Ebook: Algebraic Methods in Nonlinear Perturbation Theory

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.

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This book will be of interest for everybody working on perturbation theory in differential equations. The book requires only a standard mathematical background for engineers and does not require reference to the special literature. Topics which are covered include: matrix perturbation theory; systems of ordinary differential equations with small parameter; reconstruction and equations in partial derivatives. Boundary problems are not discussed in this volume. The reader will find many examples throughout the book.