### Curriculum

The PhD degree program requires a minimum of 26 semester hours of course work and 24 semester hours of dissertation research. The course work is comprised of the program core requirements and additional courses taken in the student's selected area of specialty.

#### Core Requirements

Each student must complete the core course requirements of the program totaling 11 credit hours. The program core has the following components:

- Mathematics - 6 credit hours
- Engineering or Science - 3 credit hours approved by both the SIUE and SIUC advisors
- Seminar - 2 credit hours

The two credit hours for the seminar, ENGR 580, must be taken over two semesters, one credit hour at a time. One of the two seminar credit hours must be taken before admission to candidacy and one after admission to candidacy.

#### Area of Concentration

In addition, a minimum of 15 credit-hours is required in the selected area of concentration to provide substantial depth relevant to the student's research interests.

No more than two courses or 6 credit hours of 400-level courses can be counted toward the requirements of the PhD degree.

ENGR 590 - Special Investigations course can only be used once for a maximum of 3 credit hours.

Applicants with master's degrees in computer science are encouraged to choose computer engineering specialization in the Cooperative PhD program.

For questions related to transfer credit please contact the Associate Dean for Research and Development.

#### Approved Mathematics Courses for the Program Core

MATH 420-3 Abstract Algebra

Standard algebraic structures and properties. Groups: subgroups, normality and quotients, isomorphism theorems, special groups. Rings: ideals, quotient rings, special rings. Fields: extensions, finite fields, geometric constructions. Prerequisite: MATH 320 or consent of instructor.

MATH 421-3 Linear Algebra II

Advanced study of vector spaces: Cayley-Hamilton Theorem, minimal and characteristic polynomials, eigenspaces, canonical forms, Lagrange-Sylvester Theorem, applications. Prerequisite: MATH 321 or consent of instructor.

MATH 423-3 Combinatorics and Graph Theory

Solving discrete problems. Counting techniques, combinatorial reasoning and modeling, generating functions and recurrence relations. Graphs: definitions, examples, basic properties, applications, and algorithms. Prerequisites: MATH 223; some knowledge of programming recommended.

MATH 435-3 Foundations for Euclidian and Non-Euclidian Geometry

Points, lines, planes, space, separations, congruence, parallelism and similarity, non-Euclidean geometries, independence of the parallel axiom. Riemannian and Bolyai-Lobachevskian geometries. Prerequisites: MATH 250; 321; MATH 320 or 350, consent of instructor.

MATH 437-3 Differential Geometry

Curve theory, surfaces in 3-dimensional space, fundamental quadratic forms of a surface, Riemannian geometry, differential manifolds. Prerequisite: MATH 250.

MATH 450-3 Real Analysis I

Differentiation and Riemann integration of functions of one variable. Taylor series. Improper integrals. Lebesgue measure and integration. Prerequisite: MATH 350.

MATH 451-3 Introduction to Complex Analysis

Analytic functions, Cauchy-Riemann equations, harmonic functions, elements of conformal mapping, line integrals, Cauchy-Goursat theorem, Cauchy integral formula, power series, the residue theorem and applications. Prerequisites: MATH 223; 250.

MATH 462-3 Engineering Numerical Analysis

Polynomial interpolation and approximations, numerical integration, differentiation, direct and iterative methods for linear systems. Numerical solutions for ODE's and PDE's. MATLAB programming required. Prerequisites: MATH 250; 305; CS 140 or 141, or consent of instructor. Not for MATH majors.

MATH 464-3 Partial Differential Equations

Partial differential equations; Fourier series and integrals; wave equation; heat equation; Laplace equation; and Sturm-Liouville theory. Prerequisites: MATH 250, 305, and 321.

MATH 465-3 Numerical Analysis

Error analysis, solution of nonlinear equations, interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, solution of linear systems of equations. Prerequisites: MATH 305; CS 140 or 141.

MATH 466-3 Numerical Linear Algebra with Applications

Direct and iterative methods for linear systems, approximation of eigenvalues, solution of nonlinear systems, numerical solution of ODE and PDE boundary value problems, function approximation. Prerequisites: MATH 305; 321; CS 140 or 141.

MATH 501-3 Differential Equations and the Fourier Analysis

Brief review of ODE. Legendre and Bessel functions. Fourier series, integrals, and transforms. Wave equation, heat equation, Laplace equation. Not for MATH majors. Prerequisite: MATH 250, MATH 305, or consent of instructor.

MATH 502-3 Advanced Calculus for Engineers

Review of vector calculus, Green's theorem, Gauss' theorem, and Stokes' theorem. Complex analysis up to contour integrals and residue theorem. Not for MATH majors. Prerequisite: MATH 250 or consent of instructor.

MATH 545-3 Real Analysis II

Riemann, Riemann-Stieltjes, and Lebesgue integrals. Differentiation of functions of n variables. Multiple integrals. Measure and probability. Differential forms, Stokes' Theorem. Prerequisites: MATH 321 and 450.

MATH 552-3 Theory of Ordinary Differential Equations

Existence and uniqueness theorem, dynamical systems, stability, bifurcation theory, boundary value problems. Prerequisites: MATH 350; 421.

MATH 555-3 Functional Analysis with Applications

Normed and Banach spaces, inner product and Hilbert spaces, Open Mapping and Closed Graph Theorem, Hahn-Banach Theorem, dual spaces and weak topology. Prerequisite: MATH 421, 450.

MATH 563-3 Optimal Control Theory (Same as ECE 563 and ME 563)

Description of system and evaluation of its performance; dynamic programming, calculus of variations and Pontryagin's minimum principle; iterative numerical techniques. Prerequisite: MATH 305 or ECE 365 or ME 450.

MATH 565-3 Advanced Numerical Analysis

Rigorous treatment of topics in numerical analysis including function approximation, numerical solutions to ordinary and partial differential equations. Convergence and stability of finite difference methods. Prerequisites: MATH 321; 350; 465; 466.

#### Engineering or Science Courses for the Program Core

Core courses are approved on a case-by-case basis. Courses may be taught by faculty at SIUC and made available at SIUE through distance education and other means. Other courses may also be taken to satisfy the Engineering or Science Core requirements subject to approval of the advisor.