21-272 - Introduction to PDEs

Overview

The aim of the course is to give a broad overview of the most important first and second order linear PDEs (namely, transport equation, Laplace equation, heat equation and wave equation) as well as of some non-linear ones. In particular, Fourier series and Fourier transform will be introduced and applied to study some linear PDEs. Time permitting, some topics from first order non-linear PDEs will be covered.

Material

• Syllabus (pdf - 77kb)

• Logbook (pdf - 58kb)

Notes

The following are the notes I used for the classes. Please take into consideration that errors may be present! Some are hand-written, some are typed in Latex.

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• Review on ODEs (pdf - 127kb)

• The transport equation (pdf - 730kb)

• Diffusion equation - derivation and derivation of the heat kernel (pdf - 404kb)

• Diffusion equation in the whole space (pdf - 308kb)

• Diffusion equation - separation of variables (pdf - 137kb)

• Wave equation - fast and loose derivation (pdf - 145kb)

• Wave equation - rigorous derivation (pdf - 409kb)

• Wave equation - d'Alembert formula (pdf - 106kb)

• Wave equation - separation of variables (pdf - 119kb)

• Energy and uniqueness (pdf - 101kb)

• Fourier series - computation of coefficients (pdf - 83kb)

• Fourier series - review (pdf - 187kb)

• Fourier series - properties (pdf - 109kb)

• Fourier series - solution of the heat equation (pdf - 123kb)

• Fourier series - solution of the wave equation (pdf - 84b)

• Forcing term in the wave equation in bounded domains (pdf - 100kb)

• The divergence theorem (pdf - 600kb)

• Laplace and Poisson equation in bouthe whole space (pdf - 134kb)

• The Laplacian in polar and spherical coordinates (pdf - 119kb)

• Laplace equation in bounded domains (pdf - 136kb)

• Poisson formula for balls (pdf - 158kb)

• Distributions (pdf - 699kb)

• Fourier transform (pdf - 525kb)

• General review (pdf - 551kb)