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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.



1. Matrix Perturbation Theory

$$X = {X_0} + \varepsilon {X_1} + {\varepsilon ^2}{X_2} + \cdots $$
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

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

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

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

Below we generalize the above formalism to make it applicable to equations in partial derivatives.
V. N. Bogaevski, A. Povzner


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