Gao / Krysko Introduction to Asymptotic Methods

Erscheinungsjahr 2010
ISBN: 978-1-4200-1173-9
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 272 Seiten

Reihe: Modern Mechanics and Mathematics

Among the theoretical methods for solving many problems of applied mathematics, physics, and technology, asymptotic methods often provide results that lead to obtaining more effective algorithms of numerical evaluation. Presenting the mathematical methods of perturbation theory, Introduction to Asymptotic Methods reviews the most important methods of singular perturbations within the scope of application of differential equations. The authors take a challenging and original approach based on the integrated mathematical-analytical treatment of various objects taken from interdisciplinary fields of mechanics, physics, and applied mathematics. This new hybrid approach will lead to results that cannot be obtained by standard theories in the field. Emphasizing fundamental elements of the mathematical modeling process, the book provides comprehensive coverage of asymptotic approaches, regular and singular perturbations, one-dimensional non-stationary non-linear waves, Padé approximations, oscillators with negative Duffing type stiffness, and differential equations with discontinuous nonlinearities. The book also offers a method of construction for canonical variables transformation in parametric form along with a number of examples and applications. The book is applications oriented and features results and literature citations that have not been seen in the Western Scientific Community. The authors emphasize the dynamics of the development of perturbation methods and present the development of ideas associated with this wide field of research.

Zielgruppe

Graduate students and researchers in mathematics, applied mathematics, and engineering.

Autoren/Hrsg.

Gao, David Y.

Krysko, Vadim A.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis

Inhaltsverzeichnis

Introduction Elements of Mathematical Modeling
Structure of a Mathematical Model
Examples of Reducing Problems to a Dimensionless Form
Mathematical Model Adequacy and Properties. Regular and Singular Perturbations
Expansion of Functions and Mathematical Methods
Expansions of Elementary Functions into Power Series
Mathematical Methods of Perturbations
Exercises Regular and Singular Perturbations
Introduction. Asymptotic Approximations with Respect to Parameter
Non-Uniformities of a Classical Perturbation Approach
Method of “Elongated” Parameters
Method of Deformed Variables
Method of Scaling and Full Approximation
Multiple Scale Methods
Variations of Arbitrary Constants
Averaging Methods
Matching Asymptotic Decompositions
On the Sources of Non-Uniformities
On the Influence of Initial Conditions
Analysis of Strongly Nonlinear Dynamical Problems
A Few Perturbation Parameters
Exercises Wave-Impact Processes
Definition of a Cylinder-Piston Wave
One-Dimensional Non-Stationary Non-Linear Waves PadÉ Approximations
Determination and Characteristics of Padé Approximations
Application of Padé Approximations
Exercises Averaging of Ribbed Plates
Averaging in the Theory of Ribbed Plates
Kantorovich-Vlasov –Type Methods
Transverse Vibrations of Rectangular Plates
Deflections of Rectangular Plates Chaos Foresight
The Analysed System
Melnikov-Gruendler’s Approach
Melnikov-Gruendler Function
Numerical Results
Continuous Approximation of Discontinuous Systems
An Illustrative Example
Higher Dimensional Systems
Nonlinear Dynamics of a Swinging Oscillator
Parametrical Form of Canonical Transformations
Function Derivative
Invariant Normalization of Hamiltonians
Algorithm of Invariant Normalization for Asymptotical Determination of the Poincaré Series
Examples of Asymptotical Solutions
Swinging Oscillator
Normal Form
Normal Form Integral
References
Bibliography