also given to problems with nonregular data.

Author : M. Grosser,M. Kunzinger,Michael Oberguggenberger,R. Steinbauer
Publisher : Springer Science & Business Media
Page : 517 pages
File Size : 50,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401598453

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Geometric Theory of Generalized Functions with Applications to General Relativity by M. Grosser,M. Kunzinger,Michael Oberguggenberger,R. Steinbauer Pdf

Over the past few years a certain shift of focus within the theory of algebras of generalized functions (in the sense of J. F. Colombeau) has taken place. Originating in infinite dimensional analysis and initially applied mainly to problems in nonlinear partial differential equations involving singularities, the theory has undergone a change both in in ternal structure and scope of applicability, due to a growing number of applications to questions of a more geometric nature. The present book is intended to provide an in-depth presentation of these develop ments comprising its structural aspects within the theory of generalized functions as well as a (selective but, as we hope, representative) set of applications. This main purpose of the book is accompanied by a number of sub ordinate goals which we were aiming at when arranging the material included here. First, despite the fact that by now several excellent mono graphs on Colombeau algebras are available, we have decided to give a self-contained introduction to the field in Chapter 1. Our motivation for this decision derives from two main features of our approach. On the one hand, in contrast to other treatments of the subject we base our intro duction to the field on the so-called special variant of the algebras, which makes many of the fundamental ideas of the field particularly transpar ent and at the same time facilitates and motivates the introduction of the more involved concepts treated later in the chapter.

Author : Michael Grosser,Eva Farkas,Michael Kunzinger,Roland Steinbauer
Publisher : American Mathematical Soc.
Page : 93 pages
File Size : 41,8 Mb
Release : 2001
Category : Mathematics
ISBN : 9780821827291

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On the Foundations of Nonlinear Generalized Functions I and II by Michael Grosser,Eva Farkas,Michael Kunzinger,Roland Steinbauer Pdf

In part 1 we construct a diffeomorphism invariant (Colombeau-type) differential algebra canonically containing the space of distributions in the sense of L. Schwartz. Employing differential calculus in infinite dimensional (convenient) vector spaces, previous attempts in this direction are unified and completed. Several classification results are achieved and applications to nonlinear differential equations involving singularities are given. Part 2 gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra ${\mathcal G}^d = {\mathcal E}_M/{\mathcal N}$ introduced in part 1 and Colombeau's original algebra ${\mathcal G}^e$.Three main results are established: first, a simple criterion describing membership in ${\mathcal N}$ (applicable to all types of Colombeau algebras) is given; second, two counterexamples demonstrate that ${\mathcal G}^d$ is not injectively included in ${\mathcal G}^e$; and finally, it is shown that in the range ""between"" ${\mathcal G}^d$ and ${\mathcal G}^e$ only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of ${\mathcal G}^d$ on manifolds are derived.

Author : Michael Grosser,Michael Kunzinger,Michael Oberguggenberger
Publisher : Unknown
Page : 524 pages
File Size : 41,9 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 9401598460

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Geometric Theory of Generalized Functions with Applications to General Relativity by Michael Grosser,Michael Kunzinger,Michael Oberguggenberger Pdf

Author : A. S. Demidov
Publisher : Nova Publishers
Page : 154 pages
File Size : 44,7 Mb
Release : 2001
Category : Mathematics
ISBN : 1560729058

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Generalized Functions in Mathematical Physics by A. S. Demidov Pdf

This important book gives an interconnected presentation of some basic ideas, concepts, results of the theory of generalised functions (first of all, in the framework of the theory of distributions) and equations of mathematical physics. A part of the material is given according to the scheme: definition -- theorem -- proof. This scheme is convenient for presenting results in clear and concentrated form. However, it seems reasonable to give a student the possibility not only to study a priori given definitions and proofs of theorems, but also to discover them while considering the problems involved. A series of sections serve this purpose. Moreover, a part of the material is given as exercises and problems.

Author : I. M. Gel′fand,G. E. Shilov
Publisher : American Mathematical Soc.
Page : 423 pages
File Size : 48,6 Mb
Release : 2016-04-19
Category : Theory of distributions (Functional analysis)
ISBN : 9781470426583

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Generalized Functions, Volume 1 by I. M. Gel′fand,G. E. Shilov Pdf

he first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel′fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. Volume 1 is devoted to basics of the theory of generalized functions. The first chapter contains main definitions and most important properties of generalized functions as functional on the space of smooth functions with compact support. The second chapter talks about the Fourier transform of generalized functions. In Chapter 3, definitions and properties of some important classes of generalized functions are discussed; in particular, generalized functions supported on submanifolds of lower dimension, generalized functions associated with quadratic forms, and homogeneous generalized functions are studied in detail. Many simple basic examples make this book an excellent place for a novice to get acquainted with the theory of generalized functions. A long appendix presents basics of generalized functions of complex variables.