This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
PrerequisitesAn introductory course on differential equations (the equivalent of 353 or 356) and basic background in multi-variable calculus (line integrals or contour integrals from complex variables).
- Introduction: Asymptotic approximations of functions. Perturbation parameters, limits, and asymptotic order relations. Asymptotic series expansions: Taylor, Laurent, Frobenius, General Asymptotic Perturbation methods for solving algebraic equations. Solution of perturbed matrix eigenvalue problems. Regular and Singular perturbation problems.
- Asymptotics for integrals: Elementary methods. Watson's lemma for Laplace-type integrals. Stationary phase for Fourier-type integrals. Steepest descent and saddle points in the complex plane.
- Asymptotics for Ordinary Differential Equations I: Local expansions of solutions: regular and irregular singular points. Irregular singular points at infinity. The WKBJ method.
- Asymptotics for Ordinary Differential Equations II: Matched asymptotic expansions. Singular perturbations: boundary layers. The method of multiple scales: nonlinear oscillators.
- Advanced Mathematical Methods for Scientists and Engineers by C. M. Bender, S. A. Orszag
- Applied Asymptotic Analysis by P. D. Miller