Integrated Applied Mathematics (IAM)

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Course numbers with the # symbol included (e.g. #400) have not been taught in the last 3 years.

IAM 830 - Graduate Ordinary Differential Equations

Credits: 3

Course is a graduate-level course on ordinary differential equations. It is designed to be accessible to first-year graduate students from math, science or engineering backgrounds who have had a first undergraduate course in differential equations, along with a standard calculus sequence. The course is designed to begin with an intensive review of undergraduate differential equations and then will proceed to handle more advanced concepts, starting with multi-dimensional coupled systems of ordinary differential equations, exponential matrix solutions, using coordinate transformations for conversion to standard forms, nonlinear systems and transform-based solutions, using coordinate tranformations for conversion to standard forms, nonlinear systems and transform-based techniques. The course will have an interdisciplinary and applied style and will cover the following topics: Intense review of undergraduate differential equations, Power Series and Fourier Series solutions, Multi-dimensional D.E.s, eigenvectors and Jordan forms, Numerical Methods, Nonlinear D.E.s Dynamical Systems and Chaos.

IAM 851 - Introduction to High-Performance Computing

Credits: 3

Course gives an introduction to select areas of high-performance computing, providing a basis for writing and working with high-performance simulation codes. The three main topics are: 1) basic software engineering, 2) high-performance and parallel programming, and 3) performance analysis and modeling. Additional topics may include hetergeneous architectures like GPUs and data analysis/visualization. Prereq: Enrollment in a CEPS graduate program, MATH 753, working knowledge of a programming language (C or Fortran), or by permission of instructor.

IAM 932 - Graduate Partial Differential Equations

Credits: 3

Graduate level introduction to the analysis of linear and nonlinear partial differential equations. topics include: seperation of variables, Fourier series, weak and strong solutions, eigenfunction expansions, the Strum-Liouville problem, Green's functions and fundamental solutions, method of characteristics, and conservation laws. Prereq: Ordinary Differential Equations and Linear Algebra.

IAM 933 - Applied Functional Analysis

Credits: 3

Introduction to rigorous mathematical analysis from the perspective of applications. Topics include: metric and normed spaces; convergence; completeness; continuity; Lebesgue measure theory; convergence theorems; Banach, Hilbert, Lp, and Sobolev spaces; orthogonality, bases, and projections; Sturm-Liouville theory; spectral theory; distributions; and weak solutions. Applications including to differential and integral equations, are presented throughout. Prereq: real analysis or graduate introductory courses in mathematical physics or applied mathematics.

IAM 940 - Asymptotic and Perturbation Methods

Credits: 3

Introduction to the asymptotic analysis of linear and nonlinear algebraic equations, ODEs, and PDEs and the to asymptotic approximation of integrals arising as transform solutions to ODEs/PDEs. Topics include: algebraic equations and dominant balance; asymptotic approximations; complex variable theory and the asymptotic evaluation of integrals via Laplace's method, stationary phase, and steepest descents; the method of matched asymptotic expansions (boundary-layer theory), coordinate straining, multiple scales, averaging, homogenization theory, and WKBJ analysis for singularly perturbed ODEs and PDEs. Prereq: MATH 527, MATH 528, MATH 644 or equivalent. Pre- or Coreq: PHYS 931.

IAM 950 - Spatiotemporal and Turbulent Dynamics

Credits: 3

Advanced graduate course on the dynamics od spatitotemporal patterns in nonlinear time-dependent PDEs. Topics include nonlinear pattern formation, bifurcationsa ans symmetry, nonlinear WKB analysis, phase diffusion/amplitude modulation theory, unstable coherent structures in turbulence, and periodic orbit theory. Example systems include 1d and 2d Swift-Hohenberg equation, the 1d Kuramoto-Sivashinsky equation, Rayleigh-Benard convection, and Navier-Stokes in plane Couette and pipe flows. Prereq: MATH 847 and IAM 932, or equivalent; or permission.

IAM 961 - Numerical Analysis I: Numerical Linear Algebra

Credits: 3

Introduction to numerical analysis and computational methods for linear systems. Topics include: IEEE floating point arithmetic; vector norms and induced norms; conditioning; projectors; LU decompositions; pivoting; Cholesky factorization; QR decompositions; Gram-Schmidt orthogonalization; Householder triangularization; Singular Value decompositions; least squares problems; stability; eigenvalue problems; power iterations; QR alogorithm; Krylov methods; Arnoldi iteration; GMRES; Lanczos iteration; Conjugate gradient algorithms; and Preconditioning. Prereq: scientific programming and linear algebra.

IAM 962 - Numerical Partial Differential Equations

Credits: 3

Numerical analysis applied to partial differential equations. Initial topics include the implementation of finite difference and spectral methods applied to the heat equation, wave equation, Burger's equation, and other model equations. The remainder of the course treats numerical analysis, starting with a brief review of function spaces. The primary topics include approximation theory for Sobolov spaces, projection operators, completeness, convergence, and error estimates. Prereq: IAM 961 or permission.