Math 550 2003-2004 Methods of Asymptotic Analysis (Part of core sequence in IAM curriculum, recommended prereq for Math 551 and Math 605E) Math 550: Prof. Rachel Kuske: rachel@math.ubc.ca This is a course in modern applied methods of asymptotic analysis, which play important roles in mathematical modeling, applied analysis, and approximations. The material provides valuable skills and resources complementary to scientific computations, mathematical modeling in applications, analysis of pde's and dynamical systems. The general concepts and methods are developed in the context of a wide variety of applications. Connections with other areas of mathematics, such as computations, analysis and modeling, are also covered. The course is designed so that the two semesters ( 550 followed by 551 and/or 605E) are complementary. Some topics are covered in both semesters (without repetition of material), including different applications of multiple scales, boundary layers, bifurcations, etc. Those who take both semesters will gain in-depth experience with these topics, as well as gaining an expertise with a broad range of techniques. Course Outline Math 550: Fall 2003 1. Introduction to asymptotic expansions 2. Dimensional analysis i) Small and large parameters/non-dimensional numbers ii) Regular expansions 3. Asymptotic approximation of integral representations of solutions i) Green's functions and Laplace-type integrals ii) Fourier integrals and stationary phase 4. Boundary layer I i) Singular Perturbation and Scales ii) Boundary value problems for ODE's iii) Convection-diffusion equations: PDE's iv) Parabolic approximations v) Flow past obstacles 5. Methods of Multiple Scales i) quasi-static approximations ii) ODE models of oscillators chemical/biological oscillators,Van der Pol and circuits,Poincare-Lindstedt iii) Envelope equations: nonlinear waves and patterns: NLS, KdV, GL, dispersion relations 6. Introduction to WKB approximations i) Eikonal and transport equations ii) Applications in quantum mechanics/waves 7. Bifurcations i) Systems of ODE's and boundary value problems ii) Turing instabilities iii) Pattern formation, Ginzburg-Landau equations iv) PDE's and Boundary value problems: elasticity, combustion... 8. Similarity Methods i) Diffusion and nonlinear diffusion ii) Free boundary problems: Stefan problem, option pricing Math 551: Spring 2003 ( We recommend taking both semesters to cover all topics. In some exceptional cases, such as a student who has a conflict with Math 550, it may be possible to take Math 551 with minimal background work, such as reading sections of the notes of Math 550.) 8. WKB Methods i) Connection formulae ii) Eigenvalue asymptotics iii) Maslow theory iv) Diffraction theory 9. Homogenization methods i) Review of method of multiple scales ii) Micro- vs. Macro-scale iii) Bubbly fluids, iv) Wave propagation on rough boundaries 10. Oscillators and dynamical systems i) Floquet theory ii) Melnikov's method iii) stick-slip problems 11. Boundary layer II i) Holes in the domain ii) Low/High Reynolds number flow iii) Exponentially ill-conditioned problems 12. Complex Variable methods i) Hele-Shaw flow and free boundaries ii) singularities in the complex plane iii) Stokes phenomenon, selection processes iv) Steepest Descent 13. Geometric Problems i) Interface motion in Multi-Dim. ii) Motion of curves by curvature iii) Geometric laws from PDE's iv) Dynamics of filaments twisting of DNA coils, super-conductivity, etc.