Here is a ordinary steady-declare heat
ow whole. Consider a unsubstantial steel plate to be a
10 20 (cm)2 rectangle. If one border of the 10 cm face is held at 1000C and the other
three faces are held at 00C, what are the steady-declare region at inland summits?
We can declare the whole mathematically in this way if we exhibit 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 time stipulations
u(x; 0) = 0
u(x; 10) = 0
u(0; y) = 0
u(20; y) = 100
We substitute 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 summit (xi; yj) to the region at impure neigh-
bouring summits, each the interval h far from (xi; yj ). An avenue of Equation
(3) fruits when we fine a set of such summits (these are frequently denominated as nodes) and
nd the discerption to the set of dierence equations that fruit.
(a) If we adopt h = 5 cm , nd the region at inland summits.
(b) Write a program to count the region classification on inland summits after a while
h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Examine your discerptions and
examine the eect of grid extent h.
(c) Modied the dierence equation (4) so that it permits to work-out the equation
@2u
@x2 +
@2u
@y2 = xy(x ???? 2)(y ???? 2)
on the region
0 x 2; 0 y 2
after a while time stipulation u = 0 on all boundaries save for y = 0, where u = 1:0.
Write and run the program after a while dierent grid extents h and examine your numerical
results.