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
Save your time - order a paper!
Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlinesOrder Paper Now
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
only in the x and y directions:
Find u(x; y) (temperature) such that
@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
h2 [ui+1;j + ui????1;j + ui;j+1 + ui;j????1 ???? 4ui;j ] = 0 (4)
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) Modied the dierence equation (4) so that it permits to solve the equation
@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
- Posted: A Day Ago
- Due: 13/01/2019
- Budget: $10