Homework 8, due November 27, 2007.

[goodwine]

1. Compute the Fourier series for
   • 
   where
   • 
   Plot f(x) and the first several terms of the series on the same graph.
2. Compute the Fourier series for
   • 
   where
   • 
   Plot f(x) and the first several terms of the series on the same graph.
3. Determine the solution to
   • 
   where
   • 
   and
   • 
   Plot the solution for t=0 and several other times to clearly demonstrate the nature of the solution.
4. Determine the solution to
   • 
   where
   • 
   and
   • 
5. Determine the solution to the wave equation where
   • 
   Plot the solution for t=0 and several other times to clearly demonstrate the nature of the solution.
   
   Extra credit: make a movie of the solution.
6. Determine the solution to the wave equation where
   • 
7. Determine the solution to
   • 
   where
   • 
   and
   • 
8. By assuming u(x,y)=X(x)Y(y), determine the solution to
   • 
   where
   • 
   Plot an approximation to the solution by using the first few terms in the series representation of the solution.
9. Consider the famous Lorenz equations
   • 
   where
   • 
   1. Use the second order Taylor series method to determine an approximate numerical solution to these equations for the time range t=0 to t=50. Submit a 3D plot of (x,y,z). Be sure to experiment with the time step to ensure that your solution is accurate. The Matlab plot3() function will probably be useful. A quick google search will probably give you an idea of what the plot should look like.
   2. Use Matlab's ode45() to compute an approximate solution and plot the result. Compare it with your plot from the first part of this problem.

[avalenzu]

When you ask us to "Plot f(x) and the first several terms of the series on the same graph," which of the following (if either) do you mean?

A)
Plot 1: f(x)
Plot 2: term1 + term2 + term3 + term4

or

B)
Plot 1: f(x)
Plot 2: term1
Plot 3: term2
Plot 4: term3
Plot 5: term4

[goodwine]

When you ask us to "Plot f(x) and the first several terms of the series on the same graph," which of the following (if either) do you mean?

A)
Plot 1: f(x)
Plot 2: term1 + term2 + term3 + term4

or

B)
Plot 1: f(x)
Plot 2: term1
Plot 3: term2
Plot 4: term3
Plot 5: term4

I mean A). I want it as 1) a check on your answer to see that it is converging to f(x) and 2) as an exercise into the nature of how it is converging. If you plot the modes separately, it will be hard, if not impossible, to see how it's converging and what it's converging to.

[colinheye]

Is the very first coefficient on problem 7 an 'a' or an alpha?

[goodwine]

Is the very first coefficient on problem 7 an 'a' or an alpha?

The very first coefficient is an "a". The coefficient of the "u" term by itself is an alpha.

[jsv2nd]

#5 asks us to plot the solution at various values for t. I am confused about how to do this.

I solved the wave equation for u(x,0) correctly. I was able to use the equation for b_n, similar to the example in class. However, when t != 0, you now need to deal with a_n as well, which is associated with the sine term. This looks very messy, and I don't know how to treat it.

Is there a way to do this that I am not thinking of. Am I missing something from my notes?

Thanks

[colinheye]

Do we need to plot the different u(x,t) plots as well as a movie? How do you want us to get the movie file to you?

[goodwine]

Do we need to plot the different u(x,t) plots as well as a movie? How do you want us to get the movie file to you?

You don't have to do a movie, but if you do it for extra credit, just email it to me. If it's too large to email, then put it in your www directory and send a link to me.

[goodwine]

Do we need to plot the different u(x,t) plots as well as a movie? How do you want us to get the movie file to you?

Just do it like I demonstrated in class on Wednesday.

[goodwine]

You may assume the initial velocity for problem 7 is zero. Alternatively, you may work it out assuming it is an unspecified g(x).

[goodwine]

I did not provide a function for the boundary condition; thus, you can't plot anything. You may skip the plotting part of the problem.