Applied and Computational Mathematics
Applicants normally submit scores in the aptitude section and an appropriate advanced section of the Educational Testing Service’s graduate record examinations. Applicants with backgrounds in areas other than mathematics (for example, a bachelor’s degree or its equivalent in engineering or physics) may be considered suitably prepared for these programs.
PhD candidates must complete a further eight graduate units beyond the MSc core course requirements shown below.
Candidates who are admitted to the PhD program without an MSc are required to obtain credit or transfer credit for an amount of course work equivalent to that obtained by students with an MSc.
Core Course Requirements
Normally courses that are cross-listed as undergraduate courses cannot be used to satisfy graduate course requirements.
Beyond all the courses the student completed for the bachelor's degree, the candidate will complete 24 units that consist of one of
Analysis and computation of classical problems from applied mathematics such as eigenfunction expansions, integral transforms, and stability and bifurcation analyses. Methods include perturbation, boundary layer and multiple-scale analyses, averaging and homogenization, integral asymptotics and complex variable methods as applied to differential equations.
First order non-linear partial differential equations (PDEs) and the method of characteristics. Hamilton-Jacobi equation and hyperbolic conservation laws; weak solutions. Second-order linear PDEs (Laplace, heat and wave equations); Green's functions. Sobolev spaces. Second-order elliptic PDEs; Lax-Milgram theorem.
and one of
Conditioning and stability of numerical methods for the solution of linear systems, direct factorization and iterative methods, least squares, and eigenvalue problems. Applications and mathematical software.
Analysis and application of numerical methods for solving partial differential equations. Potential topics include finite difference methods, spectral methods, finite element methods, and multi-level/multi-grid methods.
and one of
Basic equations governing compressible and incompressible fluid mechanics. Finite difference and finite volume schemes for hyperbolic, elliptic, and parabolic partial differential equations. Practical applications in low Reynolds number flow, high-speed gas dynamics, and porous media flow. Software design and use of public-domain codes. Students with credit for MATH 930 may not complete this course for further credit.
Analysis of models from the natural and applied sciences via analytical, asymptotic and numerical studies of ordinary and partial differential equations.
and at least one other course from the above course lists that has not already been completed
and an additional eight graduate units.
Students pass an oral candidacy exam given by the supervisory committee before the end of the fourth full time term. The exam consists of a proposed thesis topic defence and supervisory committee questions about related proposed research topics. The exam follows submission of a written PhD research proposal and is graded pass/fail. Those with a fail will complete a second exam within six months. A student failing twice will normally withdraw.
A PhD candidate must submit and defend a thesis based on his/her original work that embodies a significant contribution to mathematical knowledge.
Academic Requirements within the Graduate General Regulations
All graduate students must satisfy the academic requirements that are specified in the Graduate General Regulations, as well as the specific requirements for the program in which they are enrolled.