Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2013 | Buch

Potential Theory and Fundamental Solutions of Elliptic Equations Kapitel lesen Erstes Kapitel lesen

Distributions in the Physical and Engineering Sciences, Volume 2

Linear and Nonlinear Dynamics in Continuous Media

verfasst von: Alexander I. Saichev, Wojbor A. Woyczynski

Verlag: Springer New York

Buchreihe : Applied and Numerical Harmonic Analysis

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

Distributions in the Physical and Engineering Sciences is a comprehensive exposition on analytic methods for solving science and engineering problems. It is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important for practitioners and researchers. The goal of the books is to give the reader, specialist and non-specialist, useable and modern mathematical tools in their research and analysis.

Volume 2: Linear and Nonlinear Dynamics of Continuous Media continues the multivolume project which endeavors to show how the theory of distributions, also called the theory of generalized functions, can be used by graduate students and researchers in applied mathematics, physical sciences, and engineering. It contains an analysis of the three basic types of linear partial differential equations--elliptic, parabolic, and hyperbolic--as well as chapters on first-order nonlinear partial differential equations and conservation laws, and generalized solutions of first-order nonlinear PDEs. Nonlinear wave, growing interface, and Burger’s equations, KdV equations, and the equations of gas dynamics and porous media are also covered.

The careful explanations, accessible writing style, many illustrations/examples and solutions also make it suitable for use as a self-study reference by anyone seeking greater understanding and proficiency in the problem solving methods presented. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise.

Features

· Application oriented exposition of distributional (Dirac delta) methods in the theory of partial differential equations. Abstract formalism is keep to a minimum.

· Careful and rich selection of examples and problems arising in real-life situations. Complete solutions to all exercises appear at the end of the book.

· Clear explanations, motivations, and illustration of all necessary mathematical concepts.

Inhaltsverzeichnis

Frontmatter

Potentials, Diffusions, and Waves

Frontmatter
Chapter 9. Potential Theory and Fundamental Solutions of Elliptic Equations
Abstract
This chapter is devoted to the theory of linear elliptic partial differential equations and the related problems of potential theory. The basic concept of the Green’s function and the source solution are introduced and explored. This is followed by a detailed analysis of the Helmholtz equation in one, two, and three dimensions with applications to the diffraction problem for monochromatic waves. The inhomogeneous media case sets the stage for the Helmholtz equation with a variable coefficient and an analysis of waves in waveguides. The latter can be reduced to the celebrated Sturm–Liouville problem, and we study properties of its eigenvalues and eigenfunctions.
Alexander I. Saichev, Wojbor A. Woyczyński
Chapter 10. Diffusions and Parabolic Evolution Equations
Abstract
We begin with a study of the classic 1-D diffusion equation (also called heat equation) and its self-similar solutions. This is the simplest example of a linear parabolic partial differential equations. Well-posedness of an initial value problem with periodic data is then discussed. Subsequently, the exposition switches to the complex domain, and we introduce a simple version of the general Schrödinger equation. This makes it possible to study the diffraction problem and the so-called Fresnel zones. Multidimensional parabolic equation follow, and the general reflection method is explained. The chapter concludes with a study of the moving boundary problem and the standard physical problem of particle motion in a potential well.
Alexander I. Saichev, Wojbor A. Woyczyński
Chapter 11. Waves and Hyperbolic Equations
Abstract
Waves are everywhere, literally, and our senses are acutely attuned to waves of various types: mechanical, water, acoustic, electromagnetic, optical, etc. Moreover, quantum mechanics tells us that matter itself is, in a sense, indistinguishable from waves. In this chapter we concentrate on properties of linear waves in dispersive media; a discussion of nonlinear waves will be postponed until Chaps. 12–14. Here the methods of choice are integral transforms and asymptotic relations, especially the Fourier transform and the stationary phase method.
Alexander I. Saichev, Wojbor A. Woyczyński

Nonlinear Partial Differential Equations

Frontmatter
Chapter 12. First-Order Nonlinear PDEs and Conservation Laws
Abstract
Linear partial differential equations discussed in Part III often offer only a very simplified description of physical phenomena. To get a deeper understanding of some of them, it is necessary to move beyond the linear “universe” and consider nonlinear models, which in the case of continuous media, means nonlinear partial differential equations. Even today, their theory is far from complete and is the subject of intense study. On closer inspection, almost all physical phenomena in continuous media—from growing molecular interfaces at atomic scales to the structure of the distribution of matter in the universe at intergalactic scales—are nonlinear. The variety of nonlinear physical phenomena necessitates the use of various mathematical models and techniques to study them. In this part we shall restrict our attention to nonlinear waves of hydrodynamic type in media with weak or no dispersion. Since weak dispersion has little influence on the development of many nonlinear effects, we shall have a chance to observe typical behavior of these systems in strongly nonlinear regimes. The basic features of strongly nonlinear fields and waves are already evident in solutions of first-order nonlinear partial differential equations, and we take them as our starting point.
Alexander I. Saichev, Wojbor A. Woyczyński
Chapter 13. Generalized Solutions of First-Order Nonlinear PDEs
Abstract
Most of the equations of mathematical physics, and in particular nonlinear first-order partial differential equations, are a result of idealizing and simplifying assumptions. This approach promotes the effectiveness and elegance of mathematical models that adequately reflect some important qualitative features of the physical phenomena under consideration. However, sooner or later, one has to pay the price for the simplifying assumptions. The influence of factors not taken into account sometimes is gradual, and does not affect the qualitative picture of the physical phenomenon, but sometimes it is abrupt, and the simplified model is unable to describe the real course of events.
Alexander I. Saichev, Wojbor A. Woyczyński
Chapter 14. Nonlinear Waves and Growing Interfaces: 1-D Burgers–KPZ Models
Abstract
The present chapter studies behavior of two standard 1-D nonlinear dynamics models described by partial differential equations of order two and higher: the Burgers equation and the related KPZ model. We shall concentrate our attention on the theory of nonlinear fields of hydrodynamic type, where the basic features of the temporal evolution of nonlinear waves can be studied in the context of competition between the strengths of nonlinear and dissipative and/or dispersive effects. Apart from being model equations for specific physical phenomena, Burgers–KPZ equations are generic nonlinear equations that often serve as a testing ground for ideas for analysis of other nonlinear equations. They also produce a striking typically nonlinear phenomenon: shock formation.
Alexander I. Saichev, Wojbor A. Woyczyński
Chapter 15. Other Standard Nonlinear Models of Higher Order
Abstract
This chapter builds on the material of Chap. 14 and reviews other standard nonlinear models that can be described by partial differential equations. We begin with the model equations of gas dynamics, expand the Burgers–KPZ model to the multidimensional case and the related concentration fields, study in detail the Korteweg–de Vries (KdV) equations, in which one can observe the creation of solitary waves (solitons), and finally, discuss nonlinear flows in porous media.
Alexander I. Saichev, Wojbor A. Woyczyński
Backmatter
Metadaten
Titel
Distributions in the Physical and Engineering Sciences, Volume 2
verfasst von
Alexander I. Saichev
Wojbor A. Woyczynski
Copyright-Jahr
2013
Verlag
Springer New York
Electronic ISBN
978-0-8176-4652-3
Print ISBN
978-0-8176-3942-6
DOI
https://doi.org/10.1007/978-0-8176-4652-3