Lectures
On this page you'll find lecture recordings and summaries.
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On the rare occasions when I'm a bit slow to upload recordings, you should also be able to find them via the PDEs Blackboard > Tools > Panopto.
Lecture 20: 2024-12-06 12:10. Harmonic Functions II
Lecture 19: 2024-12-05 11:10. Harmonic Functions I
Lecture 18: 2024-11-29 12:10. More Fourier transforms
- Finishing the example of Laplace's equation in a strip.
- Solving the heat equation on an infinitely long rod.
Unfortunately, Panopto failed 33 minutes through the lecture, so the lecture's last part was not captured.
Lecture 17: 2024-11-28 11:10. Fourier transforms continued
- Functional spaces.
- Application of Fourier transforms to solve the Laplace equation in a strip.
Lecture 16: 2024-11-22 12:10. Fourier transforms
- Fourier transforms and their properties.
Lecture 15: 2024-11-21 11:10. Uniqueness of heat equation solutions
- Proof of maximum principle.
- Proof of uniqueness of heat equation solutions.
- Brief introduction to integral transforms.
Much of this lecture was on the whiteboard as the lecture room equipment malfunctioned. Only the first 15 minutes survives on tape; the riveting remainder exists only in our memories (and hopefully your lecture notes).
Lecture 14: 2024-11-15 12:10. Maximum principle
- The heat equation.
- Maximum principle proof.
Lecture 13: 2024-11-14 11:10. Finite string cont'd
- Uniqueness of wave equation Cauchy problem solutions.
Lecture 12: 2024-11-08 12:10. Wave equation on finite string
- Wave equation on a finite length string.
- Sturm-Liouville problem.
- Seeking separable solutions
Lecture 11: 2024-11-07 11:10. Inhomogeneous wave equation cont'd
- Inhomogeneous wave equation with inhomogeneous initial conditions.
Lecture 10: 2024-11-01 12:10. Duhamel's principle
- Reflection principle for semi-infinite string (continued).
- Deriving Duhamel's principle.
- In the proof, we need the Leibniz Integral Rule. While I'll never ask you to prove this rule, if you're interested in proving it, see for example its Wikipedia page or any decent Calculus textbook.
Lecture 9: 2024-10-31 11:10. Reflection and Causality
- Reflection principle for semi-infinite string.
- Causality: domains of dependence and influence.
Lecture 8: 2024-10-25 12:10. d'Alembert formula, Causality
- d'Alembert's equation
- Causality - domains of influence and dependence.
Lecture 7: 2024-10-24 11:10. The wave equation
(Don't be fooled by the fact it says Lecture_06 at the top of the recording; it really is Lecture 7!)
- Classification of second order PDEs and reduction to canonical form.
- The wave equation.
- General solution of the wave equation.
Lecture 6: 2024-10-18 12:10. Ill-posed problem. 2nd order classification
- An ill-posed problem.
- Classification of second order PDEs.
Lecture 5: 2024-10-17 11:10. Characteristics - the general case
- The method of characteristics: the general case.
Lecture 4: 2024-10-11 12:10. First order linear PDEs continued
- Boundary conditions.
- The method of characteristics: the variable coefficients case.
- The method of characteristics: the general case.
Lecture 3: 2024-10-11 09:10. First order linear PDEs
- Directional derivative.
- Solving first order linear PDEs: the method of characteristics - the constant coefficients case.
Lecture 2: 2024-10-10 11:10. Classification, Simple PDEs
- Classification: order.
- Classification: degree.
- Some simple PDEs.
Apologies for the rubbish audio quality - the mic ran out of batteries half way through and Panopto seems to have fallen back onto the webcam mic that makes it sound like I'm underwater.
Lecture 1: 2024-10-04 12:10. Introduction
- Introduction and course structure.
- Notation.
- Classification: linearity.