Math 5588 – Elementary Partial Differential Equations II

Introduction

Partial differential equations (PDE) are ubiquitous in science, engineering and physics. Applications include traffic flow, heat diffusion, wave propagation, quantum mechanics, computer vision, image processing, and optimal control theory, among many others. This course is the second part of a basic introduction to partial differential equations. Topics to be covered include calculus of variations, Fourier Transform, Green's functions and fundamental solutions, maximum principle, finite elements, waves, diffusion, scalar conservation laws, viscosity solutions, and other selected topics as time permits. The first part of this year-long course is Math 5587, which is offered in fall semesters.

The animation above is a numerical simulation of the mean curvature motion PDE, which moves a curve in the plane in the direction of its inward normal vector with a speed equal to its curvature. Mean curvature motion arises in physical systems that involve surface tension, such as soap film/bubbles and biological cell membranes, and many other fields of pure and applied mathematics. It is possible to prove that mean curvature motion evolves any simple closed curve into a convex curve, and then collapses it to a point. Along the way, the curve becomes asymptotically close to a circle.

Course Information

Instructor	Jeff Calder (Office: 538 Vincent Hall, Email: jcalder at umn dot edu)
Lectures	Tue and Thu, 4:45pm-6pm in 207 Vincent Hall
Office Hours	Mon 3pm-4pm, Tue 2:30pm-3:30pm, Wed 2pm-3pm in 538 Vincent hall
Final Exam	The final exam is schedule for Tuesday, May 9, 4:45pm-6:45pm
Midterm Exams	We will have two in-class midterm exams, scheduled for February 16th and March 30th. All midterms and exams are closed book.
Piazza	We have a Piazza website for student discussions. To sign up, click here. Rather than emailing questions to the instructor, students are encouraged to post questions on Piazza, and to participate in the discussion.
Moodle	We also have a Moodle page, which will be primarily used for posting grades. Keep in mind the grades reported on Moodle are raw scores, and each exam will have a curve that is used to assign letter grades.
Required Textbook	Olver, Peter J. Intro to Partial Differential Equations (2014) SpringerLink PDF
Recommended Textbooks	Strauss, Walter A. Partial Differential Equations: An Introduction. Shearer, M. and Levy, R. Partial Differential Equations: An Introduction to Theory and Applications.
Grades	Your final grade will be based on homework assignments (30%), midterm I (20%), midterm II (20%), and the final exam (30%). A higher score on the final exam counts in place of any lower midterm score.
Readings	Readings will be assigned on a weekly basis and posted on the schedule page on this website. It is very important to do the readings before attending the associated lecture. Additional lecture notes will be posted periodically on the schedule page under the "reading" column.
Homework	There will be weekly homework assignments posted on this website that will be due in class every Thursday. Collaboration on homework is encouraged, but you must write up the solutions in your own words and cite any class members that you worked with. Late homework will not be accepted for any reason. Your overall homework grade will be the average of your highest 9 homework scores.
Matlab	At least one homework assignment will involve some numerical computations in Matlab. All computers in Vincent Hall have Matlab, Mathematica, and Maple installed. All Linux lab computers in the college should have the same software. This includes the labs in Vincent Hall 5 (when no class is in session) and 270D. You can also download Matlab on your personal computer with a University license. The software as well as instructions are available here. Before downloading the software, you will need a CSELabs account, which you can get here. There is also a great open source alternative to Matlab called GNU Octave. The software is free and open source, and can be downloaded here.
Academic Honesty	The School of Mathematics at the University of Minnesota expects that students in mathematics courses will not engage in cheating or plagiarism. Cheating, plagiarism, and other forms of academic dishonesty will result in a grade of zero on the homework assignment or exam in question, and, in severe cases, a failing grade in the course and a referral to the Office for Student Conduct and Academic Integrity (OSCAI). Students should be familiar with the Student Code of Conduct.