丁香园AV

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 902 - Applied Complex Analysis (4)

Review of complex power series and contour integration. Conformal mapping, Schwartz-Christoffel transformation. Special functions. Asymptotic expansions. Integral transform.

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 910 - Ordinary Differential Equations (4)

The solutions and properties of ordinary differential equations and systems of ordinary differential equations in the real and complex domains.

APMA 912 - Advanced Partial Differential Equations (4)

An advanced course on partial differential equations. Potential topics include linear and non-linear elliptic equations, second-order parabolic and hyperbolic equations, calculus of variations, semigroup theory, Hamilton-Jacobi equations, hyperbolic conservation laws.

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 921 - Numerical Solution of Ordinary Differential Equations (4)

Study of the practical numerical methods for solving initial and boundary value problems for ordinary differential equations.

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 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.