Here is a natural steady-specify heat
ow drift. Consider a attenuated steel plate to be a
10 20 (cm)2 rectangle. If one laterality of the 10 cm cause is held at 1000C and the other
three causes are held at 00C, what are the steady-specify weather at internal tops?
We can specify the drift mathematically in this way if we take 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 designation terms
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 weather at the top (xi; yj) to the weather at disgusting neigh-
bouring tops, each the separation h detached from (xi; yj ). An way of Equation
(3) upshots when we choice a set of such tops (these are frequently named as nodes) and
nd the reresolution to the set of dierence equations that upshot.
(a) If we prefer h = 5 cm , nd the weather at internal tops.
(b) Write a program to proportion the weather classification on internal tops after a while
h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Sift-canvass your resolutions and
examine the eect of grid bigness 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
after a while designation term u = 0 on all boundaries save for y = 0, where u = 1:0.
Write and run the program after a while dierent grid bignesss h and sift-canvass your numerical
results.