- 7/3 We looked at the wave equation with Dirichlet-0 boundary conditions, see eq (5.4). We derived particular solutions and formal solutions to the PDE, obtained stability and uniqueness of sufficiently smooth solutions through an energy argument. Then we described an explicit numerical scheme for the PDE and derived a CFL condition under which the scheme is Neumann stable. The lecture covered content from TW 5.1-5.3.
- 8/3 We solved the exercises described in the course schedule.
- 14/3 We derived maximum principles for 1D ODE boundary-value problems and the linear heat equation with Dirichlet boundary values following TW 6.1 up to and including 6.2.3 and a maximum principle for the nonlinear heat equation in TW 6.3.
- 15/3 We derived discrete maximum principles for implicit and explicit schemes for the linear heat equation in TW 6.2 and for an explicit scheme solving the nonlinear heat equation in TW 6.3. We also solved some of the exercises listed in the schedule.
- 21/3 We described harmonic functions on bounded, connected domains and derived maximum principles for harmonic functions (solutions of the Laplace equation), and used that result to produce a uniqueness argument for solutions of the Poisson equation cf. TW 6.4. Then we looked at the discrete Laplacian and derived discrete maximum principles for discrete harmonic functions on the unit box in 2D, following TW 6.5.
- 22/3 We finished TW 6.5 on discrete harmonic functions by deriving maximum principles for the discrete Poisson equation on the unit box in 2D. Then we solved some of the exercises listed in the schedule.
- 4/4 We described how to formulate the Laplace equation in polar coordinates and how to obtain formal solutions of the Laplace equation on different types of domains in 2D: on a square, a disc, and on a wedge (using the polar coordinate formulation for the latter two domains).
- 5/4 We described how formal solutions of the Laplace equation in 2D when the domain is the exterior of a disc (so an unbounded domain), and we solved a few of the exercises listed in the schedule. (Most of the remaining ones are solved in the hint pdf)
- 11/4 We used the divergence theorem to show that the operator L = - Laplace is symmetric and positive definite on C^2_0(\Omega). And we covered mean value properties for harmonic functions.
- 12/4 We derived symmetry and positive definiteness for the discrete Laplace operator, treated truncation errors for the discrete Poisson equation and obtained a second order rate of convergence for the discrete solution of the Poisson equation (on a 2D unit box). We also solved a subset of the exercises listed in the schedule.
- 18/4 We looked at Fourier series on the interval [-1,1], periodic extension of functions, Fourier series of even and odd functions (and their relation to cosine/sine series on [0,1]). We also covered general Fourier series and took at peak at particular solutions for Sturm--Liouville problems.
- 19/4 We looked at differentiation of Fourier series, discussed piecewise continuous functions, introduced the mean square distance norm, proved the Cauchy-Schwarz inequality for the inner product associated to that norm. We also looked at Poincare's inequality and utilized this inequality to obtain a sharper enegy estimate for the heat equation with 0-valued boundary conditions. Thereafter, we went through some of the listed exercises.
- 25/4 We covered Pythagoras theorem, Bessel's inequality and Parseval's identity for general/generalized Fourier series and described three different notions of convergence for sequences of functions on an interval [a,b]: pointwise, mean square sense and uniform.
- 26/4 We proved that the truncated Fourier series S_N(f) on [-1,1] converges pointwise to a limit function on [-1,1] as N to \infty if f is piecewise continuous and one-sided differentiable, and S_N(f) converges uniformly to f on [-1,1] if f '(x) is a piecewise continuous and periodic function. We also solved exercise 8.1 and I posted the solution of 8.2 online. The rest of the homework for this week will be solved next week.
- 2/5 We proved that S_N(f) converges in mean square sense to f on [-1,1] if f is piecewise continuous on [-1,1] and we studied how smoothness and periodicity of f leads to Fourier coefficients in S_N(f) that decay quickly.
- 3/5 We studied how quickly decaying Fourier coefficients relates to the smoothness and periodicity of the limit function S_{\infty}(f) and we solved some exercises from chapter 8, as listed in the schedule.
- 10/5 We showed that as long as the initial condition is piecewise continuous, the formal solution of the heat equation (with Dirichlet 0 BC) indeed is a solution of the heat equation for all positive times. Next week we cover to what extent the formal solution also satisfies the initial condition.
- 16/5 We studied to what extent/in what sense the formal solution of the heat equation satisfies the initial condition, and showed that this relates to the regularity and boundary values of the initial condition. Then we solved the exercises listed in the schedule. This was the last exam-relevant lecture for the course.
- 23/5 We solved exercises from chapter 9, and I presented a few slides on the not exam-relevant topic of physics informed neural networks (PINNs). This is an alternative and relatively new idea for solving partial differential equations that has attracted a lot of attention in recent years.
- 24/5 We solved exercises from chapter 10, and I presented a short summary of the course.
Summary of lectures (from 7/3)
Published Mar. 8, 2024 1:36 PM
- Last modified May 16, 2024 4:08 PM