Partial differential equations F3, 2002,
home,
week 1,
Lectures:
Welcome to first weeks lectures Monday and Thursday at 10.00 in Kollektorn.
Content: Fundamental Pde-models: Navier-Stokes, Poisson, heat, wave,
convection-diffusion-reaction, Maxwell, Schrödinger.
Basic tools of Calculus: divergence theorem, Green's formula, Stokes' theorem, chain rule, potential fields.
Exercises:
E = exercise (pdef2-blandade övn), P = problem (Övningsexempel i PDE):
Demo:
Derive the 2D (or 3D) integration by parts formula int_D q.grad v = int_dD q.n v - int_D div q v from Gauss divergence theorem, with a comparison to the corresponding 1D formulas, in particular concerning boundary terms. Also, consider the particular case q = grad u (Greens formula).
Individual:
1. Prove that the (double) integral over a (2D) domain D of div q (q = (q1, q2)) equals the (curve) integral along the boundary of the domain of q.n (Gauss divergence theorem) for a) D = {x = (x1, x2) : 0 < x1 < 1, 0 < x2 < 1} b) D = {x : 0 < x1 < 2, f(x1) < x2 < g(x1)}, where n = (n1, n2) is the exterior unit normal.
2. Verify (by direct computation of the integrals involved) Gauss theorem for a) D = {x : 0 < x1 < 1, 0 < x2 < 1} and q = (x1, x2) b) D = {x : 0 < x1 < 1, 0 < x2 < x1} and q = (x2, x1 x2) c) D = {x : 0 < x1 < 1, x1^2+x2^2 < 1} and q = grad(x1^2 + x2^2).
3. Verify Greens formula for D = {x : 0 < x1 < 1, 0 < x2 < 1}, u = x1 x2, and v = x1^2 + x2^2.
4. Plot (by hand) the fields a) q = (x1, x2) b) q = (x1/(x1^2+x2^2), x2/(x1^2+x2^2)) c) q = (-x2, x1), compute the divergence of each field, and compare to your plots.
5. Determine which of the fields in 4 are irrotational and find corresponding potentials such that q = grad u.
8. E: 1
Lab work:
1. Experiment with the matlab demo tool and lab environment MultiD lab. To download the required matlab code, shift-click on MD.m, MD.mat and R2adm.m, and start the tool by giving the command MD at the matlab prompt. Do as many as possible of the exercises under MDguide.
2. To better understand the interaction of convection, diffusion and production terms in a differential equation model, and its boundary conditions, you may consult the two-point boundary value problem demo solver TwoPVPdemo, by shift-clicking to import TwoPBVdemo.fig and TwoPBVdemo_adm.m, which you start from matlab by the command open('TwoPBVdemo.fig').
Project:
Start working on project 1!
/Claes
Last modified: Mon Sept 2 00:44:19 MET DST 2002