Asymptotic & Computational Methods for Nonlinear Differential Equations
Asymptotic & Computational Methods for Nonlinear Differential Equations
"Calling a problem nonlinear is like going to the zoo and talking about all the interesting nonelephant animals you see there." Stanislaw Ulam
Most differential equations that arise in modelling scientific phenomena are nonlinear. There is no general theory of nonlinear differential equations, in the same way that there is no general biology of non-elephant animals. This project will be about exploring techniques that allow for the approximation of classes of nonlinear equations using asymptotic and numerical methods. Asymptotic methods, sometimes known as perturbation theory, exploit some small or large quantity in a problem to reduce it to an iterative method for finding increasingly accurate approximations. This has the same spirit of finding Taylor series approximations, but can be applied much more generally. Alongside these techniques, this project will develop computational techniques for solving nonlinear equations, extending techniques seen in Computational Mathematics II. This project will involve significant use of programming, as well as advanced analytical techniques for solving equations in the spirit of Calculus I and Mathematical Methods II.
Students will learn a variety of techniques through looking at example problems:
Regular and singular perturbation theory for ordinary differential equations
Variants of the method of multiple scales (WKB, Poincaré–Lindstedt, boundary layer analysis)
Higher-order and implicit methods for numerical differential equations
Spectral methods
The individual component of this project will involve either extended independent study of deeper aspects of these ideas, (e.g. exponential asymptotics, homogenization theory), or their application to a modelling problem in physics, chemistry, biology, or another topic of the student's interest.
Students ideally should have taken Computational Mathematics II and Mathematical Methods II, and be eager to explore further development of similar ideas. They should feel comfortable typesetting their work alongside graphical outputs of numerical codes.
This project is coding-intensive and will involve substantial algebraic calculations, so familiarity with computer algebra systems will be helpful . Students will read textbook material each week, work problems together, and discuss these in weekly meetings. They will present their results using graphical comparisons of exact, numerical, and asymptotic approximations to a variety of problems. Evidence of learning will be demonstrated through a written project report presenting a series of problems solved using asymptotic and numerical methods, as well as an analysis and comparison of these methods.