| name | asymptotic-perturbation-methods |
| description | Construct and validate asymptotic and perturbative approximations. Use for dominant balance, regular or singular perturbations, boundary layers, matched expansions, multiple scales, WKB, Laplace methods, and stationary phase. |
| license | MIT |
Asymptotic and Perturbation Methods
Identify the limit
State the small or large parameter, nondimensional variables, fixed quantities, domain, and observable. Rank
terms by dominant balance and identify points where the ordering changes. An expansion is incomplete unless
its limiting process and uniformity region are declared.
Construct the approximation
- Use a regular expansion only when the reduced problem preserves the needed order and boundary data.
- For singular limits, locate boundary, initial, turning-point, or internal layers and introduce stretched scales.
- Match inner and outer limits in a common overlap region, then form a composite approximation without double counting.
- Use multiple scales or averaging to remove secular terms in long-time oscillatory problems.
- For integrals and waves, identify stationary points, endpoints, saddles, and Stokes or branch structure.
Estimate the error
Retain the first neglected order, but do not equate formal order with a rigorous bound. Substitute the
approximation into the original equation and compute its residual. Track exponentially small terms when they
control switching, instability, or boundary conditions.
Verify
- Compare against an exact solution or high-accuracy numerical reference across decreasing parameter values.
- Plot or tabulate scaled error to test the predicted convergence rate.
- Check matching in the overlap and uniform accuracy near layers or turning points.
- Test conservation laws, boundary conditions, phase, and long-time drift.
- Report where the approximation fails and whether the failure is algebraic, exponential, or numerical.
Deliver
Give the distinguished limit, balances, expansion, matching, composite result, residual, observed error order,
and validity region.
Source basis
Original synthesis informed by Lebl's differential-equations text and open asymptotic-methods references listed
in ../../docs/TEXTBOOK_SOURCES.md.
