Math 272: Introduction to PDE's

Fall 2015

Course Description

A Partial Differential Equation (PDE for short), is a differential equation involving derivatives with respect to more than one variable. These arise in numerous applications from various disciplines. A prototypical example is the heat equation, governing the evolution of temperature in a conductor.

This course will serve as a first introduction to PDE’s, and will focus on the simplest model equations that arise in real life. It will study both analytical methods (e.g. separation of variables, Greens functions), numerical methods (e.g. finite elements) and the use of a computer to compute and visualize solutions. Time permitting, it will touch upon the mathematical ideas behind phenomenon observed in real life (e.g. speed of wave propagation, and or shocks in traffic flow).

Tentative Syllabus

• Physical motivation of PDE’s
  • Transport, heat, wave and Laplace equations.
  • Initial data, boundary conditions and classification of PDE’s.
• Separation of variables and Fourier series.
  • Heat, wave and Laplace equations.
• Numerics and computation.
  • Ideas behind finite differences and finite elements.
  • Using software to compute (e.g. MATLAB)
• Newton potentials and Greens functions.
  • Computation in standard domains.
  • Harmonic functions: Maximum principle and mean value property.
• Heat kernel.
• Wave equation and finite speed of propagation.
• Burgers equation, characteristics and traffic flow.

Prerequisites

Before taking this course you should have a good knowledge of ODE’s and multi-variable calculus. Few courses that cover these in sufficient detail are:

• ODE’s: 33-232, 21-260 or 21-261.
• Multi-variable calculus: 21-259, 21-268 or 21-269.

It isn’t imperative you have taken these courses; but it is imperative that you have a good understanding of the material in these courses.

References

• 372 Lecture Notes Wiki
• Introduction to Partial Differential Equations with MATLAB Corrected Edition by Jeffery M. Cooper.
• Numerical Analysis of Partial Differential Equations by Charles A. Hall and Thomas A. Porsching.
• An Introduction to Partial Differential Equations 1st Edition by Yehuda Pinchover, Jacob Rubinstein.
• Introduction to PDE by Walter Strauss.

Class Policies

Lectures

• If you must sleep, don’t snore!
• Be courteous when you use mobile devices.

Homework

• Homework solutions will be presented by students every week.
• You may work together on the homework, and coordinate amongst yourselves regarding who presents what solution. The presentation itself will not count towards your grade.
• 50% of your exams will consist of homework questions, so it is in your best interest to understand solutions to all questions.

Exams

• All exams are closed book, in class.
• No calculators, computational aids, or internet enabled devices are allowed.
• The final time will be announced by the registrar here. Be aware of their schedule before making your travel plans.

Grading

• Between 25% and 50% of your exams will consist of homework questions.
• Your better midterm will count as 30% of your grade. (Your worse midterm will not count at all.)
• Your final will count as 50% of your grade.
• Projects will count as 20% of your grade. (If, we do not cover enough material in time to successfully complete projects, then your midterm and final percentages will be scaled up appropriately.)