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Asymptotic & perturbation methods · Graduate · Math
Learning objectives from the Asymptotic & perturbation methods syllabus, grouped by unit. Click an objective for study materials.
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
Learning objectives
Click an objective for study materials.
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
Definitions and structure
- 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)
Proofs and reasoning
- 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
Abstraction and generalization
- Connection formulas (overview) — Connection formulas (overview)
- Boundary-layer theory for PDEs (introduction) — Boundary-layer theory for PDEs (introduction)
- 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
Modeling and computation
- Apply regular perturbation theory for algebraic equations in engineer... — Apply regular perturbation theory for algebraic equations in engineering contexts
- Apply poincaré–lindstedt method for periodic solutions in engineering... — Apply poincaré–lindstedt method for periodic solutions in engineering contexts
- Apply method of multiple scales for weakly nonlinear oscillators in e... — Apply method of multiple scales for weakly nonlinear oscillators in engineering contexts
- Apply averaging methods (kruskal–prudnikov / standard forms, introduc... — Apply averaging methods (kruskal–prudnikov / standard forms, introduction) in engineering contexts
- Apply renormalization group methods (introduction) in engineering con... — Apply renormalization group methods (introduction) in engineering contexts
Data and technology
- Use software to explore asymptotic & perturbation methods problems nu... — Use software to explore asymptotic & perturbation methods problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- 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
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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$1,162 · Asymptotic & perturbation methods · 18 tutoring hrs
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