Syllabus for MATH 670, SECTION 600, Fall 2004

Applied Math I

Office hours: MW 3:00-4:00; R 1:00-2:00; By appointment

Class time and place: MWF 1:50-2:40, Engineering/Physics Building (ENPH) 215.

Section web page: /~phoward/M670.html

Textbook: There is no textbook for the course, though the following texts have been set on reserve in Evans library:  F. W.  J.  Olver,  Introduction  to  asymptotics  and  special  functions,  Academic Press  1974.  E. J. Hinch, Perturbation Methods, Cambridge University Press 1991. N. G. DeBrujn, Asymptotic methods in analysis, Interscience Publishers 1961.

Prerequisites: M642 or consent of instructor.

Catalogue Description:  OLD: Mathematical tools of applied mathematics; Fredholm alternative; integral operators; Green's functions; unbounded operators; Stokes' theorem; distributions; convolutions; Fourier transforms; applications.

NEW: Math 670 is devoted to topics in the application of modern analysis to a wide range of applications in engineering, the physical sciences, the social sciences and the life sciences. The specific mathematical topics covered will vary from year to year but will be chosen from: partial differential equations, dynamical systems, integral equations and variational calculus. Examples of the application areas to be considered include: distributed control, inverse scattering, inverse problems for partial differential equations, tomography, fluid dynamics, solid mechanics, biology, ecology and physiology. The course can be taken three times for credit.

Course Goal: Our primary interests will be asymptotic analysis and perturbation methods, with special attention given to applications of these methods to ordinary and partial differential equations. Time permitting, we will also discuss relevant variational methods.

Homework Assignments: Homework assignments will be made each Friday, due the following Friday. Assignments will depreciate by 5 points for each class period they are late for up to one week, at which time a 0 will be assigned.

Exams: There will be two exams during the semester, one evening midterm and a comprehensive final.  The midterm exam will be Wednesday, October 20.  Please make a note that the final exam for this class will be on Tuesday, December 14, 3:30--5:30.

Grades: Final grades will be determined in the following manner: Homework: 50%; Exams: 25% each. Grade ranges will be standard: 89.50-100, A; 79.50-89.49, B; 69.50-79.49, C, 59.50-69.49, D; below 59.50, F.

Make-up policy: Make-ups for exams will only be given if the student can provide a documented University-approved excuse (see University Regulations). According to University Student Rules students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise the student forfeits his or her right to a make-up.

Scholastic Dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. "An Aggie does not lie, cheat, or steal or tolerate those who do." Please refer to the Honor Council Rules and Procedures, available at the Office of the Aggie Honor System.

Copyright policy: All printed materials disseminated in class or on the web are protected by copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.

Students with Disabilities: The following statement was provided by the Department of Student Life: The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accomodation of their disabilities.  If you believe you have a disability requiring an accomodation, please contact the Department of Student Life, Services for Students with Disabilities (SSD), in Room 126 of the Koldus building or call 845-1637.

Class Schedule: Roughly speaking, we should cover the following material on the following schedule:

Week of Monday	Material Covered
August  30	Algebraic perturbation theory
September 6	(Basic) Perturbation methods for ODE and PDE
September 13	Poincare's method of strained coordinates, multi-scale methods
September 20	Singular problems, boundary layer methods
September 27	WKB approximation
October 4	Topics in perturbation theory
October 11	Transform methods, integral representations
October 18	Asymptotics of real integrals (Riemann--Lebesque, Laplace's method) Midterm Wed. Oct. 20
October 25	Asymptotics of complex integrals (Watson's lemma, steepest descent methods, stationary phase)
November 1	Application to the Airy equation, WKB revisited
November 8	Applications to ODE dynamics
November 15	Applications to PDE dynamics
November 22	Introduction to energy and variational methods (Thanksgiving break, no class Fri. Nov. 26)
November 29	Minimax methods, and the Rayleigh--Ritz method
December 6	The mountain pass lemma and applications