Math 5587      Elementary Partial Differential Equations       Fall, 2004

Professor Peter J. Olver
School of Mathematics
Institute of Technology
University of Minnesota
Minneapolis, MN 55455 		540 Vincent Hall
Phone: 612-624-5534
Fax: 612-626-2017
e-mail: olver@math.umn.edu
http://www.math.umn.edu/~olver
Lectures:    T,Th 4:45-6:00pm    EE/CSCI 3125
Office Hours:   T 3:30-4:30, F 11:15-12:15, or by appointment

Course Description:   Math 5587-8 is a year course that introduces the basics of partial differential equations, guided by applications in physics and engineering. Both analytical and numerical solution techniques will be discussed. Specific topics to be covered during the year include, in rough order:

Classification of PDEs; the heat, wave, Laplace, Poisson and Helmholtz equations; characteristics; the maximum principle; separation of variables; Fourier series; harmonic functions; distributions; Green's functions and fundamental solutions; special functions, including Bessel functions and spherical harmonics; finite element method; Fourier and Laplace transforms; nonlinear PDEs; shocks and solitons. Choice of supplementary topics and applications will depend on the interests of the class.

Prerequisites: Strong background in linear algebra, multi-variable calculus and ordinary differential equations (3000 level). Some mathematical sophistication. Other topics will be introduced as needed.

Text:   Walter A. Strauss, Partial Differential Equations: an Introduction, John Wiley & Sons, New York, 1992.

Supplementary Text:   Richard Haberman, Elementary Applied Partial Differential Equations, Third Edition, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1998.

Syllabus for Fall Semester:

  1. Quick review of ordinary differential equations
  2. Solution of first order linear PDEs (Strauss 1.1, 1.2)
  3. The one-dimensional wave equation: quick derivation, d'Alembert formula, conservation of energy, finite intervals and reflection of waves (Strauss 1.3, 2.1, 2.2, 3.2)
  4. The one-dimensional heat equation: derivation, separation of variables (Strauss 1.3, 4.1)
  5. Fourier series: basic properties, convergence, Gibbs phenomena, differentiation, delta functions and distributions (Strauss 5.1--5, 12.1)
  6. Solution to heat and wave by separation of variables (Strauss 4.1, 4.2)
  7. Fourier transforms (Strauss 12.3)
  8. More on the heat equation: fundamental solution, maximum principle, well-posedness (Strauss 1.5, 2.3--5)
  9. Laplace equation in two dimensions: separation of variables for rectangle and disk, classification of 2D PDEs -- elliptic, parabolic and hyperbolic (Strauss 6.1--3, 1.6)
  10. (time permitting) Numerical methods (Strauss 8.1--5)

Homework:   Each assignment will consist of problems from the text. Assignments handed out on a Tuesday will be due the following Tuesday.

Hour Exams:   There will be two midterm exams. Make-up exams will only be given in exceptional circumstances, and then only when notice is given to me before hand and a suitable written excuse forthcoming.

First Midterm:    Thursday, October 28. Will cover sections 1.2, 2.1, 2.2, 3.2, chapter 5, 12.1.

Second Midterm:    Thursday, December 2. Will cover sections 2.3, 2.4, 2.5, 4.1, 4.2, 12.3, 12.4.

Take Home Final Exam:   Due: Thursday, December 16, 6:00pm

Grading:

Incompletes:   Only given in extreme circumstances, and only when the student has satisfactorily completed all but a small portion of the work in the course. Students must make prior arrangements with the professor well before the end of the quarter.

Grading Standards and Student Conduct:   Students are expected to be familiar with University of Minnesota policies on grading standards and student conduct, including the consequences for students who violate standards of academic honesty.

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