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-30x2;

(ii) ux(0,t) = 0, ux(1,t) = 0, u(x,0) = 20 + 20x-30x2;

(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 10th 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.