## Appl and Computational Math

- APC 503/AST 557: Analytical Techniques in Differential EquationsAsymptotic methods, Dominant balance, ODEs: initial and Boundary value problems, Wronskian, Green's functions, Complex Variables: Cauchy's theorem, Taylor and Laurent expansions, Approximate Solution of Differential Equations, singularity type, Series expansions. Asymptotic Expansions. Stationary Phase, Saddle Points, Stokes phenomena. WKB Theory: Stokes constants, Airy function, Derivation of Heading's rules, bound states, barrier transmission. Asymptotic evaluation of integrals, Laplace's method, Stirling approximation, Integral representations, Gamma function, Riemann zeta function. Boundary Layer problems, Multiple Scale Analysis
- APC 524/MAE 506/AST 506: Software Engineering for Scientific ComputingThe goal of this course is to teach basic tools and principles of writing good code, in the context of scientific computing. Specific topics include an overview of relevant compiled and interpreted languages, build tools and source managers, design patterns, design of interfaces, debugging and testing, profiling and improving performance, portability, and an introduction to parallel computing in both shared memory and distributed memory environments. The focus is on writing code that is easy to maintain and share with others. Students will develop these skills through a series of programming assignments and a group project.
- AST 559/APC 539: Turbulence and Nonlinear Processes in Fluids and PlasmasA comprehensive introduction to the theory of nonlinear phenomena in fluids and plasmas, with emphasis on turbulence and transport. Experimental phenomenology; fundamental equations, including Navier-Stokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudo-spectral and particle-in-cell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and direct-interaction approximation; PDF methods and intermittency; variational techniques. Applications from neutral fluids, fusion plasmas, and astrophysics.
- MAE 501/APC 501/CBE 509: Mathematical Methods of Engineering Analysis IMethods of mathematical analysis for the solution of problems in physics and engineering. Topics include an introduction to linear algebra, matrices and their application, eigenvalue problems,ordinary differential equations (initial and boundary value, eigenvalue problems), nonlinear ordinary differential equations, stability, bifurcations, Sturm-Liouville theory, Green's functions, elements of series solutions and special functions, Laplace and Fourier transform methods, and solutions via perturbation methods, partial differential equation including self-similar solution, separation of variables and method of characteristics.
- MAT 321/APC 321: Numerical Analysis and Scientific ComputingIntroduction to numerical methods with emphasis on algorithms, applications and numerical analysis. Topics covered include solution of nonlinear equations; numerical differentiation, integration, and interpolation; direct and iterative methods for solving linear systems; computation of eigenvectors and eigenvalues; and approximation theory. Lectures include mathematical proofs where they provide insight and are supplemented with numerical demos using MATLAB or Python.
- MAT 377/APC 377: Combinatorial MathematicsThe course covers the basic combinatorial techniques as well as introduction to more advanced ones. The topics discussed include elementary counting, the pigeonhole principle, counting spanning trees, Inclusion-Exclusion, generating functions, Ramsey Theory, Extremal Combinatorics, Linear Algebra in Combinatorics, introduction to the probabilistic method, spectral graph theory, topological methods in combinatorics.
- MAT 522/APC 522: Introduction to PDEThe course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, Fokker-Planck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the Navier-Stokes equations.