Mon 5/4: Sketch of distribution theory and fundamental solutions continued.
- Reading: Various parts of Chapter 12
- Homework: No more homework problems. Take home final exam will be distributed on Wednesday.
Thu 4/30: Sketch of distribution theory and fundamental solutions.
- Reading: Various parts of Chapter 12
- Homework: No more homework problems. Take home final exam will be distributed next week.
Mon 4/27: Maximum principles for the heat equation.
- Reading: 9.4
- Homework: (Due Thursday 4/30): No new problems.
Thu 4/23: Laplace’s equation and maximum principles.
- Reading: 9.2, 9.3
- Homework: (Due Thursday 4/30): 9.1, 9.3.
Thu 4/16: Laplace’s equation and maximum principles.
- Reading: 9.1, 9.2
- Homework: (Due Thursday 4/30) 9.2.
Mon 4/13: Fourier Series III.
- Reading: 8.5, 8.6.
- Homework: no new problems.
Thu 4/9: Fourier Series II.
- Reading: 8.2, 8.3, 8.4
- Homework: (Due Thursday 4/16): 8.3, 8.4.
Mon 4/6: Fourier Series I.
- Reading: 8.1, 8.2, 8.3
- Homework: (Due Thursday 4/9): 8.1, 8.2 (same problems).
Thu 4/2: More on Hilbert spaces.
- Reading: 7.6, 8.1, 8.2
- Homework: (Due Thursday 4/9): 8.1, 8.2.
Mon 3/30: Convergence and Completeness.
- Reading: 7.4, 7.5.
- Homework: (Due Thursday 4/2): 7.5, 7.8.
Thu 3/26: Function spaces and Lp.
- Reading: 7.1, 7.2, 7.3.
- Homework: (Due Thursday 4/2): 7.1-7.3.
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).
- Reading: No specific chapters; I will review and recontextualize some material from 4.2, 4.4, 6.3, and 6.4.
- Homework: None due this week.
Thu 3/5: The heat equation continued.
- Reading: Chapter 6.3, 6.4.
Mon 3/2: Exam 1 problems.
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.
- Reading: Chapter 6.1, 6.2.
Thu 2/27: Separation of variables: Spherical coordinates.
- Reading: Chapter 5.4.
Mon 2/24: Separation of variables: Polar and Spherical coordinates.
- Reading: Chapter 5.3, 5.4.
- Homework (due Thursday 2/27): 5.3, 5.4, 5.6.
Thu 2/20: Addendum.
- If you want to play around with the overtone series waveform generator, here is the link!
Thu 2/20: Separation of variables.
- Reading: Chapter 5.1, 5.2, 5.3.
- Homework (due Thursday 2/27): 5.1, 5.2 (these problems will not be due on 2/20 since we didn’t start Chapter 5 on Monday)
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.
- Reading: Chapter 4.6, 5.1,
5.2, 5.3. - Homework (due Thursday 2/20):
5.1, 5.2.
Thu 2/13: The wave equation continued: more space dimensions, energy and uniqueness.
- Reading: Chapter 4.5, 4.7, 4.6
- Homework (due Thursday 2/20): 4.5, 4.7.
Mon 2/10: The wave equation continued: boundary conditions and forcing terms.
- Reading: Chapter 4.3, 4.4.
- Homework (due Thursday 2/13 in addition to previous problems): 4.4.
Thu 2/6: The wave equation.
- Reading: Chapter 4.1, 4.2, 4.3.
- Homework (due Thursday 2/13): 4.1, 4.2, 4.3.
Mon 2/3: First order equations: conservation equations and characteristics continued.
- Reading: Chapter 3.3, 3.4.
- Homework (due Thursday 2/6 along with previous problems): 3.4,
3.53.6.
Thu 1/30: First order equations: conservation equations and characteristics.
- Reading: Chapter 3.1, 3.2.
- Homework (due Thursday 2/6): 3.1, 3.2, 3.3.
Mon 1/27: Introduction and generalities.
- Reading: Chapter 1
- Homework: No homework problems yet!