Below are hand-written lecture notes from a first course on partial
differential equation, which loosely follow the book "Partial
differential equations, An introduction" by W. A. Strauss. I gave
this course for fourth year undergraduate students at Linköping
university during the spring 2013.
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
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
Fulks, W. Mean
value theorem for the heat
equation. Proc. Amer. Math. Soc. 17, 1966
Evans, L. Partial differential equations. Section 2.3.2.
Plots of the kernel of the
double layer potential
Plots of the basic
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
Part 4.2: Numerical solutions
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.