Graduate Program

MATH 5315. INTRODUCTION TO 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.

MATH 5316. INTRODUCTION TO MATRIX COMPUTATION. The efficient solution of dense and sparse linear systems, least squares problems, and eigenvalue problems. Elementary and orthogonal matrix transformations provide a unified treatment. Programming is in MATLAB, with a focus on algorithms.

MATH 5331. FUNCTIONS OF A COMPLEX VARIABLE. Complex numbers, analytic functions, mapping by elementary functions, and complex integration. Cauchy-Goursat theorem and Cauchy integral formulas. Taylor and Laurent series, residues, and evaluation of improper integrals. Applications of conformal mapping and analytic functions.

MATH 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.

MATH 5353. LINEAR ALGEBRA. Spectral theory of Hermitian matrices, Jordan normal form, Perron-Frobenius theory, and convexity. Applications include image compression, Internet page ranking methods, optimization, and linear programming.

MATH 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, and the WKBJ method.

MATH 6312. ADVANCED ASYMPTOTIC AND PERTURBATION METHODS. Topics include strongly nonlinear and slowly varying oscillators, multiple scales and matched asymptotic expansions applied to partial differential equations, asymptotic evaluation of integrals and transforms, stationary phase, steepest descents, and applications.

MATH 6315. NUMERICAL METHODS I. Covers floating point arithmetic, backward stability analysis, numerical solution of dense and sparse linear systems of equations using direct and basic iterative methods, least-squares problems and eigenvalue problems, elementary and orthogonal matrix transformations, and nonlinear systems of equations.

MATH 6316. NUMERICAL METHODS II. Covers interpolation and approximation of functions, numerical differentiation and integration, basic methods for initial value problems in ordinary differential equations, and basic approximation methods for one-dimensional initial- boundary value problems. Topics focus on algorithm development and the theory underlying each method.

MATH 6318. NUMERICAL PARTIAL DIFFERENTIAL EQUATIONS. Covers finite difference methods for elliptic, parabolic, and hyperbolic problems in partial differential equations. Also, stability, consistency, and convergence results. Attention is given to computer implementations.

MATH 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.

MATH 6320. ITERATIVE METHODS. Matrix and vector norms, conditioning, iterative methods for the solution of larger linear systems and eigenvalue problems. Krylov subspace methods. Other topics to be chosen by the instructor.

MATH 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.

MATH 6324. INTRODUCTION TO DYNAMICAL SYSTEMS. Nonlinear ordinary differential equations: equilibrium, stability, phase-plane methods, limit-cycles, and oscillations. Linear systems and diagonalization. Periodic coefficients (Floquet theory) and Pioncaré map. Difference equations (maps), period doubling, bifurcations, and chaos.

MATH 6325. DYNAMICAL SYSTEMS AND CHAOS. Nonlinear differential equations. Stability and bifurcation theory of ODEs and maps. Forced oscillators. Subharmonic resonances. Melnikovcriterion for chaos. Lorenz system. Center manifolds and normal forms. Silnikov's example.

MATH 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, and method of characteristics.

MATH 6336. 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.

MATH 6337. REAL AND FUNCTIONAL ANALYSIS. Topics include continuous functions, metric and normed spaces, Banach spaces, Hilbert spaces, distributions and the Fourier transform, measure theory and function spaces, differential calculus, and variational methods.

MATH 6341. LINEAR AND NONLINEAR WAVES. 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).

MATH 6343. PHOTONICS MODELING AND SIMULATIONS. Propagation of light in photonic structures. Using asymptotic techniques and simulations, students derive and then analyze models based on ordinary and partial linear and nonlinear differential equations.

MATH 6346. ADVANCED FLUID DYNAMICS. Topics include surface waves in shallow and deep water, sound waves, Stokes flow equations and lubrication-type models, spreading of droplets, and coating flows. Other topics are chosen from dynamics of bubbles, film drainage, electroosmotic flow, electrowetting on dielectric, turbulence, and fluid mechanics of swimming and flying.

MATH 6350. MATHEMATICAL MODELS IN BIOLOGY. The mathematical analysis and modeling of biological systems, including biomedicine, epidemiology, and ecology.

MATH 6352. EPIDEMIOLOGY AND IMMUNOLOGY. Modeling and analysis of diseases from epidemiology and immunology. Considers disease dynamics modeled with delay, integral, partial, and stochastic differential equations based on susceptible-infectious-removed ODEs.

MATH 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 instructor.

MATH 6370. PARALLEL SCIENTIFIC COMPUTING. An introduction to parallel

computing in the context of scientific computation.

MATH 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.

MATH 6395. TOPICS IN COMPUTATIONAL MATHEMATICS. Selected topics of current interest, such as numerical bifurcation theory, iterative methods for linear systems, domain decomposition and multigrid methods, numerical multidimensional integration, and numerical methods for multibody problems.