MATH 400, Section 101
Session 2020W, Term 1 (Sep-Dec 2020)
Course instructor's page
Last update: 2020 Nov 30
(
preliminary lecture 35, for Dec 2
)
- Course Canvas page.
- Syllabus.
- Recommended textbook (not required): W.A. Strauss, Partial Differential Equations: An Introduction (2008).
- See the course Canvas page for more references.
Lectures: M W F 09:00--09:50 Vancouver BC Canada local time, on the course Canvas page (via Zoom). First lecture Wed Sep 9 09:00 PDT (UTC -7)
Instructor: Wayne Nagata
Email: nagata(at)math(dot)ubc(dot)ca
Office: online (via Canvas/Zoom)
Office hours: M W F 13:00-13:50 or by appointment
Preliminary Lecture Notes
These are intended be available before each online lecture so you can pre-read the material.
(Page numbers in parentheses refer to the recommended textbook.)
After the lecture, updated notes will be posted on the course Canvas page, along with recordings. SUGGESTION: print out these preliminary notes and write on them by hand during the online lectures. Doing something to keep yourself engaged will help with your learning.
- 1. for Wed Sep 9 [Simple transport (p.10), linear equations (p.6)]
- 2. for Fri Sep 11 [Solving the simple transport equation: characteristics, coordinates... (p.6)]
- 3. for Mon Sep 14 [..., general solution, signalling problem, generalized solutions (p.7)]
- 4. for Wed Sep 16 [Solving linear equations: characteristics, coordinates... (p.8)]
- 5. for Fri Sep 18 [..., general solution, things that can go wrong]
- 6. for Mon Sep 21 [Linear Second Order Equations: types of second-order equations (p.28)]
- 7. for Wed Sep 23 [Vibrating string (p.11), the wave equation (p.33)]
- 8. for Fri Sep 25 [Initial value problem (p.35), superposition (p.2)]
- 9. for Mon Sep 28 [Causality (p.39), waves with a source (p.71), reflections of waves (p.61)...]
- 10. for Wed Sep 30 [..., Heat flow (p.16)]
- 11. for Fri Oct 2 [Separation of variables, boundary conditions, Sturm-Liouville problems...]
- 12. for Mon Oct 5 [..., Sturm-Liouville theory]
- 13. for Wed Oct 7 [Eigenfuncion expansions...]
- 14. for Fri Oct 9 [...]
- 15. for Wed Oct 14 [..., nonhomogeneous problems]
- 16. for Fri Oct 16 [Steady state temperature, Laplace's equation, Laplace's equation in polar coordinates]
- 17. for Mon Oct 19 [Laplace's equation in spherical coordinates, rotationally invariant harmonic functions, Poisson's formula...]
- 18. for Wed Oct 21 [..., mean value property]
- 19. for Fri Oct 23 [examples (wedge, annulus)]
- 20. for Mon Oct 26 [PDEs in higher dimensions, Green's first identity, vibrations of a drumhead...]
- 21. for Wed Oct 28 [...]
- 22. Fri Oct 30: Midterm Test [no lecture notes]
- 23. for Mon Nov 2 [...]
- 24. for Wed Nov 4 [eigenvalues of laplacian in a ball...]
- 25. for Fri Nov 6 [..., spherical harmonics...]
- 26. for Mon Nov 9 [..., Dirichlet problem in a ball...]
- 27. for Fri Nov 13 [..., Fourier transforms...]
- 28. for Mon Nov 16 [..., source function for diffusion/heat equation]
- 29. for Wed Nov 18 [Fourier transforms and PDEs...]
- 30. for Fri Nov 20 [..., Laplace transforms...]
- 31. for Mon Nov 23 [..., Laplace transforms and PDEs...]
- 32. for Wed Nov 25 [...]
- 33. for Fri Nov 27 [..., the complex inversion formula]
- 34. for Mon Nov 30 [modal truncation]
- 35. for Wed Dec 2 [review/learning outcomes]
Supplementary Material
- Vibrating drumhead article in Wikipedia, see the animations of normal modes of vibration (spatial eigenfunctions, oscillating sinusoidally in time) at the bottom of the article. (Notation in the Wikipedia article differs from the notation used in our lectures and textbook.)
- Vibrating drumhead video on Youtube. The first normal mode seen is n=0, m=1 in notation of lectures. Then n=1, m=1, etc. (a useful exercise: identify the n, m for the normal modes seen in the video).
- For amusement: normal modes and music video on Youtube. Not exactly the vibrating drumhead, but the Chladni plate shown in the video is related. The sand collects on the nodal set of the normal mode of vibration, which doesn't move, see p. 279 in the textbook.
- To see a "normal mode" of vibration in a real physical system (with a little bit of damping), you drive the system at (or actually, near, with the exact amount determined by the damping) the pure frequency of the desired normal mode, so that the system resonates in that normal mode, and the other normal modes are damped out enough so you don't notice their effects. By using the wave equation to model the system, the mathematics is simplified(!) but we lose the damping effects. In a previous course, you probably saw the relationship between a forced simple harmonic oscillator without damping and one with light damping; here the idea is similar but how the details work out in a more accurate model that includes damping are more complicated. The model can be made even more realistic by including nonlinear effects. To get a bit philosophical: a goal of applied mathematics is to study mathematical models that are simple enough so that the mathematics can be solved, but are realistic enough so that the models introduce concepts that are useful for understanding and organizing observations of reality (or "more realistic" numerical computer simulations), useful enough to make sufficiently accurate predictions, qualitative or quantitative. From the vibrating drumhead Youtube video, we can see that our "simple" wave-equation model of a vibrating elastic membrane without damping does a reasonably good job of describing a real one.
Updated Lecture Notes and Recorded Lectures
See the course Canvas page.
Learning Outcomes
By the end of the course, a student should be able to:
- write clear explanations in English of solutions to mathematical problems, showing logical steps and arguments as well as mathematical expressions;
- for a mathematical model, interpret the mathematical solution in clear English in terms of the physical process modelled;
- adapt modelling ideas from the lectures or textbook to derive mathematical models for new situations;
- find the general solution for a linear first order PDE;
- solve a Cauchy problem for a linear first order PDE, and explain precisely where a unique solution exists, and where and why a unique solution does not exist;
- classify (subsets for) a semilinear second order PDE as elliptic, hyperbolic, or parabolic;
- transform a hyperbolic equation with constant coefficients into its canonical form and find the general solution;
- find the general solution and solve an initial value problem or initial-boundary value problem for a homogeneous or nonhomogeneous hyperbolic equation, in an unbounded 1-dimensional spatial domain;
- solve a regular or periodic Sturm-Liouville problem;
- apply definitions and theorems from Sturm-Liouville theory;
- solve an initial-boundary value problem for a parabolic or hyperbolic homogeneous or nonhomogeneous linear PDE in a bounded 1-dimensional spatial domain;
- solve a boundary value problem or an eigenvalue problem for an elliptic homogeneous or nonhomogeneous linear PDE in a bounded 2- or 3-dimensional spatial domain;
- solve an initial-boundary value problem for a parabolic or hyperbolic homogeneous or nonhomogeneous linear PDE in a bounded 2- or 3-dimensional spatial domain;
- solve a boundary value problem for an elliptic linear PDE in an unbounded 2-dimensional spatial domain, or an initial-boundary value problem for a parabolic or hyperbolic linear PDE in an unbounded 1-dimensional spatial domain;
- solve a PDE problem by applying a change of variables.
Assessment of Learning Outcomes (Grading)
See the course Canvas page.
Homework
See the course Canvas page.
Midterm Tests
See the course Canvas page.
Final Examination
See the course Canvas page.