See SimpleMind mindmap and notes and problem sets in LectureNotes

Notes from tablet

See also lectures on YB from Bender (at PI)

Perturbation methods explore the existence of a small or large parameter to derive systematically a precise approximation. More art than science, building experience is valuable.

There are two methods for obtaining precise approximations: numerical methods and analytical (asymptotic) methods. These are not in competition but complemen teach other. Perturbation methods work when some parameter is large or small. Numerical methods work best when all parameters are order one. Agreement between the two methods is reassuring when doing research. Perturbation methods often give more physical insight.

Course materials (new link), notes

See reading list there

Mathematical foundation: Asymptotic approximation

Applications

Perturbation methods for algebraic equations

Asymptotic approximation of integrals

Perturbation methods for differential equations

Local analysis

Local analysis of differential equations

(as discussed in Bender's book Part 2. Is this the same as regular perturbation methods, as discussed in Hinch's book? I think so).

Global analysis

Mostly for problems with regions of very different speed of change. These are singular perturbation problems, often when the small parameter $\epsilon$ is multiplying the highest derivative. Then the $\epsilon=0$ problem is of lower order, and will in general not be able to satisfy all the boundary conditions of the original problem

Matched asymptotic expansions

Method of multiple scales

WKB method

I wonder if there are analogous to these methods to algebraic equations. Maybe through the Perturbation methods for difference equations, which are closer to algebraic equations.

Perturbation methods for difference equations

These are described in Bender's book

Laplace's method

Books:

Hinch

Bender and Orzsag

..