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 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 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 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 2016-08-30 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from Asymptotic Methods for the Solution of Dispersive Hyberbolic Equations In recent years asymptotic methods have been developed for the solution of certain boundarynvalue problems and initial-value problems for partial differential equations. These problems involve a parameter, and the methods provide one or more terms of the asymptotic expansion, say for large values of the parameter, of the solution of the problem. They are often applicable to problems for which exact solutions are not known, and even for problems which can be solved exactly it frequently happens that only the asymptotic expansion of the solution is sufficiently simple to be useful in practical applications. Furthermore, it is invariably true that the methods which yield the asymptotic expansion directly are very much simpler than the pro cedure which involves first finding the exact solution, and then its asymp totic expansion. 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 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 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 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 Asymptotic Solution of Systems of Linear Ordinary Differential Equations With Discontinuous Coefficients Classic Reprint written by Clive R. Chester and published by . This book was released on 2016-06-26 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book On Systems of Linear Ordinary Differential Equations Classic Reprint written by Herbert B. Keller and published by Forgotten Books. This book was released on 2018-05-04 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from On Systems of Linear Ordinary Differential Equations The nearest approach to such a treatment is contained in the wkb method which yields an approximate solution of a linear second order equation as a sum of exponentials with variable coefficients and exponents. In some presentations of the method higher order approximations are also obtained, but it is generally not clear whether the successive approximations converge to a solution of the equation or not. For a large class of cond order equations involving a parameter in a particular way, Birkhoff and L ave settled this matter by showing that the web approximation is the leading term in an asymptotic expansion of the solution with respect to the parameter. 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 Dispersive and Strichatz estimates for hyperbolic equations with constant coefficients written by Michael Ruzhansky and published by Mathematical Society Of Japan Memoirs. This book was released on 2010 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this work dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients with lower order terms are considered. The global time decay estimates of Lp-Lq norms of propagators are analysed in detail and it is described how the time decay rates depend on the geometry of the problem. For these purposes, the frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are presented. Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

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 Methods for Investigating Quasiwave Equations of Hyperbolic Type written by Yuri A. Mitropolsky and published by Springer Science & Business Media. This book was released on 1997-04-30 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have been studied in classical works of Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic equations were considered by Vishik, Ladyzhenskaya, and that for general two dimensional equations were investigated by Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last decade the theory of solvability on the whole of boundary value problems for nonlinear differential equations has received intensive development. Significant results for nonlinear elliptic and parabolic equations of second order were obtained in works of Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others. Concerning the solvability in general of nonlinear hyperbolic equations, which are connected to the theory of local and nonlocal boundary value problems for hyperbolic equations, there are only partial results obtained by Bronshtein, Pokhozhev, Nakhushev.

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 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 Solution of a Class of Second Order Differential Equations Containing a Parameter Classic Reprint written by Gilbert Stengle and published by Forgotten Books. This book was released on 2017-11-06 with total page 92 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from Asymptotic Solution of a Class of Second Order Differential Equations Containing a Parameter Given a Ca) function b(t, p) leti$ denote the formal power series of at t F): 0. Since every formal power series is the power series of some CC0 function we also use circumflexed symbols to denote abstract formal power series or the formal product of such a series and a Ca) function. Since am O. 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 Exact and Asymptotic Solutions of the Cauchy Problem Classic Reprint written by Donald Ludwig and published by Forgotten Books. This book was released on 2017-12-24 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: Excerpt from Exact and Asymptotic Solutions of the Cauchy Problem Here we neither require nor expect that the series shall converge, but only that v shall be an approximate solution for sufficiently large values of a). In other words. 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 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: