Mathematics / advanced
Asymptotic and singular perturbation methods
Construct controlled approximations when exact solutions fail, including boundary layers, oscillations, and multiple scales.
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Inside this module
Module 01 advanced
Asymptotic and singular perturbation methods
Construct controlled approximations when exact solutions fail, including boundary layers, oscillations, and multiple scales.
- 01
Asymptotic sequences and Poincaré expansions
Interpret asymptotic coefficients as finite-order remainder claims, including useful expansions that diverge at every fixed nonzero parameter.
29 min - 02
Dominant balance and distinguished limits
Find hidden solution scales by balancing leading terms and checking that every discarded term is actually smaller.
29 min - 03
Regular perturbation and secular terms
Construct coefficient equations on fixed intervals and diagnose growth that destroys a long-time approximation.
30 min - 04
Boundary layers and inner–outer expansions
Resolve a lost boundary condition on a stretched coordinate and verify a uniform error bound in an exact linear example.
30 min - 05
Matched asymptotic expansions
Compare inner and outer limits in a shared overlap and subtract their common part when constructing a composite approximation.
31 min - 06
Multiple time scales and averaging
Separate fast oscillations from slow evolution and state the hypotheses needed for long-time averaging accuracy.
32 min - 07
WKB approximations and turning points
Derive phase and amplitude equations away from coefficient zeros and identify the Airy scale near a simple turning point.
33 min - 08
Laplace's method and steepest descent
Extract a Gaussian contribution near a nondegenerate minimum and track the analytic requirements behind a complex contour deformation.
33 min - 09
Stationary phase and oscillatory integrals
Locate where cancellation weakens, retain the phase of the Gaussian contribution, and distinguish stationary points from endpoints and degenerate phases.
33 min - 10
Exponentially small terms and Stokes phenomena
Understand information beyond a power expansion, least-term truncation, and convention-dependent changes in asymptotic representations across complex directions.
34 min