|The following is a selection of courses in Applied Mathematics taught by faculty of the Department of Mathematics at UBC. A more complete list, which includes the courses taught by other department members is given on the IAM home page (http://www.iam.ubc.ca/). The availability of a course on a given year depends on the faculty members teaching the course, and the number of students interested in taking the course.
MATH 402: Calculus of Variations Classical variational problems; necessary conditions of Euler, Weierstrass, Legendre, and Jacobi; Erdmann corner conditions, transversality, convex Lagrangians, fields of extremals, sufficient conditions for optimality, numerical methods; applications to classical mechanics, engineering and economics.
MATH 405: Numerical Methods for Differential Equations The primary objective of the course is to introduce the basic numerical techniques for solving ordinary and partial differential equations. The basic numerical methods (e.g. spline interpolation, numerical integration, numerical linear algebra and rootfinding), usually treated in introductory numerical method courses, are introduced by applying them to the solution of ordinary and partial differential equations. This approach, in addition to being efficient, helps to contextualize the numerical methods and enables one to focus on applications of the methods to practical problems.
MATH 519: Fluid Dynamics I Introduction to the Navier-Stokes equations. Appropriate boundary conditions. Energy estimates. Uniqueness and continuous dependence theorems. Famous open problems. Sobolev inequalities, generalized derivatives, and Sobolev spaces. Weak and strong convergence, and compact imbeddings. Representation theorems. Spectral theorem. The stokes operator. Existence theorem for steady solutions by Galerkin approximation, and its extension to large data through use of the Brouwer fixed point theorem. Regularity theorems for steady solution.
MATH 520: Fluid Dynamics II The local existence and regularity of nonstationary solutions. The problem of global continuation, often referred to as "global existence." Weak solutions. Proper posing of problems in unbounded domains. Artificial boundary conditions for restricting problems that are posed in unbounded domains to bounded domains, for computational purposes. Fine element approximation of the Navier-Stokes equations. Spatially periodic spectral computations. Stability, bifurcation, and the theory of attractors. Compressible viscous flow. Non-Newtonian flow
MATH 521: Numerical Analysis I The topic of this class is the approximation of solutions to PDE's by numerical methods. We will consider several methods (finite difference, finite element and spectral) applied to three model 1D problems corresponding to elliptic, parabolic and hyperbolic problems. Questions of stability, consistency and convergence will be addressed as well as appropriate time stepping techniques for the time-dependent problems. As a mini ``special topic" we will consider numerical methods for hyperbolic conservation laws. The solutions to these equations are discontinuous and a more involved theory of convergence for numerical methods must be developed.
MATH 550: Asymptotic Methods for Scientists and Engineers Asymptotic expansions, regular perturbation theory, and the method of dominant balance. Asymptotic evaluation of integrals, including the methods of Laplace, stationary phase, and steepest descents. Application to numerical analysis and to the physical sciences including dispersive wave propagation and wave propgation in layered media. (How to nondimensionalize equations, the importance of nondimensional parameters, basic rules for asymptotic equations, comparing convergent and divergent series, comparison of asymptotic and numerical methods,etc.)
MATH 551: Perturbation Methods for Differential Equations Regular perturbations, singular perturbations and the method of matched asymptotic expansions. Turning point problems, the WKB method, eigenvalue problems and the method of multiple-scales. Introduction to homogenization and bifurcation theory. Applications to the physical sciences, including fluid mechanics, reaction-diffusion theory and pattern formation.
MATH 552: Introduction to Dynamical Systems Ideas, methods, and applications of bifurcation theory and dynamical systems: differential and difference equations, local bifurcations, perturbation methods, chaos.
Additional courses in Applied Mathematics:
- 523 Combinatorial Optimization
- 547 Optimal Control Theory
- 553 Advanced Dynamical Systems
- 554 Symmetries and Differential Equations
- 557 Linear and Nonlinear Waves
|Check our web site for other current course listings.