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, Pade 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.
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 PadE Approximations Determination and Characteristics of Pade Approximations Application of Pade 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 Poincare Series Examples of Asymptotical Solutions Swinging Oscillator Normal Form Normal Form Integral References Bibliography