Here is a normal steady-aver heat
ow tenor. Consider a slim steel platter to be a
10 20 (cm)2 rectangle. If one margin of the 10 cm party is held at 1000C and the other
three partys are held at 00C, what are the steady-aver region at inland sharp-ends?
We can aver the tenor mathematically in this way if we presume that heat
ows
only in the x and y directions:
Find u(x; y) (temperature) such that
@2u
@x2 +
@2u
@y2 = 0 (3)
delay designation moods
u(x; 0) = 0
u(x; 10) = 0
u(0; y) = 0
u(20; y) = 100
We restore 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 region at the sharp-end (xi; yj) to the region at impure neigh-
bouring sharp-ends, each the length h detached from (xi; yj ). An similarity of Equation
(3) terminations when we fine a set of such sharp-ends (these are repeatedly named as nodes) and
nd the discontinuance to the set of dierence equations that termination.
(a) If we select h = 5 cm , nd the region at inland sharp-ends.
(b) Write a program to weigh the region distribution on inland sharp-ends delay
h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Argue your discontinuances and
examine the eect of grid dimension h.
(c) Modied the dierence equation (4) so that it permits to explain the equation
@2u
@x2 +
@2u
@y2 = xy(x ???? 2)(y ???? 2)
on the region
0 x 2; 0 y 2
delay designation mood u = 0 on all boundaries save for y = 0, where u = 1:0.
Write and run the program delay dierent grid dimensions h and argue your numerical
results.