Harj. 4

18.3. -99

References and links

[Coo] Blue book, Ch 9 (9.1 and 9.2)
[T-W] Tweito-Winther: Yellow book (Chapter 7)

[Coo] Exercises 9.2 pp. 386 - 389

These exercises are for the most part "software oriented" and make use of ready made ideas and code found in [Coo]. There should be no difficulties in completing the solutions by Fri 26.3.

Probl. 2, p. 386. Poisson kernel
Probl. 3, p. 387. Plot harmonic fns given in polar coords
Take a look at the Cooper code dirch , see also p. 387 for an explanation.
Probl. 4, p. 387
Probl. 5, p. 387-388
Probl. 7, p. 388
Probl. 8, p. 388
Write a Matlab program that solves the Dirichlet problem on a disk using Poisson's formula (and numerical integration) directly. Try to make it as vectorized as possible in order to make it efficient. Test and compare to some of the above problems.

Hints and instructions

Surface plot of a function "over a disk"

See Cooper p. 386

Case 1, function f(x,y) given in rectangular coordinates

r=linspace(0,1,20);
theta=linspace(0,2*pi);  % default of linspace is n=100
X=r'*cos(theta);
Y=r'*sin(theta);
surf(X,Y,X.*Y)  % The function f(x,y)=x*y

In general f.m is an array smart function file and the surf command is

surf(X,Y,f(X,Y))

Case 2, function h(r,theta) given in polar coordinates

r=linspace(0,1,20);
theta=linspace(0,2*pi);
R=r'*ones(size(theta));
TH=ones(size(r'))*theta;
X=r'*cos(theta);
Y=r'*sin(theta);
surf(X,Y,R.*ones(size(TH))) % Plot the function h(r,theta)=r

In general h.m is an array smart function file and the surf command is

surf(X,Y,h(R,TH))