34)
The program mynonlinheat.m has been modified to have the boundary
conditions of u(0)=0 and u(L)=10.
The modified program produces graphs of
the initial guess of the program as well as the initial guess and 400
iterations of the for loop.
Modified code of mynonlinheat.m
% mynonlinheat
% Math 344
% Homework Problem 34
% Writen by Astrom and Elbert
%
L = 4;
% Rod length (L)
n = 400;
% number of intervals (n)
h = L/n;
% Define h
hh = h^2/3;
% Define h
u0 = 0;
% u(0) boundary condition value
uL = 10;
% u(L) boundary condition value
ub = .5;
% Background temperature
ub4 = ub^4;
% Define ub to the fourth
x = 0:h:L+h;
% Vector of x values
g = sin(pi*x/4);
% Temperature function
u = zeros(1,n+2);
% Initial guess
steps = 400;
% number of times to run loop
u(1) = u0;
% Set 1st boundary condition
u(n+2) = uL;
% Set 2nd boundary condition
figure
% Individual graph
plot(x,u)
% Plot initial guess
figure
% Individual graph
plot (x,u)
% Plot initial guess
hold
on
for
j = 1:steps
% Loop
u(2:n+1) = (u(3:n+2)+u(2:n+1)+u(1:n))/3 + hh*(u(2:n+1).^4+ub4g(2:n+1));
plot(x,u)
% Plot each time
end
plot(x,u)
% Plot value after steps value
Graphs
Figure 34a – Graph of initial guess
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Figure 34a – Graph after 500 steps
****************************************************************************
35)
This program has been modified to produce both graphs with one run of
the program.
After changing d from .1 to .01 it was seen that the final
temperature at r = 0 changed from 10.06 to 10.5264.
We can see that d = 0.10
converges after around 7 iterations were as d = 0.01 take many more
iterations to converge.
This is shown in the graphs below.
The output of
the program is displayed.
The values at the beginning and the end are the
only parts of the output shown to save paper.
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 Spring '08
 Staff
 Boundary value problem, initial guess, Boundary conditions, Astrom

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