Partial Differential Equations, Spring 2020

Mon 5/4: Sketch of distribution theory and fundamental solutions continued.

Thu 4/30: Sketch of distribution theory and fundamental solutions.

Mon 4/27: Maximum principles for the heat equation.

Thu 4/23: Laplace’s equation and maximum principles.

Thu 4/16: Laplace’s equation and maximum principles.

Mon 4/13: Fourier Series III.

Thu 4/9: Fourier Series II.

Mon 4/6: Fourier Series I.

Thu 4/2: More on Hilbert spaces.

Mon 3/30: Convergence and Completeness.

Thu 3/26: Function spaces and Lp.

Mon 3/23: Fundamental solutions and Duhamel’s Principle in general.

Welcome back from spring break and to the now online version of this class! Class meetings will take place by zoom (you should have recieved a Canvas announcement with the details).

Thu 3/5: The heat equation continued.

Mon 3/2: Exam 1 problems.

Problems for Exam 1

Sign up for an oral exam time. Please email me if you have trouble with the calendar link or are not able to find a slot that works in your schedule.

Mon 3/2: The heat equation.

Thu 2/27: Separation of variables: Spherical coordinates.

Mon 2/24: Separation of variables: Polar and Spherical coordinates.

Thu 2/20: Addendum.

Thu 2/20: Separation of variables.

Wed 2/19: Homework hint for 4.7.

Based on some questions in office hours about 4.7, here is the strategy. Just like we defined energy for the wave equation and showed that it was conserved (i.e., its time derivative was 0), you are meant to define a quantity for the Schrodinger equation and show it is conserved. Here the quantity is the integral over space of the absolute value squared of u.

It is important to note that u here is complex valued, so this absolute value squared is u times its complex conjugate, and likewise for the initial condition g.

Mon 2/17: The wave equation in 3D; separation of variables.

Thu 2/13: The wave equation continued: more space dimensions, energy and uniqueness.

Mon 2/10: The wave equation continued: boundary conditions and forcing terms.

Thu 2/6: The wave equation.

Mon 2/3: First order equations: conservation equations and characteristics continued.

Thu 1/30: First order equations: conservation equations and characteristics.

Mon 1/27: Introduction and generalities.