ChEg 542

Syllabus

Fall 1998

 

Content: This is an applied math course that is intended to serve the dual role of being the only graduate mathematics class for many students (suggesting a need for breadth) but just the first of many for some students (suggesting a need for depth ... .) It will be application motivated and problem based. We will prove things only when it gives us special insight into something and thus we will assume a lot.

Topics will include a comprehensive introduction to linear algebra, a selected treatment of ordinary differential equations as they apply to important problems in engineering and science and a brief introduction to partial differential equations that arise in science and engineering. Every attempt is made to unify the linear operator concept throughout the topics of this course.

Instructor: M. J. McCready, 182A Fitzpatrick, 631-7146, mjm@nd.edu

Office Hours: stop by, or by appointment, (we may arrange a fixed group time) or you usually can get fast responses from me by email. I find that taking the time to type the question often helps to clarify the issue.

Teaching Assistant: Richard Huang, A56 Fitzpatrick, xhuang@darwin.helios.nd.edu

Text: A. Varma and M. Morbidelli

Mathematical Methods in Chemical Engineering Oxford (1997)

 

Course Grading:

Homework

25%

 

Hour tests (2)

35% (tentatively, 10/8, 11/19)

 

Final Exam

40% (12/17, 10:30 AM)

 

Homework: Homework problems are assigned as they arise in lecture. They are collected on Thursday (which allows either 7 or 9 days to complete each problem). You may discuss homework with your classmates. However, the work that you turn in should represent your understanding of the problem.

Reserve Books: See the list at http://www.nd.edu/~engrlib/reserve/inst.htm . These are available in the engineering library.

 

Topics

 

I. Applications of linear algebra (with an introduction to Mathematica)

1. Basic linear algebra:

matrix definition, transpose, symmetry, index notation, matrix multiplication, determinant, LaPlace expansion, cofactors and minors, identity matrix, rank, systems of linear algebraic equations, inverse matrix, Gaussian elimination, LU decomposition,

2. Applications of basic linear algebra

dimensional analysis, (Buckingham Pi theorem), Linear Programming, Linear least squares

3. Linear spaces and linear operators:

linear dependency, norm, inner product, adjoints, bases, orthogonality, linear transformation

4. Eigenvalue problems:

Distinct, repeated, generalized eigenvalues, eigenvectors, biorthogonality with adjoint, solution of algebraic equations by expansion of eigenvectors, similarity transform, quadratic forms

 

II. Linear ordinary differential equations

1. Initial value problems

Linearity, superposition, existence, uniqueness, inhomogeneous equations, adjoint differential operator

2. Boundary value problems

boundary conditions, Fredholm alternative and existence of solutions, inhomogeneous boundary value problems, Green's functions,

3. Differential eigenvalue problems

Properties of eigenvalues, dimension of operator, self adjoint problems, Eigenfunction expansions, Finite Fourier Transform, Series Solutions, Spectral numerical solutions,

 

III. Partial differential equations

1. Introduction to PDEs

Order and classification, separation of variables, similarity solutions, other classic solution techniques

2. General solution techniques

Finite Fourier transform