**MATH3271**

**FW03**

**Assignment
2 Due
date: Friday, Oct. 10**

1. (a) Solve the
one-dimensional wave equation for each of the following sets of initial
conditions using the method of separation of variables. Take L = 1 and c = 1.

(i) f(x) = 0.1sin(2px), g(x) = 0;

(ii) f(x) = 0, g(x) = xsin(px);

(iii) f(x) = 1 -
cos(2px), g(x) = x/4 if 0 < x £ 0.4, g(x)
= (1 - x)/6 if 0.4 < x £1.

(b) Plot the solution to each of the above
problems for a sequence of different times.
Choose six times to show the behaviour of the string through one-half
period. Plot the six curves on the same
graph and take enough terms in the fourier series to get a good representation
for the curves.

2. Use d'Alembert's formula to write down the
explicit solution to the problems given in question 1(iii). Make sure you define completely the functions
f* and G.

3.(a) Solve the one-dimensional heat equation with
the conditions given below. Take L = 1
and c = 1.

(i) u(0,t) = 20,
u(1,t) = 10, u(x,0) = 20 + 20x-30x^{2};

(ii) u_{x}(0,t)
= 0, u_{x}(1,t) = 0, u(x,0) = 20 + 20x-30x^{2};

(b) Plot the solution to each of the above
problems for a sequence of different times.
Choose times to show the behaviour of the temperature from its initial
values to times approaching the steady state solution.

4.(a) Prove the orthogonality relation _{}for a ¹ b provided
tan(a) = -ka, tan(b) = -kb.

(b) Use the MAPLE function fsolve to find the
first five positive solutions of tan(x) = -x.
Set the range option to ensure you get the desired root in each case. Use the help facikity on MAPLE to find out
how to use fsolve.

5.(a) Find the solution for a vibrating, square
membrane whose edges are of length unity and whose initial shape is given by
the function f(x,y) = sin(px) sin(2py).
The edges of the membrane are held fixed and the membrane is initially
at rest. Take c = 1.

(b) Find the temperature of a square plate with
sides of length unity if the edges are kept a zero degrees and the original
temperature is given by f(x,y) = sin(2px) sin(py)

(c) Plot the solutions of part (a) for several
values of t. Use a separate plot for
each value of t.

6. Derive the solution of the two-dimensional
heat equation for a rectangular plate if one pair of opposite edges is kept at
zero degrees while the other pair of opposite edges is insulated so that no
heat can escape.

**MORE MAPLE**

In order to be able to include a
large number of terms in a Fourier series, you can use te sum function in
MAPLE. For example, to plot the series

_{}

we can use the
sum function. The following command will
give the 10^{th} partial sum.
Note that we end the command with a colon rather than a semicolon. This supresses the printing of this large
expression.

>s:=sum(sin(n*Pi*x)*cos(n*Pi*t)/n^2,n=1..10):

Now we can define
this sum evaluated for t = 0, 0.1,0.2, 0.3, 0.4, and 0.5, for example by

>for i from 0
to 5 do

>t:=i/10;

>s||i:=s;

>od;

Now plot the
curves.

>plot([s.0,s.1,s.2,s.3,s.4,s.5],x=0..1,colour=black);

If you want to
reuse t as a variable in the nest solution you need to redefine it by

>t:='t';

We can also plot
functions in three dimensions. For
example

>f:=sin(Pi*x)*sin(2*Pi*y);

>plot3d(f,x=0..1,y=0..1);

will give a three-dimensional plot of the function f.