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# Applied Mathematics Courses

### APMA 900 - Asymptotic Analysis of Differential Equations (4)

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.

### APMA 901 - Partial Differential Equations (4)

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.

### APMA 905 - Applied Functional Analysis (4)

Infinite dimensional vector spaces, convergence, generalized Fourier series. Operator Theory; the Fredholm alternative. Application to integral equations and Sturm-Liouville systems. Spectral theory.

### APMA 920 - Numerical Linear Algebra (4)

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.

### APMA 922 - Numerical Solution of Partial Differential Equations (4)

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.

### APMA 923 - Numerical Methods in Continuous Optimization (4)

Theory and algorithms of non-linear programming with an emphasis on modern computational considerations. Topics may include: optimality conditions for unconstrained and constrained optimization, gradient methods, conjugate direction methods, Newton method, quasi-Newton methods, penalty and barrier methods, augmented Langrangian methods and interior point methods.

### APMA 929 - Selected Topics in Numerical Analysis (4)

Study of a specialized area of numerical analysis such as computational fluid dynamics, approximation theory, integral equations, integral transforms, computational complex analysis, special functions, numerical quadrature and multiple integrals, constrained optimization, finite elements methods, sparse matrix techniques, or parallel algorithms in scientific computing.

### APMA 930 - Computational Fluid Dynamics (4)

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.

### APMA 934 - Selected Topics in Fluid Dynamics (4)

Study of a specialized area of fluid dynamics such as hydrodynamic stability, multiphase flow, non-Newtonian fluids, computational fluid dynamics, boundary-layer theory, magnetic fluids and plasmas, bio- and geo-fluid mechanics, gas dynamics.

### APMA 935 - Analysis and Computation of Models (4)

Analysis of models from the natural and applied sciences via analytical, asymptotic and numerical studies of ordinary and partial differential equations.

### APMA 939 - Selected Topics in Mathematical Image Processing (4)

Study of mathematical and computational aspects of image science. Some of the main mathematical tools are partial differential equations, iterative solutions to systems of equations, filters and wavelets. Applications include deblurring, denoising, inpainting, reconstruction, registration, and segmentation. Previous course offerings focused on computational methods in medical imaging, introduction to wavelets, and mathematical image processing and analysis.

### APMA 940 - Mathematics of Data Science (4)

Theory and algorithms for problems in data science with an emphasis on mathematical aspects. Topics may include dimension reduction, supervised learning, including regression and classification, unsupervised learning, including clustering and latent variable modeling, deep learning, algorithms for big data, and foundations of learning.

### APMA 981 - Selected Topics in Continuum Mechanics (4)

### APMA 982 - Selected Topics in Mathematical Physics (4)

### APMA 990 - Selected Topics in Applied Mathematics (4)

Topics vary depending on faculty availability and student interest. Recent offerings include: geophysical fluid dynamics, adaptive numerical methods for differential equations, learning theory, and stability, pattern formation and chaos.

### APMA 995 - PhD Oral Candidacy Exam

An open oral candidacy exam given by the supervisory committee. The exam consists of a proposed thesis topic defence by the student and supervisory committee questions about related proposed research topics. The exam follows submission of a written PhD research proposal. Graded on a satisfactory/unsatisfactory basis. Students who fail will either successfully complete a second exam within six months or withdraw from the program. Prerequisite: Applied Mathematics PhD stream students only. Must be completed within first six terms of the program.