Nonlinear 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).
Homotopy Analysis Method in Nonlinear Differential Equations
Author: Shijun Liao
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-06-22
"Homotopy Analysis Method in Nonlinear Differential Equations" presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM has nothing to do with small/large physical parameters. In addition, it provides great freedom to choose the equation-type of linear sub-problems and the base functions of a solution. Above all, it provides a convenient way to guarantee the convergence of a solution. This book consists of three parts. Part I provides its basic ideas and theoretical development. Part II presents the HAM-based Mathematica package BVPh 1.0 for nonlinear boundary-value problems and its applications. Part III shows the validity of the HAM for nonlinear PDEs, such as the American put option and resonance criterion of nonlinear travelling waves. New solutions to a number of nonlinear problems are presented, illustrating the originality of the HAM. Mathematica codes are freely available online to make it easy for readers to understand and use the HAM. This book is suitable for researchers and postgraduates in applied mathematics, physics, nonlinear mechanics, finance and engineering. Dr. Shijun Liao, a distinguished professor of Shanghai Jiao Tong University, is a pioneer of the HAM.
Operator Theory in Krein Spaces and Nonlinear Eigenvalue Problems
Author: Karl-Heinz Förster
language: en
Publisher: Springer Science & Business Media
Release Date: 2006-03-16
This volume contains a collection of recent original research papers in operator theory in Krein spaces, on generalized Nevanlinna functions, which are closely connected with this theory, and on nonlinear eigenvalue problems. Key topics include: spectral theory for normal operators; perturbation theory for self-adjoint operators in Krein spaces; and, models for generalized Nevanlinna functions.