Asymptotic and Perturbation Methods

Table of Contents

• 0.1 About this book
• 0.2. What one needs to know?

Chapter 1. Introduction

• 1.1. Introduction
• 1.2. Asymptotic series etc

Chapter 2. Asymptotic expansion of integrals

• 2.1. Laplace integrals
• 2.2. Laplace integrals. II. Multidimensional theory
• 2.3. Oscillatory integrals
• 2.4. Oscillatory integrals. II. Multidimensional theory
• 2.5. Method of the steepest descent
• 2.P. Problems

Chapter 3. Asymptotic Solutions of Linear ODEs

• 3.1. Introduction and classification
• 3.2. Ordinary points of differential equation
• 3.3. Regular singular points of differential equation
• 3.4. Irregular singular points of differential equation
• 3.5. Some examples of nonlinear differential equations
• 3.P. Problems

Chapter 4. Perturbation methods for ODEs and PDEs

• 4.1. Regular perturbation theory
• 4.A. Perturbation theory for celestial mechanics
• 4.B. Perturbation theory for celestial mechanics. 2
• 4.2. Singular perturbation theory
• 4.3. Singular perturbation theory. II
• 4.4. Singular perturbation theory for PDEs
• 4.P. Problems

Chapter 5. Semiclassical and high frequency asymptotics

• 5.1. Preliminaries
• 5.2. Eikonal and Hamilton-Jacobi equations
• 5.3. Transport equations
• 5.4. Initial and initial-boundary value problems
• 5.5. Elasticity system
• 5.A. Elasticity system. Derivation
• 5.B. Maxwell system
• 5.P. Problems

Chapter 6. WKB in dimension $1$

• 6.1. Preliminaries
• 6.2. Global theory
• 6.3. Bohr-Sommerfeld formula
• 6.P. Problems

Chapter 7. WKB in dimension $d>1$

• 7.1. Preliminaries
• 7.2. Global theory
• 7.3. Geometry of rays and caustics
• 7.P. Problems

Chapter 8. Multiple-scale analysis

• 8.1. Secular terms
• 8.2. Derivative expansion
• 8.3. Two-variable expansion
• 8.4. Rayleigh oscillator
• 8.P. Problems

Chapter 9. Burgers Equation

• 9.1. Burgers equation. 1
• 9.2. Burgers equation. Initial data already has a jump
• 9.3. Burgers equation. Shock formation

Appendix A. Perturbation of eigenvalues and eigenvectors of matrices

• A.1. Roots of the polynomials
• A.2. Eigenvalues and eigenvectors

Extra textbooks may be useful, especially because they contain extensive bibliography

• Asymptotic Methods, Henry van Roessel and John C. Bowman, University of Alberta, (2012).
• Advanced Mathematical Methods for Scientists and Engineers; Asymptotic Methods and Perturbation Theory, by Carl M. Bender and Steven A. Orszag, 1st edition, McGraw-Hill, 1978, 2nd ed., Springer 1999.
• Applied Asymptotic Analysis, by Peter Miller, AMS, Grad. Studies in Math, Vol 75.