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1991 | Book

Matrix Perturbation Theory Read chapter Read first chapter

Algebraic Methods in Nonlinear Perturbation Theory

Authors: V. N. Bogaevski, A. Povzner

Publisher: Springer New York

Book Series : Applied Mathematical Sciences

Included in: Professional Book Archive

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About this book

Many books have already been written about the perturbation theory of differential equations with a small parameter. Therefore, we would like to give some reasons why the reader should bother with still another book on this topic. Speaking for the present only about ordinary differential equations and their applications, we notice that methods of solutions are so numerous and diverse that this part of applied mathematics appears as an aggregate of poorly connected methods. The majority of these methods require some previous guessing of a structure of the desired asymptotics. The Poincare method of normal forms and the Bogolyubov-Krylov­ Mitropolsky averaging methods, well known in the literature, should be mentioned specifically in connection with what will follow. These methods do not assume an immediate search for solutions in some special form, but make use of changes of variables close to the identity transformation which bring the initial system to a certain normal form. Applicability of these methods is restricted by special forms of the initial systems.

Table of Contents

Frontmatter
1. Matrix Perturbation Theory
Abstract
$$X = {X_0} + \varepsilon {X_1} + {\varepsilon ^2}{X_2} + \cdots $$
(1.1.1)
which acts in the n-dimensional (complex) vector space R.
V. N. Bogaevski, A. Povzner
2. Systems of Ordinary Differential Equations with a Small Parameter
Abstract
In this chapter we construct an analogue of the matrix perturbation theory for systems of the form
$${\varepsilon ^\alpha }\frac{{d{x_i}}}{{dt}} = {a_{0i}}\left( x \right) + \varepsilon {a_{1i}}\left( x \right) + {\varepsilon ^2}{a_{2i}}\left( x \right) + \cdots = {f_i}\left( {x;\varepsilon } \right),$$
where \(x = \left( {{x_1}, \ldots ,{x_n}} \right),i = 1, \ldots ,\) n, or, in vector notation,
$${\varepsilon ^\alpha }\frac{{dx}}{{dt}} = {a_0}\left( x \right) + \varepsilon {a_1}\left( x \right) + {\varepsilon ^2}{a_2}\left( x \right) + \cdots = f\left( {x;\varepsilon } \right).$$
.
V. N. Bogaevski, A. Povzner
3. Examples
Abstract
We will begin with the now classical example of motion of the pendulum of variable length. This examples illustrates the simplest and at the same time the most essential methods of computation.
V. N. Bogaevski, A. Povzner
4. Reconstruction
Abstract
In various problems we must employ variable transformations degenerate at e ε = 0. One such example is the case of a nilpotent X0 considered in Sections 1.5 and 2.5, where a special (shearing) transformation reconstructs X so that another operator, different from X0, becomes the leading one.
V. N. Bogaevski, A. Povzner
5. Equations in Partial Derivatives
Abstract
Below we generalize the above formalism to make it applicable to equations in partial derivatives.
V. N. Bogaevski, A. Povzner
Backmatter
Metadata
Title
Algebraic Methods in Nonlinear Perturbation Theory
Authors
V. N. Bogaevski
A. Povzner
Copyright Year
1991
Publisher
Springer New York
Electronic ISBN
978-1-4612-4438-7
Print ISBN
978-1-4612-8770-4
DOI
https://doi.org/10.1007/978-1-4612-4438-7