Christopher S. Bretherton
Survey of practical solution techniques for ordinary differential equations. Linear systems of equations including nondiagonable case. Nonlinear systems; stability phase plane analysis. Asymptotic expansions. Regular and singular perturbations. Recommended: AMATH 402 or equivalent. Offered: W.
Approximate solution strategies exploiting small or large parameters or variables and their applications to ordinary differential equations. Regular and singular perturbations. Boundary layers and multiple scales. Frobenius expansion about regular singular points. Asymptotic expansions around irregular singular points. Application to Bessel, Airy and error functions. WKB theory.
Note: The Amath 568 catalog description includes topics now instead covered in Amath 402/502 (dynamical systems), notably analysis of linear and autonomous nonlinear systems of ODEs. Amath 568 is fully independent of Amath 402/502 and does not require material from that course.
Student learning goals
Understand the utility of exploiting small and large parameters for solving differential equations commonly arising in applications, and why this is a useful complement to numerical solution techniques
Recognize regular vs. singular perturbation problems and understand strategies for approximate solutions for each class of problem
Appreciate the meaning of an asymptotic solution or asymptotic series.
General method of instruction
Undergraduate ODEs and working knowledge of Matlab
Class assignments and grading
Weighted average of homeworks and an in-class midterm and final, both open-book.