Courses offered at Penn State in
Computational and Applied Mathematics

Computational Applied Mathematics Related Courses

Undergraduate/Graduate (400) Level Courses in Math

*satisfy requirements only for the M.Ed. degree.

401. Introduction to Analysis I (3)

403. Classical Analysis I (3)

404. Classical Analysis II (3)

405.* Advanced Calculus for Engineers and Scientists I (3)

406. Advanced Calculus for Engineers and Scientists II (3)

411. Ordinary Differential Equations (3)

412. Fourier Series and Partial Differential Equations (3)

414. (Stat 414) Introduction to Probability Theory (3)

415. (Stat 415) Introduction to Mathematical Statistics (3)

416. (Stat 416) Stochastic Modeling (3)

417. Qualitative Theory of Differential Equations (3)

418. (Stat 418) Probability (3)

419. (Phys 419) Theoretical Mechanics (3)

421. Complex Analysis (3)

436. Linear Algebra (3)

441.* Matrix Algebra (3)

451. (CSE 451) Numerical Computations (3)

455. (CSE 455) Introduction to Numerical Analysis I (3)

456. (CSE 456) Introduction to Numerical Analysis II (3)

461. (Phys 461) Theoretical Mechanics (3)

485. Graph Theory (3)

486. Mathematical Theory of Games (3)

496. Independent Studies (3)

497. Special Topics (3)

499. Foreign Studies (1-12)

Graduate Level (500) Courses in Math

501-502. Real and Complex Analysis (3 each) Cauchy's theorem, Laurent expansion, residue calculus, harmonic functions, conformal mapping, measure and integration, convergence theorems, Lp spaces, Hilbert spaces. Fourier analysis, Fubini's theorem, Hahn-Banach theorem, open mapping theorem, uniform boundedness principle, dual spaces, selected topics from functional analysis. Prerequisite: MATH 404 (for MATH 501 only); MATH 501 (for MATH 502 only).

503. Functional Analysis (3) Topological vector spaces, completeness, convexity, duality, Banach algebras, bounded operators on Hilbert space, the spectral theorem, unbounded operators, applications. Prerequisite: MATH 502.

504. Analysis in Euclidean Space (3) The Fourier transform in L1 and L2 and applications, interpolation of operators, Riesz and Marcinkiewicz theorems, singular integral operators. Prerequisite: MATH 502.

505. Mathematical Fluid Mechanics (3) Kinematics, balance laws, constitutive equations. Ideal fluids, viscous flows, boundary layers, lubrication. Gas dynamics. Prerequisite: MATH 402 or 404.

506. Ergodic Theory (3) Measure-preserving transformations and flows, ergodic theorems, ergodicity, mixing, weak mixing, spectral invariants, measurable partitions, entropy, Ornstein isomorphism theory. Prerequisite: MATH 502.

507. Dynamical Systems I (3) Fundamental concepts. Extensive survey of examples. Equivalence and classification of dynamical systems, principal classes of asymptotic invariants, circle maps. Prerequisite: MATH 502.

508. Dynamical Systems II (3) Hyperbolic theory. Stable manifolds, hyperbolic sets, attractors, Anosov systems, shadowing, structural stability, entropy, pressure, Lyapunov characteristic exponents and non-uniform hyperbolicity. Prerequisite: MATH 507.

509. Linear Analysis and Applications I (3) Vector spaces, linear transformations, integration, Fourier and Laplace transforms, distributions, differential operators. Prerequisite: MATH 401 or 411 or 412.

510. Linear Analysis and Applications II (3) Integral equations, compact operators, variational methods, partial differential equations. Prerequisite: MATH 509.

511. Ordinary Differential Equations I (3) Existence and uniqueness, linear systems, series methods, Poincare-Bendixson theory, stability. Prerequisite: MATH 411 or 412.

512. Ordinary Differential Equations II (3) Floquet theory, regular and singular boundary value problems, Green's functions, eigenfunction expansions. Prerequisite: MATH 511.

513. Partial Differential Equations I (3) First order equations, the Cauchy problem, Cauchy- Kowalevski theorem, Laplace equation, wave equation, heat equation. Prerequisite: MATH 411 or 412.

514. Partial Differential Equations II (3) Sobolev spaces and elliptic boundary value problems, Schauder estimates, quasilinear symmetric hyperbolic systems, conservation laws. Prerequisite: MATH 502 and 513.

515. Classical Mechanics and Variational Methods (3) Introduction to the calculus of variations, variational formulation of Lagrangian mechanics, symmetry in mechanical systems, Legendre transformation, Hamiltonian mechanics, completely integrable systems. Prerequisite: Math 401 or 411 or 412.

516. (STAT 516) Stochastic Processes (3) Markov Chains; generating functions; limit theorems; continuous time and renewal processes; martingales, submartingales and supermartingales; diffusion processes; applications. Prerequisite: MATH (STAT) 416.

517. (STAT 517) Probability Theory I (3) Measure theoretic foundations of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics. Prerequisite: MATH 502.

518. (STAT 518) Probability Theory II (3) Measure theoretic foundations of probability, distribution functions and laws, types of convergence, central limit problem, conditional probability, special topics. Prerequisite: MATH 517.

519. (STAT 519) Topics in Stochastic Processes (3) Selected topics in Stochastic processes, including Markov and Wiener processes; stochastic integrals, optimization and control; optimal filtering. Prerequisite: MATH (STAT) 516,517.

521. Complex Analysis: Theory and Applications I (3) Conformal mappings, Schwarz-Cristoffel transformations, Dirichlet and Neumann problems, electrostatics and fluid flow, transform methods, asymptotic methods, Runge approximation theorems. Prerequisite: MATH 502.

522. Complex Analysis: Theory and Applications II (3) Factorization theorems, prime number theorem, Mittag-Leffler theorem, Nevanlinna theory, Riemann surfaces, Hartog's theorems, holomorphic mappings and automorphisms of bounded domains. Prerequisite: MATH 521.

523. Numerical Analysis I (3) Approximation and interpolation, numerical quadrature, direct methods of numerical linear algebra, numerical solutions of nonlinear systems and optimizati. Prerequisite: MATH 456 or equivalent.

524. Numerical Analysis II (3) Iterative methods in linear algebra, numerical solution of ordinary and partial differential equations. Prerequisite: MATH 523.

525. Theory of Functions of Several Complex Variables (3) Fundamental properties of holomorphic functions, reproducing kernels, integral representations, domain of holomorphy and pseudoconvexity, Weierstrass preparation theorem, complex manifolds. Prerequisite: MATH 502.

549. Mathematical Programming (3) Quadratic and convex programming, Integer and combinatorial programming, dynamic and stochastic programming. Prerequisite: MATH 484.

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550. (CSE 550) Numerical Linear Algebra (3) Solution of linear systems, sparse matrix techniques, linear least squares, singular value decomposition, numerical computation of eigenvalues and eigenvectors. Prerequisite: MATH (CSE) 456 or MATH 441.

551. (CSE 551) Numerical Solutions of Ordinary Differential Equations (3) Methods for initial value and boundary value problems. convergence and stability analysis, automatic error control, stiff systems, boundary value problems. Prerequisite: MATH (CSE) 451 or 456, MATH 411.

552. (CSE 552) Numerical Solution of Partial Differential Equations (3) Finite difference methods for elliptic, parabolic, and hyperbolic differential equations. Solutions techniques for discretized systems. Finite element methods for elliptic problems. Prerequisite: MATH 402 or 404; MATH (CSE) 451 or 456.

553. (CSE 553) Introduction to Approximation Theory (3) Interpolation; remainder theory; approximation of functions; error analysis; orthogonal polynomials; approximation of linear functionals; functional analysis applied to numerical analysis. Prerequisite: MATH 401 and 3 credits of computer science and engineering.

554. Approximation Theory (3) Approximation in normed spaces; existence, uniqueness, characterization, computation of best approximations; error bounds; degree of approximation; approximation of linear functionals. Prerequisite: MATH (CSE) 451 or 456, MATH 501.

555. (CSE 555) Numerical Optimization Techniques (3) Unconstrained and constrained optimization methods, linear and quadratic programming, software issues, ellipsoid and Karmarkar's algorithm, global optimization, parallelism in optimization. Prerequisite: MATH (CSE) 456.

556. (CSE 556) Finite Element Methods (3) Sobolev spaces, variational formulations of boundary value problems; piecewise polynomial approximation theory, convergence and stability, special methods and applications. Prerequisite: MATH 502 and 552.

577. (ME 577) Stochastic Systems for Science and Engineering (3) Develops the theory of stochastic processes and linear and non-linear stochastic differential equations for applications to science and engineering. Prerequisite: MATH(STAT) 414 or 418, and MATH 501 or ME 550.

580. Introduction to Applied Mathematics I (3) A graduate course of fundamental techniques including tensor, ordinary and partial differential equations, and linear transforms. Prerequisite: Basic knowledge of linear algebra, vector calculus and ODE, MATH 405.

581. Introduction to Applied Mathematics II (3) A graduate course of fundamental techniques including Ordinary, Partial, and Stochastic Differential Equations, Wavelet Analysis, and Perturbation Theory. Prerequisite:  MATH 580, or consent of instructor

590. Colloquium (1-3)

596. Individual Studies (1-9)

597. Special Topics (1-9)

598. Special Topics (1-9)

CAM related courses offered outside math dept

Courses for the computational sciences minor graduate option

CAM Students are also encouraged to take course in other department on campus to receive a broad training in various science and engineering subjects

For additional information, please contact individual CAM faculty or :
           Graduate Studies Office, Department of Mathematics
           224 McAllister Building, University Park, PA 16802
           (814) 865-7529, gradstudies@math.psu.edu
      Computational and Applied Mathematics at PSU       Department of Mathematics