MAPL -- Applied Mathematics

MAPL 460 Computational Methods (3) Prerequisites: {a grade of C or better in MATH 240 and MATH 241}; and {CMSC 104 or CMSC 113}. Also offered as CMSC 460. Credit will be granted for only one of the following: MAPL/CMSC 460 or MAPL/CMSC 466. Basic computational methods for interpolation, least squares, approximation, numerical quadrature, numerical solution of polynomial and transcendental equations, systems of linear equations and initial value problems for ordinary differential equations. Emphasis on methods and their computational properties rather than their analytic aspects. Intended primarily for students in the physical and engineering sciences.

MAPL 466 Introduction to Numerical Analysis I (3) Prerequisites: {a grade of C or better in MATH 240 and MATH 241}; and CMSC 104. Also offered as CMSC 466. Credit will be granted for only one of the following: MAPL/CMSC 460 or MAPL/CMSC 466. Floating point computations, direct methods for linear systems, interpolation, solution of nonlinear equations.

MAPL 467 Introduction to Numerical Analysis II (3) Prerequisites: MAPL/CMSC 466 with a grade of C or better. Also offered as CMSC 467. Credit will be granted for only one of the following: CMSC 467 or MAPL 467. Advanced interpolation, linear least squares, eigenvalue problems, ordinary differential equations, fast Fourier transforms.

MAPL 472 Methods and Models in Applied Mathematics I (3) Prerequisites: {MATH 241; and MATH 246; and MATH 240; and PHYS 161 or 171} or permission of department. Recommended: one of the following: MATH 410, MATH 414, MATH 415, MATH 462, MATH 463, PHYS 262, PHYS 273. Also offered as MATH 472. Credit will be granted for only one of the following: MATH 472 and MAPL 472. Mathematical models in fluid dynamics and elasticity, both linear and non-linear partial differential equations, variational characterizations in eigenvalue problems, numerical algorithms. Additional optional topics as time permits. Some examples are Hamiltonian systems, Maxwell's equations, non-linear programming.

MAPL 473 Methods and Models in Applied Mathematics II (3) Prerequisite: MAPL 472 or permission of department. Also offered as MATH 473. Credit will be granted for only one of the following: MAPL 473 or MATH 473. Continuation of the two-semester sequence, MAPL 472 and MAPL 473.

MAPL 477 Optimization (3) Prerequisites: (CMSC/MAPL 460, or CMSC/MAPL 466 or CMSC/MAPL 467) with a grade of C or better. Also offered as CMSC 477. Credit will be granted for only one of the following: CMSC 477 or MAPL 477. Linear programming including the simplex algorithm and dual linear programs, convex sets and elements of convex programming, combinatorial optimization, integer programming.

MAPL 498 Selected Topics in Applied Mathematics (1-3) Repeatable to 6 credits if content differs. Topics in applied mathematics of special interest to advanced undergraduate students.

MAPL 600 Advanced Linear Numerical Analysis (3) Prerequisite: CMSC/MAPL 666 or permission of instructor. Also offered as CMSC 760. Advanced topics in numerical linear algebra, such as dense eigenvalue problems, sparse elimination, iterative methods, and other topics.

MAPL 604 Numerical Solution of Nonlinear Equations (3) Prerequisites: CMSC/MAPL 666 and CMSC/MAPL 667; or permission of instructor. Also offered as CMSC 762. Numerical solution of nonlinear equations in one and several variables. Existence questions. Minimization methods. Selected applications.

MAPL 607 Advanced Numerical Optimization (3) Prerequisites: MATH 410; and MAPL/CMSC 477; or equivalent. Modern numerical methods for solving unconstrained and constrained nonlinear optimization problems in finite dimensions. Design of computational algorithms and the analysis of their properties.

MAPL 610 Numerical Solution of Ordinary Differential Equations (3) Prerequisites: a two semester course in numerical analysis and a one semester advanced undergraduate course in ordinary differential equations; or permission of instructor. Numerical methods for solving initial value problems in ordinary differential equations. Single step and multi-step methods, stability and convergence theory, adaptive methods, methods for stiff systems. Shooting methods for boundary value problems.

MAPL 612 Numerical Methods in Partial Differential Equations (3) Prerequisite: a graduate level one semester course in partial differential equations or a theoretical graduate level course in applied field such as fluid mechanics; or permission of instructor. Finite difference methods for elliptic, parabolic, and hyperbolic partial differential equations. Additional topics such as spectral methods, variational methods for elliptic problems, stability theory for hyperbolic initial-boundary value problems, and solution methods for conservation laws.

MAPL 614 Mathematics of the Finite Element Method (3) Prerequisite: one semester graduate level course in partial differential equations; or permission of instructor. Variational formulations of linear and nonlinear elliptic boundary value problems; formulation of the finite element method; construction of finite element subspaces; error estimates; eigenvalue problems; time dependent problems.

MAPL 655 Asymptotic Analysis and Special Functions I (3) Prerequisite: MATH 413 or MATH 463. Also offered as MATH 655. Transcendental equations, gamma function, orthogonal polynomials, Bessel functions, integral transforms, Watson's lemma, Laplace's method, stationary phase, analytic theory of ordinary differential equations, Liouville-Green (or WKBJ) approximation.

MAPL 656 Asymptotic Analysis and Special Functions II (3) Prerequisite: MATH/MAPL 655. Also offered as MATH 656. Steepest descents, coalescing saddle-points, singular integral equations, irregular singularities, Bessel, hypergeometric, and Legendre functions, Euler-Maclaurin formula, Darboux's method, turning points, phase shift.

MAPL 660 Scientific Computing I (3) Prerequisite: MAPL 460 or MAPL 466, or knowledge of basic numerical analysis (linear equations, nonlinear equations, integration, interpolation) with permission of instructor. Also offered as CMSC 660. Credit will be granted for only one of the following: MAPL 660 or CMSC 660. Monte Carlo simulation, fast Fourier transform and applications, nonlinear systems and continuation method, optimization, sparse matrices. Fundamental techniques in scientific computation with an introduction to the theory of each topic.

MAPL 661 Scientific Computing II (3) Prerequisite: MAPL 460 or MAPL 466, or knowledge of basic numerical analysis (linear equations, nonlinear equations, integration, interpolation) with permission of instructor. Knowledge of Fortran. Also offered as CMSC 661. Credit will be granted for only one of the following: MAPL 661 or CMSC 661. Finite element methods, numerical methods for time dependent PDEs, numerical linear algebra and libraries, parallel programming with message passing systems, data parallel programming, parallel libraries for numerical linear algebra, parallel programs for PDE problems. Techniques for high performance scientific computation with a brief introduction to the theory of each topic. Course is part of a two course sequence (660 and 661), but can be taken independently.

MAPL 666 Numerical Analysis I (3) Prerequisites: MAPL 466; and MATH 410. Also offered as CMSC 666. Iterative methods for linear systems, piecewise interpolation, eigenvalue problems, numerical integration.
MAPL 667 Numerical Analysis II (3) Prerequisite: MAPL 666. Also offered as CMSC 667. Nonlinear systems of equations, ordinary differential equations, boundary value problems.

MAPL 670 Ordinary Differential Equations I (3) Prerequisite: MATH 405; and MATH 410 or equivalent. Also offered as MATH 670. Existence and uniqueness, linear systems usually with Floquet theory for periodic systems, linearization and stability, planar systems usually with Poincare-Bendixson theorem.

MAPL 671 Ordinary Differential Equations II (3) Prerequisite: MATH 630; and MATH/MAPL 670 or equivalent. Also offered as MATH 671. The content of this course varies with the interests of the instructor and the class. Stability theory, control, time delay systems, Hamiltonian systems, bifurcation theory, and boundary value problems.

MAPL 673 Partial Differential Equations I (3) Prerequisite: MATH 411 or equivalent. Also offered as MATH 673. Credit will be granted for only one of the following: MAPL 673 or MATH 673. Analysis of boundary value problems for Laplace's equation, initial value problems for the heat and wave equations. Fundamental solutions, maximum principles, energy methods. First order nonlinear PDE, conservation laws. Characteristics, shock formation, weak solutions. Distributions, Fourier transform.

MAPL 674 Partial Differential Equations II (3) Prerequisite: MATH/MAPL 673 or permission of instructor. Also offered as MATH 674. Credit will be granted for only one of the following: MAPL 674 or MATH 674. Boundary value problems for elliptic partial differential equations via operator-theoretic methods. Hilbert spaces of functions. Duality, weak convergence. Sobolev spaces. Spectral theory of compact operators. Eigenfunction expansions.

MAPL 680 Eigenvalue and Boundary Value Problems I (3) Prerequisite: MATH 405 and MATH 410 or equivalent. Also offered as MATH 680. Operational methods applied to ordinary differential equations. Introduction to linear spaces, compact operators in Hilbert space, study of eigenvalues.

MAPL 681 Eigenvalue and Boundary Value Problems II (3) Prerequisite: MATH/MAPL 680. Also offered as MATH 681. Boundary value problems for linear differential equations. Method of energy integrals applied to Laplace's equation, heat equation and the wave equation. Study of eigenvalues.

MAPL 685 Modern Methods in Partial Differential Equations I (3) Prerequisite: MATH 630 and MATH 631. Also offered as MATH 685. Spaces of distributions, Fourier transforms, concept of weak and strong solutions. Existence, uniqueness and regularity theory for elliptic and parabolic problems using methods of functional analysis.

MAPL 686 Modern Methods in Partial Differential Equations II (3) Prerequisite: MATH/MAPL 685. Also offered as MATH 686. Emphasis on nonlinear problems. Sobolev embedding theorems, methods of monotonicity, compactness, applications to elliptic, parabolic and hyperbolic problems.

MAPL 698 Advanced Topics in Applied Mathematics (1-4) Repeatable if content differs.

MAPL 699 Applied Mathematics Seminar (1-3) Repeatable if content differs. Seminar to acquaint students with a variety of applications of mathematics and to develop skills in presentation techniques.

MAPL 701 Introduction to Continuum Mechanics (3) Background from algebra and geometry, kinematics of deformation. Stress equations of motion, thermodynamics of deforming continua. Theory of constitutive relations. Materials with memory. Initial boundary value problems of nonlinear solid and fluid thermomechanics. Boundary value problems of linear theories of solids and fluids.

MAPL 710 Linear Elasticity (3) Prerequisite: MAPL 701. Formulation of the equations. Compatability, uniqueness, existence, representation and qualitative behavior of solutions. Variational principles. St. Venant beam problems, plane strain and plane stress, half-space problems, contact problems, vibration problems, wave propagation. Emphasis is placed on formulation and technique rather than on specific examples.

MAPL 711 Non-linear Elasticity (3) Prerequisite: MAPL 701. Formulation of initial boundary value problems. Constitutive restrictions. Special solutions. Perturbation methods and their validity. Theories of rods and shells. Buckling and stability. Shock propagation.

MAPL 720 Fluid Dynamics I (3) A mathematical formulation and treatment of problems arising in the theory of incompressible, compressible and viscous fluids.

MAPL 721 Fluid Dynamics II (3) A continuation of MAPL 720.

MAPL 799 Master's Thesis Research (1-6)

MAPL 899 Doctoral Dissertation Research (1-8)