Part 1: Introduction to PDEs

The three basic PDEs. Notion of well posedness.

Part 2: Evolution problems for heat- and wave equation on R^n

Computation of heat kernel and the Riemann function through partial Fourier transform. Propagation speed, well posedness, smoothing and reversibility.

Part 3: Boundary value problems for the Laplace equation

Computation of fundamental solution, Green's 3 identities, Poisson kernel, Green's function. An alternative to Green's function: the double layer potential. Properties of harmonic functions: mean value property, maximum principle, smoothness. Existence and uniqueness for the Dirichlet problem, Dirichlet's principle.

A mean value formula and the maximum principle for the heat equation

Inspiration:

Fulks, W. Mean value theorem for the heat equation. Proc. Amer. Math. Soc. 17,

Evans, L. Partial differential equations. Section 2.3.2.

Plots of the kernel of the double layer potential

Plots of the basic functions

Heat kernel, Riemann functions in 1,2,3D, fundamental solution, Green's function, Poisson kernel

Part 4.1: Eigenfunctions to Laplace in one dimension

Eigenfunction expansions with periodic, Dirichlet, Neumann and mixed boundary conditions.

Part 4.2: Numerical solutions of PDEs

FEM for the Poisson problem, BEM for the Dirichlet problem using the double layer potential.

Matlab code for FEM and BEM

Part 4.3: Eigenfunction expansions in dimension 2 and 3

Rayleigh quatients, Rayleigh-Ritz method, minimum and maximin principles, proof of asymptotic formula for eigenvalues and completeness of eigenfucntions.

Part 5.1: Systems of linear first order PDEs. 5.2: Examples of non-linear PDEs

Comparison between analytic functions in the plane and Maxwell's equations from electromagnetics.

Calculus of variations, Euler-Lagrange equations, p-Laplace equation, minimal surface equation.

Derivation of the Navier-Stokes equations.