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MATHEMATICS

Professor Peter Moore, Department Chair

Professors: John Chen, Ian Gladwell, Richard Haberman, George Reddien, Douglas Reinelt, Lawrence Shampine, Richard Williams; Associate Professors: Thomas Carr, Robert Davis, Mogens Melander, Montie Monzingo, Johannes Tausch; Assistant Professors: Vladimir Ajaev, Yeojin Chung, Sheng Xu, Yunkai Zhou.

The department of mathematics offers M.S. and Ph.D. degrees in computational and applied mathematics.

Admission Requirements

Minimum requirements for admission to the graduate programs in mathematics are 18 hours in college-level mathematics courses beyond first- and second-year calculus (including differential equations, linear algebra and statistics). Undergraduate courses in numerical methods, partial differential equations, physics and computer science are particularly helpful, as would be familiarity with programming, specifically MATLAB. There is no foreign language requirement.

Both the M.S. and Ph.D. degree programs require Graduate Record Examination aptitude test scores. Two letters of recommendation are required.

Financial aid is available in the form of teaching assistantships, which include the waiver of tuition and fees.

Degree of Master of Science

Degree Requirements

A total of 33 semester hours of graduate course credit beyond the bachelor’s degree (usually 11 graduate courses) are required for the master’s degree, including at least 18 hours at the 6000 level (at least 12 of these hours to be taken in the department of mathematics). Candidates must complete two courses in computational mathematics (Math 5315, 6316) and two courses in differential equations and their applications (Math 5334, 6324). Of the remaining courses, a maximum of three approved courses can be taken from outside the department. An oral examination is required for graduation.

Degree of Doctor of Philosophy

Degree Requirements

To qualify for the Ph.D. degree, the student must fulfill the following requirements:

1. Satisfy all curricular requirements as specified by the departmental faculty

2. Pass comprehensive written and oral examinations

3. Complete a minimum of three years of graduate academic work, including at least one year in full-time residence on the SMU campus or at a research facility approved by the departmental faculty and the dean of the graduate program

4. Write and make a successful defense of a dissertation

Course requirements for the Ph.D. are flexible but must include the equivalent of 51 semester hours of graduate course credit beyond the bachelor’s degree (excluding dissertation work) and at least six credit hours of dissertation. The Ph.D. qualifying examination consists of a written examination based on individualized concentration courses in computational and applied mathematics and presentation of a paper (usually based on a reading course with a faculty member).

The Courses (MATH)

5315 (CSE 7365). 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. Prerequisites: MATH 3315/ CSE 3365, MATH 2343 and a programming course (e.g., MATLAB, C or FORTRAN).

5316. Introduction to Matrix Computation. The efficient solution of dense and sparse linear systems, least squares and eigenvalue problems. Elementary and orthogonal matrix transformations provide a unified treatment. Programming will be in MATLAB with a focus on algorithms. Prerequisites: MATH 3353 and MATH 3315/CSE 3365.

5331. Functions of a Complex Variable. Complex numbers, analytic functions, mapping by elementary functions, 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. Prerequisite: MATH 3337.

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 1338, 2339, 3353 and MATH 3315/CSE 3365.

5334. Introduction to Partial Differential Equations. Elementary partial differential equations of applied mathematics: heat, wave and Laplace’s equations. Physical derivations, separation of variables, Fourier series, Sturm-Liouville eigenvalue problems and Bessel functions. Prerequisite: MATH 3337.

5353. Linear Algebra. Spectral theory of Hermitian matrices, Jordan normal form, Perron-Frobenius theory and convexity. Applications include image compression, Internet page rank methods, optimization and linear programming. Prerequisite: MATH 3353.

6311. Methods of Applied Mathematics – 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 averaging, and WKBJ method. Prerequisite: MATH 2343. (MATH 5334 also recommended.)

6312. Advanced Perturbation Methods. 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, 6311.

6313. Asymptotic Expansions and Integral Transforms. Fourier and Laplace transforms. Asymptotic expansions with applications to integrals. Includes integration by parts, Watson’s lemma, Laplace’s method, stationary phase, steepest descents and 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.

6316 (CSE 7366). Numerical linear algebra. The efficient solution of dense and sparse linear systems, least squares problems and eigenvalue problems. Elementary and orthogonal matrix transformations provide a unified treatment. In addition to algorithm development, the course will emphasize the theory underlying the methods. Prerequisites:. MATH 3353 or an equivalent undergraduate course in linear algebra and MATH 5315/CSE 7365 or consent of the instructor.

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 5365/CSE 7365 and MATH 6316/ /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 6316/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 and MATH 5315/CSE 7365.

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 Poincaré map. Difference equations (maps), period doubling, bifurcations and chaos. Prerequisites: MATH 2343, 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 6324.

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. Prerequisite: MATH 5334.

6336 (ME 5336/ME 7336). Fluid Dynamics. Preliminaries and 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 or MATH 5334.

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

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. Includes nonlinear hyperbolic waves (characteristics, breaking waves, shock fitting and Burger’s equation) and linear dispersive waves (method of stationary phase, group velocity and 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. Waves: group velocity and dispersion. Viscous flow theory: flow past a sphere and lubrication theory. Two-phase flows: dynamics of bubbles, instabilities of thin films and liquid jets. Vortex dynamics: point vortices and Crow instability. Turbulence. Prerequisites: MATH 6336/ME 5336/ME 7336.

6347. Vortex Dynamics. Vorticity transport equation. Rectilinear vorticies as a Hamiltonian system. Elliptical vortices-moment model. Vortex rings. Swirling Flows. Vortices near boundaries. Pairing. Reconnection. Prerequisites: MATH 5331, MATH 5315/CSE 7365 and MATH 6324. (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 and similarity theories. Prerequisites: MATH 6324, MATH 5315/CSE 7365, MATH 5331 and MATH 5332 (or 5334). (MATH 6336 useful, but not necessary)

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

6360 (EE 8332). Computational Electromagnetics. Numerical methods for electromagnetics, with emphasis on practical applications. Numerical discretizations including 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. Fundamentals of Multiphase Flows in Porous Media. 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. Prequisites: 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. Prerequisites: MATH 5315/CSE 7365, MATH 6316/CSE 7366 and MATH 5334.

6370. Parallel Scientific Computing. An introduction to parallel computing in the context of scientific computation. Prerequisites: MATH 5315/CSE 7365 and MATH 6316/CSE 7366.

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

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 the 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 the instructor.

8398. Dissertation.