Eigenvalue Problems On Infinite Intervals
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Nonlinear Eigenvalue Problems on Infinite Intervals
This paper is concerned with nonlinear eigenvalue problems of boundary value problems for ordinary differential equation posed on an infinite interval. It is shown that under certain analyticity assumptions - a domain in the complex plain can be identified, in which all eigenvalues are isolated. An intriguing way to solve such problems is to cut the infinite interval at a finite but large enough point and to impose additional, so called asymptotic boundary conditions at this far end. The obtained eigenvalue problem for the two point boundary value problem on this finite but large interval can be solved by an appropriate code. In this paper suitable asymptotic boundary conditions are devised and the order of convergence, as the length of the interval, on which these approximating problems are posed, converges to infinity, is investigated. Exponential convergence is shown for well posed approximating problems. (Author).
Eigenvalue Problems of Infinite Intervals
This paper is concerned with eigenvalue problems for boundary value problems of ordinary differential equations posed on an infinite interval. Problems of that kind occur for example in fluid mechanics when the stability of laminar flows is investigated. Characterizations of eigenvalues and spectral subspaces are given and the convergence of approximating problems which are derived by reducing the infinite interval to a finite but large one and by imposing additional boundary conditions at the far end is proved. Exponential convergence is shown for a large class of problems. (Author).
Spectral Methods for Non-Standard Eigenvalue Problems
Author: Călin-Ioan Gheorghiu
language: en
Publisher: Springer Science & Business
Release Date: 2014-04-22
This book focuses on the constructive and practical aspects of spectral methods. It rigorously examines the most important qualities as well as drawbacks of spectral methods in the context of numerical methods devoted to solve non-standard eigenvalue problems. In addition, the book also considers some nonlinear singularly perturbed boundary value problems along with eigenproblems obtained by their linearization around constant solutions. The book is mathematical, poising problems in their proper function spaces, but its emphasis is on algorithms and practical difficulties. The range of applications is quite large. High order eigenvalue problems are frequently beset with numerical ill conditioning problems. The book describes a wide variety of successful modifications to standard algorithms that greatly mitigate these problems. In addition, the book makes heavy use of the concept of pseudospectrum, which is highly relevant to understanding when disaster is imminent in solving eigenvalue problems. It also envisions two classes of applications, the stability of some elastic structures and the hydrodynamic stability of some parallel shear flows. This book is an ideal reference text for professionals (researchers) in applied mathematics, computational physics and engineering. It will be very useful to numerically sophisticated engineers, physicists and chemists. The book can also be used as a textbook in review courses such as numerical analysis, computational methods in various engineering branches or physics and computational methods in analysis.