Asymptotic & perturbation methods
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Asymptotic & Perturbation Methods — scope drawn from open asymptotics notes and typical US graduate asymptotics syllabi.
Perturbation series
- Regular perturbation theory for algebraic equations — Regular perturbation theory for algebraic equations
- Poincaré–Lindstedt method for periodic solutions — Poincaré–Lindstedt method for periodic solutions
- Method of multiple scales for weakly nonlinear oscillators — Method of multiple scales for weakly nonlinear oscillators
- Averaging methods (Kruskal–Prudnikov / standard forms, introduction) — Averaging methods (Kruskal–Prudnikov / standard forms, introduction)
- Renormalization group methods (introduction) — Renormalization group methods (introduction)
Boundary layers and matched asymptotics
- Singular perturbations — Singular perturbations
- Inner/outer expansions — Inner/outer expansions
- Method of matched asymptotic expansions — Method of matched asymptotic expansions
- WKB theory for linear ODEs with large parameters — WKB theory for linear ODEs with large parameters
- Turning points — Turning points
- Connection formulas (overview) — Connection formulas (overview)
- Boundary-layer theory for PDEs (introduction) — Boundary-layer theory for PDEs (introduction)
Integral asymptotics
- Laplace's method and Watson's lemma — Laplace's method and Watson's lemma
- Method of stationary phase — Method of stationary phase
- Steepest descent — Steepest descent
- Saddle-point methods (introduction) — Saddle-point methods (introduction)
- Asymptotics of special functions — Asymptotics of special functions
- Summation of divergent series: Borel summability (introduction) — Summation of divergent series: Borel summability (introduction)
Applications across domains
- Fluid mechanics — Fluid mechanics
- Lubrication theory — Lubrication theory
- Quantum mechanics: semiclassical limits and WKB — Quantum mechanics: semiclassical limits and WKB
- Heat transfer at high/low Biot numbers — Heat transfer at high/low Biot numbers
- Epidemic models with multiple time scales — Epidemic models with multiple time scales
- Elasticity — Elasticity
- Shell theories from asymptotic reduction — Shell theories from asymptotic reduction
Computation and validation
- Comparing asymptotic predictions to numerical simulation — Comparing asymptotic predictions to numerical simulation
- Identifying small parameters — Identifying small parameters
- Regime diagrams — Regime diagrams
- Failure of naive expansions — Failure of naive expansions
- Resummation strategies — Resummation strategies
- Software for symbolic perturbation series — Software for symbolic perturbation series
- Research problems from student thesis areas — Research problems from student thesis areas
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Perturbation series
Regular perturbation theory for algebraic equations
Coming soon - Boundary layers and matched asymptotics
Singular perturbations
Coming soon - Integral asymptotics
Laplace's method and Watson's lemma
Coming soon - Applications across domains
Fluid mechanics
Coming soon - Computation and validation
Comparing asymptotic predictions to numerical simulation
Coming soon
Notes
Graduate Asymptotic & Perturbation Methods — scope drawn from open asymptotics notes and typical US graduate asymptotics syllabi. Topic outline: `content/topics/graduate/asymptotic_methods.json`.