MATLAB.8
                                                                        
Partial Differential Equations
Solving the Heat Equation .

Let f(x)=0 if 0<x<1,
        =1  if 1<x<2.
                                                                 
Consider the problem

d^2u/dxdx=du/dt        0<x<2, t>0
u(0,t)=u(2,t)=0               t>0,
u(x,0)=f(x) .

The Fourier sine coefficients for f are
b(m)=(cos(m*pi/2)-cos(m*pi))*2/m/pi

The M-file computing the first n terms is

*****************************************************************
function w=U(x,t,n)
w=0;
for m=1:n
w=w+2*(cos(m*pi/2)-cos(m*pi))/m/pi*exp(-m^2*pi^2*t/4).*sin(m*pi*x/2);
end
***************************************************************

Notice the  .*  above. We will do a 3-dimensional plot and U.m needs to
be array smart for this.
Let's set n=50.
To  plot the temperature distribution at different times enter

*****************************************************************
hold on
fplot('U(x,0,50)',[0,2])
fplot('U(x,.1,50)',[0,2])
fplot('U(x,.5,50)',[0,2])
fplot('U(x,1,50)',[0,2])
fplot('U(x,2,50)',[0,2])
fplot('U(x,5,50)',[0,2])
hold off
*************************************************************
To do a 3-D plot on the x-t rectangle 0<x<2,0<t<1 you subdivide it into 
rectangles of side length dx by dt and plot the u at the verticies.
Here we choose dx=.05 and dt=.05.
Enter

************************************************
x=0:.05:2;
t=0:.05:1;
[x,t]=meshgrid(x,t);
mesh(x,t,U(x,t,50))
*********************************************** 

ASSIGNMENT 8  :

1. HEAT EQUATION

Let u model the temperature in a rod and solve 

d^2u/dxdx=du/dt  0<x<40, t>0

u(0,t)=u(40,t)=0  t>0,

u(x,0)=x   for 0<x<40 .

a)  Find the series solution and consider the partial sum U=U(x,t,50).

b)   Plot U versus x for t=5,10,20,40,100 and 200.

c)  For each value of t used in b) estimate the value of x for which the 
    value of U is greatest . (See Lab Assignment.2 .) Plot these values 
    versus t to
    see how  the warmest point in the rod changes with time.

d)  Plot U versus t for x=10,20 and 30. The command fplot only recognizes  
    "x" as an independent variable. For example if you have an M-file for 
    a function  g(x,t), and you want the graph of g(3,t) for  1<t<3 type :
       fplot('g(3,x)',[1,3]) .

e)  Do a 3-D plot of U versus x and t.

f)  How long does it take for the entire rod to cool off to a temperature
    of no more than 2 degrees ?
        
2. WAVE EQUATION

Let u model the displacement of a vibrating string and solve 

d^2u/dxdx=d^2u/dtdt  0<x<10, t>0

u(0,t)=u(10,t)=0  t>0,

u(x,0)=1   for 4<x<6
          =0    for 0<x<4 & 6<x<10 

du/dt(x,0)=0  for 0<x<10 .


a)  Find the series solution and consider the partial sum U=U(x,t,50).

b)   Plot U versus x for t=0,2,4,...18,20 .Each on its own axes.

c)  Plot U versus t for 0<t<20 for x= 5 and x=2.5 on separate axes.

d)  Describe the motion of the string in a few sentences.