MATH 703: Methods of Applied Mathematics I, Fall 2020

Course Details

Time: Asynchronous

Office hours: MW 9:55am-10:45am

Instructor: Saverio Spagnolie

Course website: https://canvas.wisc.edu/courses/212819 (for assignment uploads, grades, etc.)

Piazza: https://piazza.com/wisc/fall2020/703/home (for announcements, discussions, etc.)

Texts: The following books will be useful as supplementary texts.

• Nayfeh, Introduction to Perturbation Techniques (Wiley)
• Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers I (Springer)
• Stakgold and Holst, Green's Functions and Boundary Value Problems (Wiley)
• Gelfand and Fomin, Calculus of Variations (Dover)

Course Content:

This is a first-year course for all incoming PhD and Master students interested in pursuing research in applied mathematics. The course introduces advanced mathematical methods with a particular focus on asymptotic analysis of algebraic, differential, and integral equations, including the methods of stationary phase, steepest descent, multiple scales, matched asymptotics, and WKB approximations. Other topics include divergent series, Fourier transform, complex integration, dimensional analysis, phase-plane analysis, Floquet theory, Green's functions, self-adjoint operators, Sturm-Liouville problems, and calculus of variations.

• Week 1: Asymptotic series and expansions, algebraic equations
• Week 2: Generalized Laplace integrals, Laplace's method
• Week 3: Generalized Fourier integrals, method of stationary phase
• Week 4: Fourier series, Fourier transform
• Week 5: Complex variables, complex integration, Cauchy's integral formula
• Week 6: Complex integration, method of steepest descents, dimensional analysis
• Week 7: Differential equations; Duffing equation, Lindstedt-Poincare technique, renormalization, averaging, method of multiple scales
• Week 8: Self-excited oscillators, weakly-damped Duffing equation, linear stability of systems of ODEs
• Week 9: Phase-plane analysis, Mathieu equation, detuning
• Week 10: Fundamental matrix, monodromy matrix, Floquet multipliers/exponents
• Week 11: Boundary layer theory, WKBJ approximation
• Week 12: Variational calculus, Euler-Lagrange equation, canonical form, Hamilton's principle
• Week 13: Symmetric systems in finite dimensions, adjoint operators, boundary conditions, self-adjoint operators
• Week 14: Green's functions, solvability/alternative theorems
• Week 15: Sturm-Liouville eigenvalue problem, examples