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Author | : Michael Oberguggenberger |
Publisher | : CRC Press |
Total Pages | : 396 |
Release | : 1999-03-16 |
Genre | : Mathematics |
ISBN | : 9780849306495 |
Download Nonlinear Theory of Generalized Functions Book in PDF, ePub and Kindle
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 |
Total Pages | : 226 |
Release | : 2006-11-14 |
Genre | : Mathematics |
ISBN | : 3540469818 |
Download A Nonlinear Theory of Generalized Functions Book in PDF, ePub and Kindle
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 | : Springer |
Total Pages | : 704 |
Release | : 1990 |
Genre | : Mathematics |
ISBN | : |
Download A Nonlinear Theory of Generalized Functions Book in PDF, ePub and Kindle
Author | : M. Oberguggenberger |
Publisher | : |
Total Pages | : 304 |
Release | : 1998-11-01 |
Genre | : |
ISBN | : 9780582356825 |
Download Nonlinear Theory of Generalized Functions Book in PDF, ePub and Kindle
Author | : Hebe Azevedo Biagioni |
Publisher | : |
Total Pages | : 228 |
Release | : 1988 |
Genre | : Nonlinear theories |
ISBN | : |
Download Introduction to a Nonlinear Theory of Generalized Functions Book in PDF, ePub and Kindle
Author | : Ram P. Kanwal |
Publisher | : Springer Science & Business Media |
Total Pages | : 474 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1468400355 |
Download Generalized Functions Theory and Technique Book in PDF, ePub and Kindle
This second edition of Generalized Functions has been strengthened in many ways. The already extensive set of examples has been expanded. Since the publication of the first edition, there has been tremendous growth in the subject and I have attempted to incorporate some of these new concepts. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12 of this edition by a judicious process of elimination and addition of the subject matter. The new Chapter 13 explains the interplay between the theories of moments, asymptotics, and singular perturbations. Similarly, some sections of Chapter 15 have been revised and included in earlier chapters to improve the logical flow of ideas. However, two sections are retained. The section dealing with the application of the probability theory has been revised, and I am thankful to Professor Z.L. Crvenkovic for her help. The new material included in this chapter pertains to the modern topics of periodic distributions and microlocal theory. I have demonstrated through various examples that familiarity with the generalized functions is very helpful for students in physical sciences and technology. For instance, the reader will realize from Chapter 6 how the generalized functions have revolutionized the Fourier analysis which is being used extensively in many fields of scientific activity.
Author | : M. Grosser |
Publisher | : Springer Science & Business Media |
Total Pages | : 517 |
Release | : 2013-04-17 |
Genre | : Mathematics |
ISBN | : 9401598452 |
Download Geometric Theory of Generalized Functions with Applications to General Relativity Book in PDF, ePub and Kindle
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 | : E.E. Rosinger |
Publisher | : Elsevier |
Total Pages | : 429 |
Release | : 1987-11-01 |
Genre | : Science |
ISBN | : 0080872573 |
Download Generalized Solutions of Nonlinear Partial Differential Equations Book in PDF, ePub and Kindle
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 | : Jean F. Colombeau |
Publisher | : Springer |
Total Pages | : 193 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 3540475109 |
Download Multiplication of Distributions Book in PDF, ePub and Kindle
This book presents recent and very elementary developments of a theory of multiplication of distributions in the field of explicit and numerical solutions of systems of PDEs of physics (nonlinear elasticity, elastoplasticity, hydrodynamics, multifluid flows, acoustics). The prerequisites are kept to introductory calculus level so that the book remains accessible at the same time to pure mathematicians (as a smoothand somewhat heuristic introdcution to this theory) and to applied mathematicians, numerical engineers and theoretical physicists (as a tool to treat problems involving products of distributions).
Author | : M Nedeljkov |
Publisher | : CRC Press |
Total Pages | : 172 |
Release | : 1998-05-20 |
Genre | : Mathematics |
ISBN | : 9780582356832 |
Download Linear Theory of Colombeau Generalized Functions Book in PDF, ePub and Kindle
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.