Download or read book Asymptotic Expansions of Solutions of Initial Boundary Value Problems for a Dispersive Hyperbolic Equation Classic Reprint written by Norman Bleistein and published by Forgotten Books. This book was released on 2018-10-10 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from Asymptotic Expansions of Solutions of Initial-Boundary Value Problems for a Dispersive Hyperbolic Equation Recently, the ray method has been applied to time dependent problems and in particular, dispersive hyperbolic equations A particular equation of this type, the klein-gordon equation, may be used to describe the propagation of electromagnetic waves in certain plasmas Initial-boundary value problems for this equation are presently being studied, since it is the simplest example of a large class of dispersive hyperbolic equations. As of now, there is no general proof that the ray method actually yields the asymptotic expansion of the exact solution. The purpose of the research in this paper is to compare the leading term of the asymptotic expansion of the exact solution and the ray solution for particular problems. The results are shown to agree and thereby add to the overwhelming evidence that the ray method is valid. We consider initial - boundary value problems in two space dimensions for the klein-gordon equation. The boundary is a parabola and.we require that the solution satisfy a homogeneous condition. The total solution is assumed to consist of a sum of a primary field and a reflected field. The associated rays are called primary and reflected rays and are straight lines in space time. The primary field must satisfy the equation and the initial conditions. The reflected field is taken to be zero initially and is determined at the boundary in such a way that the total solution satisfies the given boundary condition. The primary rays are emitted from the plane t O in space-time. For each primary ray incident on the boundary, a reflected ray is produced. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Download or read book Asymptotic Expansions of Solutions of Initial Boundary Value Problems for a Dispersive Hyperbolic Equation written by Norman Bleistein and published by Sagwan Press. This book was released on 2015-08-23 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work.This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
Download or read book Asymptotic Expansions of Solutions of Initial Boundary Value Problems for a Dispersive Hyperbolic Equation Primary Source Edition written by Norman Bleistein and published by Nabu Press. This book was released on 2013-12 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book.
Download or read book Asymptotic Solution of a Dispersive Hyperbolic Equation With Variable Coefficients Classic Reprint written by Norman Bleistein and published by Forgotten Books. This book was released on 2018-02-07 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from Asymptotic Solution of a Dispersive Hyperbolic Equation With Variable Coefficients In the past few years there has been a great deal of interest in the asymptotic expansion of solutions of boundary value problems and initial boundary value problems for partial differential equations. These expansions are obtained for large distance and/or time or for large values of a parameter which may appear naturally in the equation and/or initial conditions, boundary conditions, etc. Many of these problems can be treated by a method in which special curves, called rays, in space or space-time plan an essential role. The asymptotic solution is determined in terms of amplitude and phase functions which satis fy ordinary differential equations on these rays. These equations can often be solved explicitly, so that the solution to the original problem is known if the initial values on the rays of the amplitude and phase functions are known. In some cases these initial values can be determined directly from the data initial conditions, boundary conditions, etc.) of the given problem, while in other situations an indirect method is necessary. This method involves the exact solution of a canonical problem i.a., a problem simpler than that given originally but having the same local features as the given problem. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Download or read book Technical Abstract Bulletin written by and published by . This book was released on with total page 908 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Asymptotic Solution of a Dispersive Hyperbolic Equation with Variable Coefficients written by Norman Bleistein and published by . This book was released on 1965 with total page 80 pages. Available in PDF, EPUB and Kindle. Book excerpt: Initial-boundary value problems are considered for an energy conserving dispersive hyperbolic equation, the Klein-Gordon equation. This equation contains the main feature of dispersion: The speed of propagation depends on the frequency. The asymptotic expansion of solutions obtained by a technique which we call the ray method is compared with the asymptotic expansion of the exact solution. In every case considered, the solutions agree. Solutions are obtained for a series of initial-boundary value problems in one space dimension with variable coefficients. A new feature which is called space-time diffraction is found. This phenomenon has the following physical interpretation: A portion of the energy of a wave reaches a boundary surface and then gradually leaks off, leaving a diminishing residue on the boundary for all time. (Author).
Download or read book Matching of Asymptotic Expansions of Solutions of Boundary Value Problems written by A. M. Ilʹin and published by American Mathematical Soc.. This book was released on 1992 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the solution of singularly perturbed boundary value problems for differential equations. It presents, for the first time, a detailed and systematic treatment of the version of the matching method developed by the author and his colleagues. A broad class of problems is considered from a unified point of view, and the procedure for constructing asymptotic expansions is discussed in detail. The book covers formal constructions of asymptotic expansions and provides rigorous justifications of these asymptotics. One highlight is a complete asymptotic analysis of Burger's equation with small diffusion in the neighborhood of the gradient catastrophe point. The book is suitable as a text for graduate study in asymptotic methods in calculus and singularly perturbed equations.
Download or read book Asymptotic Methods for the Solution of Initial boundary Value Problems for Dispersive Hyperbolic Equations written by Norman Bleistein and published by . This book was released on 1965 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Asymptotic Expansions for Ordinary Differential Equations written by Wolfgang Wasow and published by Courier Dover Publications. This book was released on 2018-03-21 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: This outstanding text concentrates on the mathematical ideas underlying various asymptotic methods for ordinary differential equations that lead to full, infinite expansions. "A book of great value." — Mathematical Reviews. 1976 revised edition.
Download or read book Asymptotic Methods for the Solution of Dispersive Hyperbolic Equations written by Robert Mervin Lewis and published by . This book was released on 1964 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: A general method is presented for finding asymptotic solutions of initial-boundary value problems for linear hyperbolic partial differential equations. A large parameter lambda appears in the equation, multiplying a lower order (dispersive) term. It also may appear in the initial data and the inhomogeneous (source) term, in a variety of ways. This gives rise to a variety of different types of asymptotic solutions. The expansion procedure is a ray method, i.e., all of the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. These rays do not lie on characteristic hypersurfaces, but instead fill out the interior of the characteristic hypercone. They are associated with an appropriately defined group velocity. The details of the expansion are presented for a simple second order hyperbolic equation. The applicability of the method to symmetric hyperbolic systems and to the integro-differential equations for dispersive dielectrics is discussed. (Author).
Download or read book Composite Asymptotic Expansions written by Augustin Fruchard and published by Springer. This book was released on 2012-12-15 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of these lecture notes is to develop a theory of asymptotic expansions for functions involving two variables, while at the same time using functions involving one variable and functions of the quotient of these two variables. Such composite asymptotic expansions (CAsEs) are particularly well-suited to describing solutions of singularly perturbed ordinary differential equations near turning points. CAsEs imply inner and outer expansions near turning points. Thus our approach is closely related to the method of matched asymptotic expansions. CAsEs offer two unique advantages, however. First, they provide uniform expansions near a turning point and away from it. Second, a Gevrey version of CAsEs is available and detailed in the lecture notes. Three problems are presented in which CAsEs are useful. The first application concerns canard solutions near a multiple turning point. The second application concerns so-called non-smooth or angular canard solutions. Finally an Ackerberg-O’Malley resonance problem is solved.
Download or read book Asymptotic Methods for the Solution of Dispersive Hyberbolic Equations Classic Reprint written by Robert M. Lewis and published by Forgotten Books. This book was released on 2018-01-30 with total page 82 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from Asymptotic Methods for the Solution of Dispersive Hyberbolic Equations An important class of asymptotic methods is characterized by the fact that certain curves, often called rays, play a central role in the theory. The rays are of fundamental importance because all of the functions which make up the various terms of the expansion can be shown to satisfy ordinary differential equations along these curves. Thus, in a sense, the method is one which reduces partial differential equations to ordinary differential equations. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Download or read book Uniform Asymptotic Solutions of Discontinuous Initial Value Problems for Dispersive Hyperbolic Equations written by J. Boersma and published by . This book was released on 1967 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Asymptotic Solutions of Differential Equations and Their Applications written by Calvin Hayden Wilcox and published by . This book was released on 1964 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Asymptotic Theory of Dynamic Boundary Value Problems in Irregular Domains written by Dmitrii Korikov and published by Springer Nature. This book was released on 2021-04-01 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book considers dynamic boundary value problems in domains with singularities of two types. The first type consists of "edges" of various dimensions on the boundary; in particular, polygons, cones, lenses, polyhedra are domains of this type. Singularities of the second type are "singularly perturbed edges" such as smoothed corners and edges and small holes. A domain with singularities of such type depends on a small parameter, whereas the boundary of the limit domain (as the parameter tends to zero) has usual edges, i.e. singularities of the first type. In the transition from the limit domain to the perturbed one, the boundary near a conical point or an edge becomes smooth, isolated singular points become small cavities, and so on. In an "irregular" domain with such singularities, problems of elastodynamics, electrodynamics and some other dynamic problems are discussed. The purpose is to describe the asymptotics of solutions near singularities of the boundary. The presented results and methods have a wide range of applications in mathematical physics and engineering. The book is addressed to specialists in mathematical physics, partial differential equations, and asymptotic methods.
Download or read book An Asymptotic Solution of Linear Second Order Hyperbolic Differential Equations Classic Reprint written by Morris Kline and published by Forgotten Books. This book was released on 2017-11-18 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from An Asymptotic Solution of Linear Second-Order Hyperbolic Differential Equations Cambridge Research Laboratories or Communications on Pure and Applied Mathematics, as opposed to the system of first-order Maxwell equations. While the general principles underlying this development are the same as those previously employed. The details of proof are quite different and the resulting ordinary differential equations for the coefficients of the expansion are also quite different in form. It is hoped that the present material will be easier to apply because the ordinary differential equations to be solved are simpler. For example, each ordinary differ ential equation here is the analogue of two sets of three ordinary differential equa tions in the electromagnetic case. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Download or read book Notes on Time Decay and Scattering for Some Hyperbolic Problems written by Cathleen S. Morawetz and published by SIAM. This book was released on 1975-01-01 with total page 86 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solutions of the wave equation or Maxwell's equations in boundary value and free space problems are analyzed. Hyperbolic systems in domains going off to infinity are studied. New results on Maxwell's equations and non-star shaped reflecting bodies are included.
