Perturbation Methods for Differential Equations

1 Asymptotic Series and Expansions.- 1.1 Introduction.- 1.2 Taylor Series Expansions.- 1.3 Gauge Functions.- 1.4 Asymptotic Series and Expansions.- 1.5 Asymptotic Solutions of Differential Equations.- 1.6 Exercises.- 2 Regular Perturbation Methods.- 2.1 Introduction.- 2.2 Algebraic Equations.- 2.3 Ordinary Differential Equations.- 2.4 Partial Differential Equations.- 2.5 Applications to Fluid Dynamics: Decay of a Line Vortex.- 2.6 Exercises.- 2.7 Appendix. Review of Partial Differential Equations.- 3 The Method of Strained Coordinates/Parameters.- 3.1 Introduction.- 3.2 Poincaré-Lindstedt-Lighthill Method of Perturbed Eigenvalues.- 3.3 Eigenfunction Expansion Method.- 3.4 Lighthill’s Method of Shifting Singularities.- 3.5 Pritulo’s Method of Renormalization.- 3.6 Wave Propagation in an Inhomogeneous Medium.- 3.7 Applications to Solid Mechanics: Nonlinear Buckling of Elastic Columns.- 3.8 Applications to Fluid Dynamics.- 3.9 Applications to Plasma Physics.- 3.10 Limitations of the Method of Strained Parameters.- 3.11 Exercises.- 3.12 Appendix 1. Fredholm’s Alternative Theorem.- 3.13 Appendix 2. Floquet Theory.- 3.14 Appendix 3. Bifurcation Theory.- 4 Method of Averaging.- 4.1 Introduction.- 4.2 Krylov-Bogoliubov Method of Averaging.- 4.3 Krylov-Bogoliubov-Mitropolski Generalized Method of Averaging.- 4.4 Whitham’s Averaged Lagrangian Method.- 4.5 Hamiltonian Perturbation Method.- 4.6 Applications to Fluid Dynamics: Nonlinear Evolution of Modulated Gravity Wave Packet on the Surface of a Fluid.- 4.7 Exercises.- 4.8 Appendix 1. Review of Calculus of Variations.- 4.9 Appendix 2. Hamilton-Jacobi Theory.- 5 The Method of Matched Asymptotic Expansions.- 5.1 Introduction.- 5.2 Physical Motivation.- 5.3 The Inner and Outer Expansions.- 5.4 Hyperbolic Equations.- 5.5Elliptic Equations.- 5.6 Parabolic Equations.- 5.7 Interior Layers.- 5.8 Latta’s Method of Composite Expansions.- 5.9 Turning Point Problems.- 5.10 Applications to Fluid Dynamics: Boundary-Layer Flow Past a Flat Plate.- 5.11 Exercises.- 5.12 Appendix 1. Initial-Value Problem for Partial Differential Equations.- 5.13 Appendix 2. Review of Nonlinear Hyperbolic Equations.- 6 Method of Multiple Scales.- 6.1 Introduction.- 6.2 Differential Equations with Constant Coefficients.- 6.3 Struble’s Method.- 6.4 Differential Equations with Slowly Varying Coefficients.- 6.5 Generalized Multiple-Scale Method.- 6.6 Applications to Solid Mechanics: Dynamic Buckling of a Thin Elastic Plate.- 6.7 Applications to Fluid Dynamics.- 6.8 Applications to Plasma Physics.- 6.9 Exercises.- 7 Miscellaneous Perturbation Methods.- 7.1 A Quantum-Field-Theoretic Perturbative Procedure.- 7.2 A Perturbation Method for Linear Stochastic Differential Equations.- 7.3 Exercises.