Igor V. Andrianov obtained his Master (1971) and PhD (1975) degrees from the Dnepropetrovsk State University (Ukraine). He obtained the Doctor of Sciences degree in Mechanics of Solids from the Moscow State Institute of Electronics and Mathematics (Russia) in 1990. During 1974-1977, he was a Research Scientist at the Dnipropetrovs’k State University, during 1977-1990, an Associated Professor, and during 1990-1997, a Full Professor of Mathematics in the Dnepropetrovsk Civil Engineering Institute. Currently he is a Research Scientist in the RWTH Aachen University in Germany. Prof. Andrianov is the author or co-author of 12 books and over 250 papers in refereed journals and conference proceedings. He has presented papers at numerous International Conferences, and he has supervised 25 PhD Theses. His research interests are in Mechanics of Solids, Nonlinear Dynamics, and Asymptotic Methods.Jan Awrejcewicz graduated from Lodz University of Technology in 1977 (Mechanics) and from the University of Lodz in 1978 (Philosophy). He obtained his PhD (Habilitation) in 1981 (1990), and become a Full Professor in 1997. He has authored and/or co-authored 17 monographs in English; 2 textbooks; 12 edited conference proceedings; 275 journal papers; 340 conference papers; 18 chapters in books. He has served as an editor of 9 books, and as a Guest-Editor of 15 journal special issues. His research includes Nonlinear Mechanics, Mechatronics and Control, and Biomechanics. He is a recipient of the Humboldt Research Award.Vladyslav V. Danishevs'kyy obtained his Masters (1996), Ph.D. (1999) degrees, and Doctor of Sciences degree in Structural Mechanics (2008) from the Prydniprovska State Academy of Civil Engineering and Architecture, Ukraine. He is a Professor at this State Academy. He has authored 2 monographs and over 70 refereed papers. Among his awards are the Soros Post-Graduate Student’s Award (1997), Prize of the National Academy of Sciences of Ukraine for the best academic achievement among young scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO Research Fellowship (2003), NATO Reintegration Grant (2005), and institutional academic co-operation grant of the Alexander von Humboldt Foundation (2007). He conducted research at the Institute of General Mechanics in the RWTH Aachen University, Germany (2001–2002) and at the Group of Physics of Materials in the University of Rouen, France (2003–2004). His research interests are in asymptotic methods, nonlinear dynamics, and heterogeneous materials and structures.Andrey O. Ivankov, PhD is the author of more than author of more than 30 research publications. His main areas of research include ODE, PDE, Mixed BVPs and Padé approximants.

• Preface ixList of Abbreviations xiii1 Asymptotic Approaches 11.1 Asymptotic Series and Approximations 11.1.1 Asymptotic Series 11.1.2 Asymptotic Symbols and Nomenclatures 51.2 Some Nonstandard Perturbation Procedures 81.2.1 Choice of Small Parameters 81.2.2 Homotopy Perturbation Method 101.2.3 Method of Small Delta 131.2.4 Method of Large Delta 171.2.5 Application of Distributions 191.3 Summation of Asymptotic Series 211.3.1 Analysis of Power Series 211.3.2 Padé Approximants and Continued Fractions 241.4 Some Applications of PA 291.4.1 Accelerating Convergence of Iterative Processes 291.4.2 Removing Singularities and Reducing the Gibbs-Wilbraham Effect 311.4.3 Localized Solutions 321.4.4 Hermite-Padé Approximations and Bifurcation Problem 341.4.5 Estimates of Effective Characteristics of Composite Materials 341.4.6 Continualization 351.4.7 Rational Interpolation 361.4.8 Some Other Applications 371.5 Matching of Limiting Asymptotic Expansions 381.5.1 Method of Asymptotically Equivalent Functions for Inversion of Laplace Transform 381.5.2 Two-Point PA 411.5.3 Other Methods of AEFs Construction 431.5.4 Example: Schrödinger Equation 451.5.5 Example: AEFs in the Theory of Composites 461.6 Dynamical Edge Effect Method 491.6.1 Linear Vibrations of a Rod 491.6.2 Nonlinear Vibrations of a Rod 511.6.3 Nonlinear Vibrations of a Rectangular Plate 541.6.4 Matching of Asymptotic and Variational Approaches 581.6.5 On the Normal Forms of Nonlinear Vibrations of Continuous Systems 601.7 Continualization 611.7.1 Discrete and Continuum Models in Mechanics 611.7.2 Chain of Elastically Coupled Masses 621.7.3 Classical Continuum Approximation 641.7.4 “Splashes” 651.7.5 Envelope Continualization 661.7.6 Improvement Continuum Approximations 681.7.7 Forced Oscillations 691.8 Averaging and Homogenization 711.8.1 Averaging via Multiscale Method 711.8.2 Frozing in Viscoelastic Problems 741.8.3 The WKB Method 751.8.4 Method of Kuzmak-Whitham (Nonlinear WKB Method) 771.8.5 Differential Equations with Quickly Changing Coefficients 791.8.6 Differential Equation with Periodically Discontinuous Coefficients 841.8.7 Periodically Perforated Domain 881.8.8 Waves in Periodically Nonhomogenous Media 92References 952 Computational Methods for Plates and Beams with Mixed Boundary Conditions 1052.1 Introduction 1052.1.1 Computational Methods of Plates with Mixed Boundary Conditions 1052.1.2 Method of Boundary Conditions Perturbation 1072.2 Natural Vibrations of Beams and Plates 1092.2.1 Natural Vibrations of a Clamped Beam 1092.2.2 Natural Vibration of a Beam with Free Ends 1142.2.3 Natural Vibrations of a Clamped Rectangular Plate 1182.2.4 Natural Vibrations of the Orthotropic Plate with Free Edges Lying on an Elastic Foundation 1232.2.5 Natural Vibrations of the Plate with Mixed Boundary Conditions “Clamping-Simple Support” 1282.2.6 Comparison of Theoretical and Experimental Results 1332.2.7 Natural Vibrations of a Partially Clamped Plate 1352.2.8 Natural Vibrations of a Plate with Mixed Boundary Conditions “Simple Support-Moving Clamping” 1402.3 Nonlinear Vibrations of Rods, Beams and Plates 1442.3.1 Vibrations of the Rod Embedded in a Nonlinear Elastic Medium 1442.3.2 Vibrations of the Beam Lying on a Nonlinear Elastic Foundation 1532.3.3 Vibrations of the Membrane on a Nonlinear Elastic Foundation 1552.3.4 Vibrations of the Plate on a Nonlinear Elastic Foundation 1582.4 SSS of Beams and Plates 1602.4.1 SSS of Beams with Clamped Ends 1602.4.2 SSS of the Beam with Free Edges 1632.4.3 SSS of Clamped Plate 1662.4.4 SSS of a Plate with Free Edges 1702.4.5 SSS of the Plate with Mixed Boundary Conditions “Clamping–Simple Support” 1722.4.6 SSS of a Plate with Mixed Boundary Conditions “Free Edge–Moving Clamping” 1802.5 Forced Vibrations of Beams and Plates 1842.5.1 Forced Vibrations of a Clamped Beam 1842.5.2 Forced Vibrations of Beam with Free Edges 1892.5.3 Forced Vibrations of a Clamped Plate 1902.5.4 Forced Vibrations of Plates with Free Edges 1942.5.5 Forced Vibrations of Plate with Mixed Boundary Conditions “Clamping-Simple Support” 1972.5.6 Forced Vibrations of Plate with Mixed Boundary Conditions “Free Edge – Moving Clamping” 2022.6 Stability of Beams and Plates 2072.6.1 Stability of a Clamped Beam 2072.6.2 Stability of a Clamped Rectangular Plate 2092.6.3 Stability of Rectangular Plate with Mixed Boundary Conditions “Clamping-Simple Support” 2112.6.4 Comparison of Theoretical and Experimental Results 2192.7 Some Related Problems 2212.7.1 Dynamics of Nonhomogeneous Structures 2212.7.2 Method of Ishlinskii-Leibenzon 2242.7.3 Vibrations of a String Attached to a Spring-Mass-Dashpot System 2302.7.4 Vibrations of a String with Nonlinear BCs 2332.7.5 Boundary Conditions and First Order Approximation Theory 2382.8 Links between the Adomian and Homotopy Perturbation Approaches 2402.9 Conclusions 263References 264Index 269

Igor V. Andrianov obtained his Master (1971) and PhD (1975) degrees from the Dnepropetrovsk State University (Ukraine). He obtained the Doctor of Sciences degree in Mechanics of Solids from the Moscow State Institute of Electronics and Mathematics (Russia) in 1990. During 1974-1977, he was a Research Scientist at the Dnipropetrovs’k State University, during 1977-1990, an Associated Professor, and during 1990-1997, a Full Professor of Mathematics in the Dnepropetrovsk Civil Engineering Institute. Currently he is a Research Scientist in the RWTH Aachen University in Germany. Prof. Andrianov is the author or co-author of 12 books and over 250 papers in refereed journals and conference proceedings. He has presented papers at numerous International Conferences, and he has supervised 25 PhD Theses. His research interests are in Mechanics of Solids, Nonlinear Dynamics, and Asymptotic Methods.Jan Awrejcewicz graduated from Lodz University of Technology in 1977 (Mechanics) and from the University of Lodz in 1978 (Philosophy). He obtained his PhD (Habilitation) in 1981 (1990), and become a Full Professor in 1997. He has authored and/or co-authored 17 monographs in English; 2 textbooks; 12 edited conference proceedings; 275 journal papers; 340 conference papers; 18 chapters in books. He has served as an editor of 9 books, and as a Guest-Editor of 15 journal special issues. His research includes Nonlinear Mechanics, Mechatronics and Control, and Biomechanics. He is a recipient of the Humboldt Research Award.Vladyslav V. Danishevs'kyy obtained his Masters (1996), Ph.D. (1999) degrees, and Doctor of Sciences degree in Structural Mechanics (2008) from the Prydniprovska State Academy of Civil Engineering and Architecture, Ukraine. He is a Professor at this State Academy. He has authored 2 monographs and over 70 refereed papers. Among his awards are the Soros Post-Graduate Student’s Award (1997), Prize of the National Academy of Sciences of Ukraine for the best academic achievement among young scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO Research Fellowship (2003), NATO Reintegration Grant (2005), and institutional academic co-operation grant of the Alexander von Humboldt Foundation (2007). He conducted research at the Institute of General Mechanics in the RWTH Aachen University, Germany (2001–2002) and at the Group of Physics of Materials in the University of Rouen, France (2003–2004). His research interests are in asymptotic methods, nonlinear dynamics, and heterogeneous materials and structures.Andrey O. Ivankov, PhD is the author of more than author of more than 30 research publications. His main areas of research include ODE, PDE, Mixed BVPs and Padé approximants.

• Preface ixList of Abbreviations xiii1 Asymptotic Approaches 11.1 Asymptotic Series and Approximations 11.1.1 Asymptotic Series 11.1.2 Asymptotic Symbols and Nomenclatures 51.2 Some Nonstandard Perturbation Procedures 81.2.1 Choice of Small Parameters 81.2.2 Homotopy Perturbation Method 101.2.3 Method of Small Delta 131.2.4 Method of Large Delta 171.2.5 Application of Distributions 191.3 Summation of Asymptotic Series 211.3.1 Analysis of Power Series 211.3.2 Padé Approximants and Continued Fractions 241.4 Some Applications of PA 291.4.1 Accelerating Convergence of Iterative Processes 291.4.2 Removing Singularities and Reducing the Gibbs-Wilbraham Effect 311.4.3 Localized Solutions 321.4.4 Hermite-Padé Approximations and Bifurcation Problem 341.4.5 Estimates of Effective Characteristics of Composite Materials 341.4.6 Continualization 351.4.7 Rational Interpolation 361.4.8 Some Other Applications 371.5 Matching of Limiting Asymptotic Expansions 381.5.1 Method of Asymptotically Equivalent Functions for Inversion of Laplace Transform 381.5.2 Two-Point PA 411.5.3 Other Methods of AEFs Construction 431.5.4 Example: Schrödinger Equation 451.5.5 Example: AEFs in the Theory of Composites 461.6 Dynamical Edge Effect Method 491.6.1 Linear Vibrations of a Rod 491.6.2 Nonlinear Vibrations of a Rod 511.6.3 Nonlinear Vibrations of a Rectangular Plate 541.6.4 Matching of Asymptotic and Variational Approaches 581.6.5 On the Normal Forms of Nonlinear Vibrations of Continuous Systems 601.7 Continualization 611.7.1 Discrete and Continuum Models in Mechanics 611.7.2 Chain of Elastically Coupled Masses 621.7.3 Classical Continuum Approximation 641.7.4 “Splashes” 651.7.5 Envelope Continualization 661.7.6 Improvement Continuum Approximations 681.7.7 Forced Oscillations 691.8 Averaging and Homogenization 711.8.1 Averaging via Multiscale Method 711.8.2 Frozing in Viscoelastic Problems 741.8.3 The WKB Method 751.8.4 Method of Kuzmak-Whitham (Nonlinear WKB Method) 771.8.5 Differential Equations with Quickly Changing Coefficients 791.8.6 Differential Equation with Periodically Discontinuous Coefficients 841.8.7 Periodically Perforated Domain 881.8.8 Waves in Periodically Nonhomogenous Media 92References 952 Computational Methods for Plates and Beams with Mixed Boundary Conditions 1052.1 Introduction 1052.1.1 Computational Methods of Plates with Mixed Boundary Conditions 1052.1.2 Method of Boundary Conditions Perturbation 1072.2 Natural Vibrations of Beams and Plates 1092.2.1 Natural Vibrations of a Clamped Beam 1092.2.2 Natural Vibration of a Beam with Free Ends 1142.2.3 Natural Vibrations of a Clamped Rectangular Plate 1182.2.4 Natural Vibrations of the Orthotropic Plate with Free Edges Lying on an Elastic Foundation 1232.2.5 Natural Vibrations of the Plate with Mixed Boundary Conditions “Clamping-Simple Support” 1282.2.6 Comparison of Theoretical and Experimental Results 1332.2.7 Natural Vibrations of a Partially Clamped Plate 1352.2.8 Natural Vibrations of a Plate with Mixed Boundary Conditions “Simple Support-Moving Clamping” 1402.3 Nonlinear Vibrations of Rods, Beams and Plates 1442.3.1 Vibrations of the Rod Embedded in a Nonlinear Elastic Medium 1442.3.2 Vibrations of the Beam Lying on a Nonlinear Elastic Foundation 1532.3.3 Vibrations of the Membrane on a Nonlinear Elastic Foundation 1552.3.4 Vibrations of the Plate on a Nonlinear Elastic Foundation 1582.4 SSS of Beams and Plates 1602.4.1 SSS of Beams with Clamped Ends 1602.4.2 SSS of the Beam with Free Edges 1632.4.3 SSS of Clamped Plate 1662.4.4 SSS of a Plate with Free Edges 1702.4.5 SSS of the Plate with Mixed Boundary Conditions “Clamping–Simple Support” 1722.4.6 SSS of a Plate with Mixed Boundary Conditions “Free Edge–Moving Clamping” 1802.5 Forced Vibrations of Beams and Plates 1842.5.1 Forced Vibrations of a Clamped Beam 1842.5.2 Forced Vibrations of Beam with Free Edges 1892.5.3 Forced Vibrations of a Clamped Plate 1902.5.4 Forced Vibrations of Plates with Free Edges 1942.5.5 Forced Vibrations of Plate with Mixed Boundary Conditions “Clamping-Simple Support” 1972.5.6 Forced Vibrations of Plate with Mixed Boundary Conditions “Free Edge – Moving Clamping” 2022.6 Stability of Beams and Plates 2072.6.1 Stability of a Clamped Beam 2072.6.2 Stability of a Clamped Rectangular Plate 2092.6.3 Stability of Rectangular Plate with Mixed Boundary Conditions “Clamping-Simple Support” 2112.6.4 Comparison of Theoretical and Experimental Results 2192.7 Some Related Problems 2212.7.1 Dynamics of Nonhomogeneous Structures 2212.7.2 Method of Ishlinskii-Leibenzon 2242.7.3 Vibrations of a String Attached to a Spring-Mass-Dashpot System 2302.7.4 Vibrations of a String with Nonlinear BCs 2332.7.5 Boundary Conditions and First Order Approximation Theory 2382.8 Links between the Adomian and Homotopy Perturbation Approaches 2402.9 Conclusions 263References 264Index 269

Igor V. Andrianov obtained his Master (1971) and PhD (1975) degrees from the Dnepropetrovsk State University (Ukraine). He obtained the Doctor of Sciences degree in Mechanics of Solids from the Moscow State Institute of Electronics and Mathematics (Russia) in 1990. During 1974-1977, he was a Research Scientist at the Dnipropetrovs’k State University, during 1977-1990, an Associated Professor, and during 1990-1997, a Full Professor of Mathematics in the Dnepropetrovsk Civil Engineering Institute. Currently he is a Research Scientist in the RWTH Aachen University in Germany. Prof. Andrianov is the author or co-author of 12 books and over 250 papers in refereed journals and conference proceedings. He has presented papers at numerous International Conferences, and he has supervised 25 PhD Theses. His research interests are in Mechanics of Solids, Nonlinear Dynamics, and Asymptotic Methods.Jan Awrejcewicz graduated from Lodz University of Technology in 1977 (Mechanics) and from the University of Lodz in 1978 (Philosophy). He obtained his PhD (Habilitation) in 1981 (1990), and become a Full Professor in 1997. He has authored and/or co-authored 17 monographs in English; 2 textbooks; 12 edited conference proceedings; 275 journal papers; 340 conference papers; 18 chapters in books. He has served as an editor of 9 books, and as a Guest-Editor of 15 journal special issues. His research includes Nonlinear Mechanics, Mechatronics and Control, and Biomechanics. He is a recipient of the Humboldt Research Award.Vladyslav V. Danishevs'kyy obtained his Masters (1996), Ph.D. (1999) degrees, and Doctor of Sciences degree in Structural Mechanics (2008) from the Prydniprovska State Academy of Civil Engineering and Architecture, Ukraine. He is a Professor at this State Academy. He has authored 2 monographs and over 70 refereed papers. Among his awards are the Soros Post-Graduate Student’s Award (1997), Prize of the National Academy of Sciences of Ukraine for the best academic achievement among young scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO Research Fellowship (2003), NATO Reintegration Grant (2005), and institutional academic co-operation grant of the Alexander von Humboldt Foundation (2007). He conducted research at the Institute of General Mechanics in the RWTH Aachen University, Germany (2001–2002) and at the Group of Physics of Materials in the University of Rouen, France (2003–2004). His research interests are in asymptotic methods, nonlinear dynamics, and heterogeneous materials and structures.Andrey O. Ivankov, PhD is the author of more than author of more than 30 research publications. His main areas of research include ODE, PDE, Mixed BVPs and Padé approximants.

• Preface ixList of Abbreviations xiii1 Asymptotic Approaches 11.1 Asymptotic Series and Approximations 11.1.1 Asymptotic Series 11.1.2 Asymptotic Symbols and Nomenclatures 51.2 Some Nonstandard Perturbation Procedures 81.2.1 Choice of Small Parameters 81.2.2 Homotopy Perturbation Method 101.2.3 Method of Small Delta 131.2.4 Method of Large Delta 171.2.5 Application of Distributions 191.3 Summation of Asymptotic Series 211.3.1 Analysis of Power Series 211.3.2 Padé Approximants and Continued Fractions 241.4 Some Applications of PA 291.4.1 Accelerating Convergence of Iterative Processes 291.4.2 Removing Singularities and Reducing the Gibbs-Wilbraham Effect 311.4.3 Localized Solutions 321.4.4 Hermite-Padé Approximations and Bifurcation Problem 341.4.5 Estimates of Effective Characteristics of Composite Materials 341.4.6 Continualization 351.4.7 Rational Interpolation 361.4.8 Some Other Applications 371.5 Matching of Limiting Asymptotic Expansions 381.5.1 Method of Asymptotically Equivalent Functions for Inversion of Laplace Transform 381.5.2 Two-Point PA 411.5.3 Other Methods of AEFs Construction 431.5.4 Example: Schrödinger Equation 451.5.5 Example: AEFs in the Theory of Composites 461.6 Dynamical Edge Effect Method 491.6.1 Linear Vibrations of a Rod 491.6.2 Nonlinear Vibrations of a Rod 511.6.3 Nonlinear Vibrations of a Rectangular Plate 541.6.4 Matching of Asymptotic and Variational Approaches 581.6.5 On the Normal Forms of Nonlinear Vibrations of Continuous Systems 601.7 Continualization 611.7.1 Discrete and Continuum Models in Mechanics 611.7.2 Chain of Elastically Coupled Masses 621.7.3 Classical Continuum Approximation 641.7.4 “Splashes” 651.7.5 Envelope Continualization 661.7.6 Improvement Continuum Approximations 681.7.7 Forced Oscillations 691.8 Averaging and Homogenization 711.8.1 Averaging via Multiscale Method 711.8.2 Frozing in Viscoelastic Problems 741.8.3 The WKB Method 751.8.4 Method of Kuzmak-Whitham (Nonlinear WKB Method) 771.8.5 Differential Equations with Quickly Changing Coefficients 791.8.6 Differential Equation with Periodically Discontinuous Coefficients 841.8.7 Periodically Perforated Domain 881.8.8 Waves in Periodically Nonhomogenous Media 92References 952 Computational Methods for Plates and Beams with Mixed Boundary Conditions 1052.1 Introduction 1052.1.1 Computational Methods of Plates with Mixed Boundary Conditions 1052.1.2 Method of Boundary Conditions Perturbation 1072.2 Natural Vibrations of Beams and Plates 1092.2.1 Natural Vibrations of a Clamped Beam 1092.2.2 Natural Vibration of a Beam with Free Ends 1142.2.3 Natural Vibrations of a Clamped Rectangular Plate 1182.2.4 Natural Vibrations of the Orthotropic Plate with Free Edges Lying on an Elastic Foundation 1232.2.5 Natural Vibrations of the Plate with Mixed Boundary Conditions “Clamping-Simple Support” 1282.2.6 Comparison of Theoretical and Experimental Results 1332.2.7 Natural Vibrations of a Partially Clamped Plate 1352.2.8 Natural Vibrations of a Plate with Mixed Boundary Conditions “Simple Support-Moving Clamping” 1402.3 Nonlinear Vibrations of Rods, Beams and Plates 1442.3.1 Vibrations of the Rod Embedded in a Nonlinear Elastic Medium 1442.3.2 Vibrations of the Beam Lying on a Nonlinear Elastic Foundation 1532.3.3 Vibrations of the Membrane on a Nonlinear Elastic Foundation 1552.3.4 Vibrations of the Plate on a Nonlinear Elastic Foundation 1582.4 SSS of Beams and Plates 1602.4.1 SSS of Beams with Clamped Ends 1602.4.2 SSS of the Beam with Free Edges 1632.4.3 SSS of Clamped Plate 1662.4.4 SSS of a Plate with Free Edges 1702.4.5 SSS of the Plate with Mixed Boundary Conditions “Clamping–Simple Support” 1722.4.6 SSS of a Plate with Mixed Boundary Conditions “Free Edge–Moving Clamping” 1802.5 Forced Vibrations of Beams and Plates 1842.5.1 Forced Vibrations of a Clamped Beam 1842.5.2 Forced Vibrations of Beam with Free Edges 1892.5.3 Forced Vibrations of a Clamped Plate 1902.5.4 Forced Vibrations of Plates with Free Edges 1942.5.5 Forced Vibrations of Plate with Mixed Boundary Conditions “Clamping-Simple Support” 1972.5.6 Forced Vibrations of Plate with Mixed Boundary Conditions “Free Edge – Moving Clamping” 2022.6 Stability of Beams and Plates 2072.6.1 Stability of a Clamped Beam 2072.6.2 Stability of a Clamped Rectangular Plate 2092.6.3 Stability of Rectangular Plate with Mixed Boundary Conditions “Clamping-Simple Support” 2112.6.4 Comparison of Theoretical and Experimental Results 2192.7 Some Related Problems 2212.7.1 Dynamics of Nonhomogeneous Structures 2212.7.2 Method of Ishlinskii-Leibenzon 2242.7.3 Vibrations of a String Attached to a Spring-Mass-Dashpot System 2302.7.4 Vibrations of a String with Nonlinear BCs 2332.7.5 Boundary Conditions and First Order Approximation Theory 2382.8 Links between the Adomian and Homotopy Perturbation Approaches 2402.9 Conclusions 263References 264Index 269