Introduction to Perturbation Methods

1: Introduction to Asymptotic Approximations.- 1.1 Introduction.- 1.2 Taylor's Theorem and l'Hospital's Rule.- 1.3 Order Symbols.- 1.4 Asymptotic Approximations.- 1.4.1 Asymptotic Expansions.- 1.4.2 Accuracy versus Convergence of an Asymptotic Series.- 1.4.3 Manipulating Asymptotic Expansions.- 1.5 Asymptotic Solution of Algebraic and Transcendental Equations.- 1.6 Introduction to the Asymptotic Solution of Differential Equations.- 1.7 Uniformity.- 1.8 Symbolic Computing.- 2: Matched Asymptotic Expansions.- 2.1 Introduction.- 2.2 Introductory Example.- 2.3 Examples with Multiple Boundary Layers.- 2.4 Interior Layers.- 2.5 Corner Layers.- 2.6 Partial Differential Equations.- 2.7 Difference Equations.- 3: Multiple Scales.- 3.1 Introduction.- 3.2 Introductory Example.- 3.3 Slowly Varying Coefficients.- 3.4 Forced Motion Near Resonance.- 3.5 Boundary Layers.- 3.6 Introduction to Partial Differential Equations.- 3.7 Linear Wave Propagation.- 3.8 Nonlinear Waves.- 3.9 Difference Equations.- 4: The WKB and Related Methods.- 4.1 Introduction.- 4.2 Introductory Example.- 4.3 Turning Points.- 4.4 Wave Propagation and Energy Methods.- 4.5 Wave Propagation and Slender Body Approximations.- 4.6 Ray Methods.- 4.7 Parabolic Approximations.- 4.8 Discrete WKB Method.- 5: The Method of Homogenization.- 5.1 Introduction.- 5.2 Introductory Example.- 5.3 Multidimensional Problem: Periodic Substructure.- 5.4 Porous Flow.- 6: Introduction to Bifurcation and Stability.- 6.1 Introduction.- 6.2 Introductory Example.- 6.3 Analysis of a Bifurcation Point.- 6.4 Linearized Stability.- 6.5 Relaxation Dynamics.- 6.6 An Example Involving a Nonlinear Partial Differential Equation.- 6.7 Bifurcation of Periodic Solutions.- 6.8 Systems of Ordinary Differential Equations.- Appendix AI: Solution and Properties of Transition Layer Equations.- A1.1 Airy Functions.- A1.2 Confluent Hypergeometric Functions.- A1.3 Higher-Order Turning Points.- Appendix A2: Asymptotic Approximations of Integrals.- A2.1 Introduction.- A2.2 Watson's Lemma.- A2.3 Laplace's Approximation.- A2.4 Stationary Phase Approximation.- Appendix A3: Numerical Solution of Nonlinear Boundary-Value Problems.- A3.1 Introduction.- A3.2 Examples.- A3.3 Computer Code.- References.