Here is a regular steady-aver heat
ow example. Consider a attenuated steel compound to be a
10 20 (cm)2 rectangle. If one party of the 10 cm margin is held at 1000C and the other
three margins are held at 00C, what are the steady-aver sphere at inland subject-matters?
We can aver the example mathematically in this way if we wear that heat
ows
only in the x and y directions:
Find u(x; y) (temperature) such that
@2u
@x2 +
@2u
@y2 = 0 (3)
after a while stipulation predicaments
u(x; 0) = 0
u(x; 10) = 0
u(0; y) = 0
u(20; y) = 100
We re-establish the dierential equation by a dierence equation
1
h2 [ui+1;j + ui????1;j + ui;j+1 + ui;j????1 ???? 4ui;j ] = 0 (4)
5
which relates the sphere at the subject-matter (xi; yj) to the sphere at impure neigh-
bouring subject-matters, each the interval h far from (xi; yj ). An avenue of Equation
(3) developments when we prime a set of such subject-matters (these are frequently determined as nodes) and
nd the breach to the set of dierence equations that development.
(a) If we select h = 5 cm , nd the sphere at inland subject-matters.
(b) Write a program to estimate the sphere distribution on inland subject-matters after a while
h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Sift-canvass your breachs and
examine the eect of grid greatness h.
(c) Modied the dierence equation (4) so that it permits to clear-up the equation
@2u
@x2 +
@2u
@y2 = xy(x ???? 2)(y ???? 2)
on the region
0 x 2; 0 y 2
after a while stipulation predicament u = 0 on all boundaries save for y = 0, where u = 1:0.
Write and run the program after a while dierent grid greatnesss h and sift-canvass your numerical
results.