School of Science and Engineering: Mathematics
2008-2009 Academic Year
666
MATH 665, 666 Differential Equations I, II (3, 3)
Staff. ODE: existence and uniqueness, stability and linearized stability, phase
plane analysis, bifurcation and chaos. PDE: heat, wave, and Laplace equations,
functional analytic (Sobolev space) and geometric (characteristic) methods.
Maximum principle. Introduction to nonlinear PDE's.
MATH 671, 672 Analysis I and II (3, 3)
Staff. Prerequisites: MATH 305, 309, and 406. Lebesgue measure on R.
Measurable functions (including Lusin's and Egoroff's theorems). The Lebesgue
integral. Monotone and dominated convergence theorems. Radon-Nikodym
Theorem. Differentiation: bounded variation, absolute continuity, and the
fundamental theorem of calculus. Measure spaces and the general Lebesgue
integral (including summation and topics in Rn such as the Lebesgue
differentiation theorem). Lp spaces and Banach spaces. Hahn-Banach, open
mapping, and uniform boundedness theorems. Hilbert space. Representation of
linear functionals. Completeness and compactness. Compact operators, integral
equations, applications to differential equations, self-adjoint operators, unbounded
operators.
MATH 675, 676 Computation I, II (3, 3)
Staff. Floating point arithmetic (limitations and pitfalls). Numerical linear algebra,
solving linear systems by direct and iterative methods, eigenvalue problems,
singular value decompositions, numerical integration, interpolation. Iterative
solution of nonlinear equations. Unconstrained optimization. Solution of ODE,
both initial and boundary value problems. Numerical PDE. Introduction to fluid
dynamics and other areas of application.
MATH 681, 682 Applied Mathematics I and II (3, 3)
Staff. Prerequisites: MATH 305, 309, 347, and 406. Formulating mathematical
models. Introduction to differential equations and integral equations. Fourier
series and transforms, Laplace transforms. Generating functions. Dimensional
analysis and scaling. Regular and singular perturbations. Asymptotic expansions.
Boundary layers. The calculus of variations and optimization theory. Similarity
solutions. Difference equations. Stability and bifurcation. Introduction to
probability and statistics, and applications.
Note: Mathematics 651, 652, 655, 656, 661, 662, 665, 666, 671, 672, 675, 676,
681, 682 are particularly recommended for students planning to do graduate work
in mathematics.
MATH 684 Numerical Methods in Partial Differential Equations (3)
Staff. Prerequisites: MATH 331 and 347 or approval of instructor. This course
will present a detailed analysis of the methods for numerically approximating the
solution of ordinary and partial differential equations typically encountered in
applications from engineering and physics. Mathematical theory, practical
implementation and applications will be emphasized equally. Typical applications
to be discussed include population dynamics, particle dynamics, waves, diffusion
processes.