Asymptotic & perturbation methods
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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$1,162 · Asymptotic & perturbation methods · 18 tutoring hrs
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Topics typically covered
Standard syllabus
Perturbation series
- Regular perturbation theory for algebraic equations
- Poincaré–Lindstedt method for periodic solutions
- Method of multiple scales for weakly nonlinear oscillators
- Averaging methods (Kruskal–Prudnikov / standard forms, introduction)
- Renormalization group methods (introduction)
Boundary layers and matched asymptotics
- Singular perturbations and inner/outer expansions
- Method of matched asymptotic expansions
- WKB theory for linear ODEs with large parameters
- Turning points and connection formulas (overview)
- Boundary-layer theory for PDEs (introduction)
Integral asymptotics
- Laplace's method and Watson's lemma
- Method of stationary phase
- Steepest descent and saddle-point methods (introduction)
- Asymptotics of special functions
- Summation of divergent series: Borel summability (introduction)
STEM / applied
Applications across domains
- Fluid mechanics: thin-film and lubrication theory
- Quantum mechanics: semiclassical limits and WKB
- Heat transfer at high/low Biot numbers
- Epidemic models with multiple time scales
- Elasticity: plate and shell theories from asymptotic reduction
Computation and validation
- Comparing asymptotic predictions to numerical simulation
- Identifying small parameters and regime diagrams
- Failure of naive expansions and resummation strategies
- Software for symbolic perturbation series
- Research problems from student thesis areas
Notes
Topics reflect common graduate asymptotic and perturbation methods syllabi at US universities. Prerequisites include graduate ODEs and complex analysis.