5315 (CSE 7365). Numerical Analysis. Numerical solution of linear and nonlinear equations, interpolation and approximation of functions, numerical integration, floating-point arithmetic, and the numerical solution of initial value problems in ordinary differential equations. Student use of the computer is emphasized. Requirements: Either graduate standing or, Prerequisites: MATH 2343 and MATH 3315 or CSE 3365; and a programming course (e.g., C, FORTRAN, or MATLAB). Graduate students who have doubts about their preparedness for this course should consult the course instructor before enrolling.

6316 (CSE 7366). Numerical Linear Algebra. The efficient solution of linear systems by both direct and iterative methods and least-squares problems by direct methods. Elementary and orthogonal matrix transformations provide a unified treatment of direct methods. Stationary and conjugate direction methods for efficiently solving sparse linear systems. (Credit will not be given for this course and for Math 5316) Prerequisites: A programming course (e.g., C, FORTRAN, or MATLAB), MATH 3353, and MATH 3315/CSE 3365 or MATH 5315.

5331. Functions of a Complex Variable. Complex numbers, analytic functions, mapping by elementary functions, complex integration. Cauchy-Goursat theorem, Cauchy integral formulas. Taylor and Laurent series, residues, evaluation of improper integrals. Applications of conformal mapping and analytic functions. Requirements: Either graduate standing or, Prerequisite: MATH 3337. Graduate students who have doubts about their preparedness for this course should consult the course instructor before enrolling.

5332. Wavelet Transforms. A mathematical introduction to sampling, data compression, multiresolution analysis, Fourier analysis, and wavelet theory, including biorthogonal wavelets and spline wavelets. Prerequisites: MATH 3337, 3353, and MATH 3315 or CSE 3365. Graduate students who have doubts about their preparedness for this course should consult the course instructor before enrolling.

5334. Introduction to Partial Differential Equations. Elementary partial differential equations of applied mathematics: heat, wave, and Laplace's equations. Topics include physical derivations, separation of variables, Fourier series, Sturm-Liouville eigenvalue problems, and Bessel functions. Requirements: Either graduate standing or Prerequisite: MATH 3337. Graduate students who have doubts about their preparedness for this course should consult the course instructor before enrolling.

5353. Linear Algebra. Spectral theory of Hermitian matrices, Jordan normal form, Perron-Frobenius theory, convexity. Applications include image compression, internet page rank methods, optimization, linear programming. Requirements: Either graduate standing or Prerequisite: MATH 3353

6311. Perturbation Methods. Solving differential equations with a small parameter by asymptotic techniques: weakly nonlinear oscillators, perturbed eigenvalue problems, boundary layers, method of multiple scales, WKBJ method. Prerequisite: MATH 2343 (MATH 5334 also recommended).

6312. Advanced Perturbation Methods. Topics include Kuzmak's theory of strongly nonlinear slowly varying oscillators and the methods of multiple scales and matched asymptotic expansions applied to partial differential equations such as those describing fluid dynamics and wave phenomena. Prerequisites: MATH 5334 and 6311.

6313. Asymptotic Expansions and Integral Transforms. Fourier and Laplace transforms. Asymptotic expansions with applications to integrals. Topics include integration by parts, Watson's lemma, Laplace's method, stationary phase, steepest descents, uniform expansions. Applications and examples from physical problems. Prerequisite: MATH 5331.

6315. Numerical Solution of Partial Differential Equations. Finite difference methods for elliptic, parabolic, and hyperbolic problems in partial differential equations. Stability, consistency, and convergence results are given. Attention is given to computer implementations. Prerequisites: MATH 5315/CSE 7365 and MATH 5334.

6319. Finite Element Analysis. Finite element method for elliptic problems, theory, practice and applications, finite element spaces, curved elements and numerical integration, minimization algorithms and iterative methods. Prerequisites: MATH 5315/CSE 7365 and MATH 5316/CSE 7366.

6320. Iterative Methods. Matrix and vector norms, conditioning, iterative methods for the solution of large linear systems and eigenvalue problems. Krylov subspace methods. Other topics to be chosen by the instructor. Prerequisites: MATH 5316/CSE 7366 and some programming experience.

6321. Numerical Solution of Ordinary Differential Equations. Numerical methods for initial value problems and boundary value problems for ordinary differential equations. Emphasizes practical solution of problems using Matlab. Prerequisites: MATH 2343, MATH 5315/CSE 7365.

6324. Introduction to Dynamical Systems. Nonlinear ordinary differential equations: equilibrium, stability, phase-plane methods, limit-cycles, and oscillations. Linear systems, diagonalization. Periodic coefficients (Floquet theory), Poincaré map. Difference equations (maps), period doubling, bifurcations, chaos. Prerequisites: MATH 2343 and 3353.

6325. Nonlinear Dynamical Systems and Chaos. Nonlinear differential equations. Stability and bifurcation theory of ODEs and maps. Forced oscillators. Subharmonic resonances. Melnikov criterion for chaos. Lorenz system. Center manifolds and normal forms. Silnikov's example. Prerequisite: MATH 5325.

6333. Partial Differential Equations. Method of eigenfunction expansion for nonhomogeneous problems. Green's functions for the heat, wave, and Laplace equations. Dirac delta functions, Fourier and Laplace transform methods, method of characteristics. Prerequisite: MATH 5334.

6336 (ME 5336/ME 7355). Fluid Dynamics. Preliminaries, concepts from vector calculus. The transport theorem, the Navier-Stokes and other governing equations. Dynamical similarity and Reynolds number. Vorticity theorems. Ideal and potential flow. The influence of viscosity and the boundary layer approximation. Prerequisite: MATH 3337.

6337. Real analysis. Real and functional analysis, including the Lebesgue integral, Fourier series, Fourier integrals, Banach and Hilbert spaces. Prerequisite: MATH 4338 or instructor approval.

6341. Linear and Nonlinear Wave Phenomena. The mathematical theory of linear and nonlinear waves. Applications from water waves, traffic flow, gas dynamics, and various other fields. Topics include nonlinear hyperbolic waves (characteristics, breaking waves, shock fitting, Burger's equation) and linear dispersive waves (method of stationary phase, group velocity, wave patterns). Prerequisite: MATH 5334.

6342. Solitons and the Inverse Scattering Transform. Nonlinear dispersive waves. The use of the direct and inverse scattering of the Schrödinger eigenvalue problem to obtain solitons and multiply-interacting solitons for the Korteweg-de Vries equation. Also the Zakharov-Shabat eigenvalue problem for the nonlinear Schrödinger (envelope solitons) and sine-Gordon (kinks) equations. Prerequisite: MATH 6341.

6346. Advanced Fluid Dynamics. Topics include waves (group velocity, dispersion); viscous flow theory (flow past a sphere, lubrication theory); two-phase flows (dynamics of bubbles, instabilities of thin films and liquid jets); vortex dynamics (point vortices, Crow instability); turbulence. Prerequisite: MATH 6336

6347. Vortex Dynamics. Vorticity transport equation. Rectilinear vorticies as Hamiltonian system. Elliptical vortices-moment model. Vortex rings. Swirling flows. Vortices near boundaries. Pairing. Reconnection. Prerequisites: MATH 5331, 6324, 5315/CSE 7365. (MATH 6336 useful but not essential)

6348. Turbulence in Fluids. A mathematical introduction to turbulence - the last great problem of classical physics according to Feynman. Kolmogorov's 1941 theory, closures theories, shell models, similarity theories. Prerequisites: MATH 6324, 5315, 5331, 5332 (or 5334). (MATH 6336 useful, not necessary)

6350. Mathematical Models in Biology. The mathematical analysis and modeling of biological systems, including biomedicine, epidemiology and ecology. Prerequisite: Consent of instructor.

6360. Computational Electromagnetics. Numerical methods for electromagnetics, with emphasis on practical applications. Numerical discretizations covered include the method of moments, finite differences, finite elements, boundary elements, and fast multipole methods. Prerequisites: EE 7330 or MATH 5334 and proficiency in one computer language (e.g. FORTRAN) or permission of the instructor.

6361. Multiphase Flows. Flow and transport equations for single-phase, two-phase, black oil, compositional, and thermal flows in porous media. Introduction to conservation equations of mass and energy and Darcy's law. Prerequisite: MATH 5334

6362. Numerical Reservoir Simulation. Numerical simulation of flow and transport problems for single-phase, two-phase, black oil, compositional, and thermal flows in porous media. Introduction of finite difference and finite element methods and linear solvers to reservoir simulation.

6370. Parallel scientific computing. An introduction to parallel computing in the context of scientific computation. Prerequisites: MATH 5315/CSE 7365 and MATH 5316/CSE 7366.

6371. Numerical Bifurcation Theory. A survey of basic nonlinear phenomena is given, including simple bifurcations, Hopf and Turing bifurcations, and bifurcation of periodic orbits in differential equations. Prerequisites: MATH 6337 or instructor approval.

6391. Topics in Applied Mathematics. Selected topics in the application of mathematical analysis to such fields as differential, integral, and functional equations; mechanics; hydrodynamics; mathematical biology; and economics. Prerequisite: Permission of instructor.

6395. Topics in Computational Mathematics. Selected topics of current interest. For example: numerical bifurcation theory, iterative methods for linear systems, domain decomposition and multigrid methods, numerical multidimensional integration, and numerical methods for multi-body problems. Prerequisite: Permission of instructor.

8398. Dissertation.