Cover -- Title Page -- Copyright -- Contents -- Preface -- List of Abbreviations -- Chapter 1 Asymptotic Approaches -- 1.1 Asymptotic Series and Approximations -- 1.1.1 Asymptotic Series -- 1.1.2 Asymptotic Symbols and Nomenclatures -- 1.2 Some Nonstandard Perturbation Procedures -- 1.2.1 Choice of Small Parameters -- 1.2.2 Homotopy Perturbation Method -- 1.2.3 Method of Small Delta -- 1.2.4 Method of Large Delta -- 1.2.5 Application of Distributions -- 1.3 Summation of Asymptotic Series -- 1.3.1 Analysis of Power Series -- 1.3.2 Padé Approximants and Continued Fractions -- 1.4 Some Applications of PA -- 1.4.1 Accelerating Convergence of Iterative Processes -- 1.4.2 Removing Singularities and Reducing the Gibbs-Wilbraham Effect -- 1.4.3 Localized Solutions -- 1.4.4 Hermite-Padé Approximations and Bifurcation Problem -- 1.4.5 Estimates of Effective Characteristics of Composite Materials -- 1.4.6 Continualization -- 1.4.7 Rational Interpolation -- 1.4.8 Some Other Applications -- 1.5 Matching of Limiting Asymptotic Expansions -- 1.5.1 Method of Asymptotically Equivalent Functions for Inversion of Laplace Transform -- 1.5.2 Two-Point PA -- 1.5.3 Other Methods of AEFs Construction -- 1.5.4 Example: Schrödinger Equation -- 1.5.5 Example: AEFs in the Theory of Composites -- 1.6 Dynamical Edge Effect Method -- 1.6.1 Linear Vibrations of a Rod -- 1.6.2 Nonlinear Vibrations of a Rod -- 1.6.3 Nonlinear Vibrations of a Rectangular Plate -- 1.6.4 Matching of Asymptotic and Variational Approaches -- 1.6.5 On the Normal Forms of Nonlinear Vibrations of Continuous Systems -- 1.7 Continualization -- 1.7.1 Discrete and Continuum Models in Mechanics -- 1.7.2 Chain of Elastically Coupled Masses -- 1.7.3 Classical Continuum Approximation -- 1.7.4 "Splashes'' -- 1.7.5 Envelope Continualization.

1.7.6 Improvement Continuum Approximations -- 1.7.7 Forced Oscillations -- 1.8 Averaging and Homogenization -- 1.8.1 Averaging via Multiscale Method -- 1.8.2 Frozing in Viscoelastic Problems -- 1.8.3 The WKB Method -- 1.8.4 Method of Kuzmak-Whitham (Nonlinear WKB Method) -- 1.8.5 Differential Equations with Quickly Changing Coefficients -- 1.8.6 Differential Equation with Periodically Discontinuous Coefficients -- 1.8.7 Periodically Perforated Domain -- 1.8.8 Waves in Periodically Nonhomogenous Media -- References -- Chapter 2 Computational Methods for Plates and Beams with Mixed Boundary Conditions -- 2.1 Introduction -- 2.1.1 Computational Methods of Plates with Mixed Boundary Conditions -- 2.1.2 Method of Boundary Conditions Perturbation -- 2.2 Natural Vibrations of Beams and Plates -- 2.2.1 Natural Vibrations of a Clamped Beam -- 2.2.2 Natural Vibration of a Beam with Free Ends -- 2.2.3 Natural Vibrations of a Clamped Rectangular Plate -- 2.2.4 Natural Vibrations of the Orthotropic Plate with Free Edges Lying on an Elastic Foundation -- 2.2.5 Natural Vibrations of the Plate with Mixed Boundary Conditions "Clamping-Simple Support'' -- 2.2.6 Comparison of Theoretical and Experimental Results -- 2.2.7 Natural Vibrations of a Partially Clamped Plate -- 2.2.8 Natural Vibrations of a Plate with Mixed Boundary Conditions "Simple Support-Moving Clamping'' -- 2.3 Nonlinear Vibrations of Rods, Beams and Plates -- 2.3.1 Vibrations of the Rod Embedded in a Nonlinear Elastic Medium -- 2.3.2 Vibrations of the Beam Lying on a Nonlinear Elastic Foundation -- 2.3.3 Vibrations of the Membrane on a Nonlinear Elastic Foundation -- 2.3.4 Vibrations of the Plate on a Nonlinear Elastic Foundation -- 2.4 SSS of Beams and Plates -- 2.4.1 SSS of Beams with Clamped Ends -- 2.4.2 SSS of the Beam with Free Edges -- 2.4.3 SSS of Clamped Plate.

2.4.4 SSS of a Plate with Free Edges -- 2.4.5 SSS of the Plate with Mixed Boundary Conditions "Clamping-Simple Support'' -- 2.4.6 SSS of a Plate with Mixed Boundary Conditions "Free Edge-Moving Clamping'' -- 2.5 Forced Vibrations of Beams and Plates -- 2.5.1 Forced Vibrations of a Clamped Beam -- 2.5.2 Forced Vibrations of Beam with Free Edges -- 2.5.3 Forced Vibrations of a Clamped Plate -- 2.5.4 Forced Vibrations of Plates with Free Edges -- 2.5.5 Forced Vibrations of Plate with Mixed Boundary Conditions "Clamping-Simple Support'' -- 2.5.6 Forced Vibrations of Plate with Mixed Boundary Conditions "Free Edge-Moving Clamping'' -- 2.6 Stability of Beams and Plates -- 2.6.1 Stability of a Clamped Beam -- 2.6.2 Stability of a Clamped Rectangular Plate -- 2.6.3 Stability of Rectangular Plate with Mixed Boundary Conditions "Clamping-Simple Support'' -- 2.6.4 Comparison of Theoretical and Experimental Results -- 2.7 Some Related Problems -- 2.7.1 Dynamics of Nonhomogeneous Structures -- 2.7.2 Method of Ishlinskii-Leibenzon -- 2.7.3 Vibrations of a String Attached to a Spring-Mass-Dashpot System -- 2.7.4 Vibrations of a String with Nonlinear BCs -- 2.7.5 Boundary Conditions and First Order Approximation Theory -- 2.8 Links between the Adomian and Homotopy Perturbation Approaches -- 2.9 Conclusions -- References -- Index.