### Partial differential equations F, 2001,

home,
week 1,
week 2,
week 3,

### Lectures:

Lectures Monday 10-12 and Thusday 8-10 in Kollektorn as usual.

Content: Variational formulation of Poisson's equation. Finite
element methods. Read Chapter 8 and 15 of CDE or
Fem in 1d,
Fem in 23d

### Exercises:

Recommended, where Lx:y = exercise y under lecture x, E = exercise (pdef2-blandade övn), P = problem (Övningsexempel i PDE):

**Demo:**

For the solution u of the heat equation with u_0=0 and f a unit mass point load at x=0, compute lim_{t->oo}u(x,t). Compare to the fundamental solution of Poissons equation.

Show that for a solution to Poissons eq with homogeneous Dirichlet boundary conditions one has F(u)=-1/2\int_\Omega fu.

L5:1,2.

**Individual work:**

L4:1 compute c in fundamental solution.

L4:2 sol of non-homogeneous heat eq with unit mass load at x=0 and t=0.

L4:4 sol of non-homogeneous wave eq

L4:5 sol of wave eq with c^2.

L4:7 wave eq with boundary condition u(0,t)=0.

L4:8,9 the exploding ballon problem

P: 3, 6, 23

L5:3.

u=x^2+y^2-1 is the solution of a Poissons eq with homogeneous Dirichlet boundary conditions on the unit disc in 2D. Compute the total energy F(u) of the system. Consider some other function v satisfying the same boundary conditions, and verify that indeed F(v)>F(u).

/Claes

Last modified: Tue Sep 18 08:26:48 MET DST 2001