Here is a typical steady-state heat

ow problem. Consider a thin steel plate to be a

10  20 (cm)2 rectangle. If one side of the 10 cm edge is held at 1000C and the other

three edges are held at 00C, what are the steady-state temperature at interior points?

We can state the problem mathematically in this way if we assume that heat

ows

only in the x and y directions:

Find u(x; y) (temperature) such that

@2u

@x2 +

@2u

@y2 = 0 (3)

with boundary conditions

u(x; 0) = 0

u(x; 10) = 0

u(0; y) = 0

u(20; y) = 100

We replace 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 temperature at the point (xi; yj) to the temperature at four neigh-

bouring points, each the distance h away from (xi; yj ). An approximation of Equation

(3) results when we select a set of such points (these are often called as nodes) and

nd the solution to the set of dierence equations that result.

(a) If we choose h = 5 cm , nd the temperature at interior points.

(b) Write a program to calculate the temperature distribution on interior points with

h = 2:5, h = 0:25, h = 0:025 and h = 0:0025 cm. Discuss your solutions and

examine the eect of grid size h.

(c) Modied the dierence equation (4) so that it permits to solve the equation

@2u

@x2 +

@2u

@y2 = xy(x ???? 2)(y ???? 2)

on the region

0  x  2; 0  y  2

with boundary condition u = 0 on all boundaries except for y = 0, where u = 1:0.

Write and run the program with dierent grid sizes h and discuss your numerical

results.