9: School of Graduate Engineering and Applied Science

General Information | Degree Programs | Program Descriptions | Course Descriptions | Faculty

Aerospace Engineering | Applied Mathematics | Applied Mechanics | Biomedical Engineering
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Engineering Physics | Materials Science and Engineering | Mechanical and Aerospace Engineering
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Applied Mathematics

Applied Mathematics faculty

APMA 507 - (3) (Y)
Numerical Methods
Prerequisites: Two years of college mathematics, including some linear algebra and differential equations, and the ability to write computer programs
Introduction to techniques used in obtaining numerical solutions, with emphasis on error estimation. Areas of application studied include approximation and integration of functions, solution of algebraic and differential equations.

APMA 513 - (3) (Y)
Vector Calculus and Complex Variables
Prerequisite: Two years of undergraduate mathematics
Plane and three-dimensional curves, directional derivative, gradient. Line integrals, conservative force fields. Surface and volume integrals, divergence and Stokes' theorems. The geometry and algebra of complex numbers. The exponential, logarithm, and other elementary functions. Analytic functions, the Cauchy-Riemann equations, Laplace's equation. Fluid flow. Contour integrals, Cauchy's theorem and integral formula. The residue theorem, Laurent's expansion, and the evaluation of real-valued, definite integrals.

APMA 541 - (3) (Y-SS)
Engineering Mathematics
Prerequisites: Four semesters of calculus including ordinary differential equations
Solution of the heat, potential, and wave equation in rectangular and polar coordinates. Separation of variables and eigenfunction expansion techniques for nonhomogeneous boundary-value problems.

APMA 602 - (3) (Y)
Continuum Mechanics With Applications
Prerequisite: Permission of instructor
Introduction to continuum mechanics and mechanics of deformable solids. Vectors and Cartesian tensors, stress, strain, deformation, equations of motion, constitutive laws, introduction to elasticity, thermal elasticity, viscoelasticity, plasticity, and fluids. Cross-listed as AM 602, CE 602, MAE 602.

APMA 613 - (3) (Y)
Mathematical Foundations of Continuum Mechanics
Prerequisites: Linear Algebra, Vector Calculus, Elementary PDE (or concurrently)
Describes the mathematical foundations of continuum mechanics from a unified viewpoint. Review of relevant concepts from linear algebra, vector calculus and Cartesian tensors. Kinematics of finite deformations and motions; finite strain measures; linearization. Concept of stress; conservation laws of mechanics, and equations of motion and equilibrium. Constitutive theory, constitutive laws for nonlinear elasticity; generalized Hooke's law for a linearly elastic solid. Constitutive laws for Newtonian and non-Newtonian fluids. Basic problems of continuum mechanics as boundary-value problems for partial differential equations. Cross-listed as AM 613.

APMA 615 - (3) (Y)
Linear Algebra
Prerequisite: Three years of college mathematics or permission of instructor
Systems of linear equations, least squares procedures for solving over-determined systems, finite dimensional vector spaces, linear transformations and their representation by matrices, determinants, Jordan canonical form, unitary reduction of symmetric and Hermitian forms, eigenvalues and invariant subspaces.

APMA 618 - (3) (SI)
Reliability and Risk Analysis
Prerequisite: APMA 310 or permission of instructor
Probability and statistics applied to analyzing reliability and/or availability of engineering components and systems; probability distributions for failures and failure times of engineering components; fault tree analysis of designed systems for probability of failure and associated risk predictions. Cross-listed as NE 618.

APMA 634 - (3) (Y)
Numerical Analysis
Prerequisites: Two years of college mathematics, including some linear algebra, and the ability to write computer programs
Solution of systems of linear and nonlinear equations, calculations of matrix eigenvalues, least squares problems, boundary value problems in ordinary and partial differential equations.

APMA 637 - (3) (O)
Singular Perturbation Theory
Prerequisites: APMA 315, 513 or equivalent
Regular perturbations, roots of polynomials. Singular perturbations in ODE's. Periodic solutions of simple nonlinear differential equations. Multiple-Scales method. WKBJ approximation. Turning-point problems, Langer's method of uniform approximation. Asymptotic behavior of integrals, Laplace Integrals, stationary phase, steepest descents. Examples drawn from physical systems. Cross-listed as MAE 637.

APMA 642 - (3) (Y)
Engineering Mathematics II
Prerequisites: APMA 341 or APMA 541, and complex variables
Methods of solution of linear partial differential equations, Green's functions, transform methods, asymptotic expansions, and the solution of a variety of linear boundary/initial value problems in various coordinate systems.

APMA 643 - (3) (SI)
Statistics for Engineers and Scientists
Prerequisite: Admission to graduate studies
Role of statistics in science, hypothesis tests of significance, confidence intervals, design of experiments, regression, correlation analysis, analysis of variance, and introduction to statistical computing with statistical software libraries.

APMA 648 - (3) (SI)
Special Topics in Applied Mathematics
Prerequisite: Permission of instructor
The material which varies from year to year is selected to fill special needs of graduate students.

APMA 672 - (3) (Y)
Computational Fluid Dynamics I
Prerequisite: MAE 631 or permission of instructor
Solution of flow and heat transfer problems involving steady and transient convective and diffusive transport. Superposition and panel methods for inviscid flow, finite-difference methods for elliptic, parabolic and hyperbolic partial differential equations, elementary grid generation for odd geometries, primitive variable and vorticity-steamfunction algorithms for incompressible, multidimensional flows. Extensive use of personal computers/workstations including graphics. Cross-listed as MAE 672.

APMA 693 - (Credit as arranged) (SI)
Independent Study
Detailed study of graduate-level material on an independent basis under the guidance of a faculty member.

APMA 695 - (Credit as arranged) (Y)
Supervised Project Research
Formal record of student commitment to project research under the guidance of a faculty advisor. Registration may be repeated as necessary.

APMA 702 - (3) (Y)
Applied Partial Differential Equations I
Prerequisites: APMA 642 or equivalent
First order partial differential equations (linear, quasilinear, nonlinear). Classification of equations and characteristics. Well-posedness of initial and boundary value problems.

APMA 703 - (3) (Y)
Applied Partial Differential Equations II
Prerequisites: APMA 702, linear algebra, real analysis
Operator foundations for eigenfunction expansions for heat, wave, and plate problems. Cauchy-Kovalevsky and Holmgren theorems. Duhamel's Principle. Laplace equation and harmonic functions (mean value theorem, maximum principle, Harnack's inequality, etc.). Wave propagation (plane and spherical waves, energy estimates, Kirchhoff's formula and the method of descent, Huygens' principle). Maximum Principles.

APMA 706 - (3) (O)
Dynamical Systems and Control
Prerequisites: Undergraduate differential equations, linear algebra, real analysis or advanced calculus
Classical theory of ordinary differential and difference equations and some new directions that the subject has taken in recent years: existence and uniqueness of solutions, dependence on parameters, fundamental matrices, Floquet theory, Lyapunov stability, limit cycles, Poincare-Bendixson theorem, the invariance principle, limit sets, Lie theory and exact computer solutions using MATHEMATICA, chaotic dynamics, Lyapunov exponents, homoclinic points, controllability of linear and nonlinear equations, modern feedback theory of pole placement, state decoupling and reduction of control dimension.

APMA 708 - (3) (SI)
Inelastic Solid Mechanics
Prerequisite: AM 602
Primary emphasis on the formulation of a variety of nonlinear models. Specific topics to be discussed include nonlinear elasticity, creep, visco-elasticity, and elasto-plasticity. Solutions to boundary value problems of practical interest will be presented in the context of these various theories in order to illustrate the differences in stress distributions caused by different types of material nonlinearities. Cross-listed as AM 708.

APMA 714 - (3) (Y)
Nonlinear Elasticity Theory
Prerequisite: AM/APMA 602
This course describes the theory of finite (nonlinear) elasticity governing large deformations of highly deformable elastic solids. New features not present in the linear theory are emphasized. These include instabilities (both material and geometric), normal stress effects, non-uniqueness, bifurcations and stress singularities. A variety of illustrative boundary value problems will be discussed which exhibit some of the foregoing features. Both physical and mathematical implications will be considered. The results are applicable to rubber-like and biological materials and the theory serves as a prototype for more elaborate nonlinear theories of mechanics of continuous media. Cross-listed as AM 714.

APMA 734 - (3) (Y)
Numerical Solution of Partial Differential Equations
Prerequisite: One or more graduate courses in applied mathematics or mathematics
Numerical solution of elliptic equations by finite element methods. Solution of time dependent problems by finite element and finite difference methods. Stability and convergence results for the methods presented.

APMA 738 - (3) (SI)
Nonlinear Dynamics and Waves
Prerequisites: APMA/MAE 637 or permission of instructor
Application of phase plane methods, singular perturbation theory, inverse scattering, and related techniques to the study of free and forced nonlinear vibrations, nonlinear stability and bifurcations, and nonlinear wave motions. Examples will be drawn from mechanics and fluid dynamics. Cross-listed as MAE 738.

APMA 747, 748 - (3) (Y)
Selected Topics in Applied Mathematics
Prerequisite: Permission of instructor
Course content varies from year to year and depends on the interest and needs of our students. Topics suitable for a semester's work could be wave propagation theory, shell theory, control theory, or advanced numerical analysis, etc.

APMA 767 - (3) (SI)
Micromechanics of Heterogeneous Media
Prerequisite: APMA 602
Averaging principles, equivalent homogencity, effective moduli, bounding principles, self-consistent schemes, composite spheres, concentric cylinders, three phase model, repeating cell models, inelastic and nonlinear effects, thermal effects, isotropic and anisotropic media, strength and fracture. Cross-listed as AM 767, CE 767.

APMA 772 - (3) (Y)
Computational Fluid Dynamics II
Prerequisite: APMA 672 or equivalent
A continuation of APMA 672. More advanced methods for grid generation, transformation of governing equations for odd geometries, methods for compressible flows, methods for parabolic flows, calculations using vector and parallel computers. Use of personal computers/workstations/supercomputer including graphics. Cross-listed as MAE 772.

APMA 773 - (3) (O)
Combinatorics and Graph Theory
Prerequisite: Permission of instructor
Combinations and permutations, generating functions, Polya's counting theorem. Maximal matching theorems. Graph enumeration. Directed graphs, convex and acyclic subgraphs, graph topologies. Material may vary with interests of instructor. Cross-listed as CS 773.

APMA 775 - (3) (Y)
Introduction to Modern Partial Differential Equations I
Prerequisites: APMA 702, real analysis
The main goal of the one-year sequence APMA 775-776 is to present a modern theory of partial differential equations, based on functional analytic techniques which will focus on the qualitative theory (existence, uniqueness, regularity, stability, etc.) of large classes of well-established partial differential equations which arise in the engineering and physical sciences. Topics include the following. Distributions and Fourier Transform. Sobolev spaces and their properties (imbedding, compactness, trace theory, etc.). Weak solutions of elliptic boundary value problems including regularity of solutions, maximal principles. Introduction to Galerkin and Finite Element Methods.

APMA 776 - (3) (Y)
Introduction to Modern Partial Differential Equations II
Prerequisite: APMA 775 or equivalent
Introduction to semi-group theory of operators and applications to heat, wave, plate, Schrodinger equations. Some techniques from nonlinear analysis (fixed point theorems, monotone and variational methods), nonlinear elliptic equations, and nonlinear conservation laws.

APMA 792 - Credit as arranged (SI)
Independent Study
Detailed study of advanced graduate-level material on an independent basis under the guidance of a faculty member.

APMA 847, 848 - (3) (SI)
Advanced Topics in Applied Mathematics
Prerequisite: Permission of instructor
Course content varies from year to year and depends on interests and needs of our students. See APMA 747 for possible topics.

APMA 895 - (Credit as arranged) (S-SS)
Supervised Project Research
Formal record of student commitment to project research for Master of Applied Mathematics degree under the guidance of a faculty advisor. Registration may be repeated as necessary.

APMA 897 - (Credit as arranged) (S)
Graduate Teaching Instruction
For master's students.

APMA 898 - (Credit as arranged) (S-SS)
Formal record of student commitment to master's thesis research under the guidance of a faculty advisor. Registration may be repeated as necessary.

APMA 997 - (Credit as arranged) (S)
Graduate Teaching Instruction
For doctoral students.

APMA 999 - (Credit as arranged) (S-SS)
Formal record of student commitment to doctoral research under the guidance of a faculty advisor. Registration may be repeated as necessary.

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