Home Page

FINAL EXAM INFO:
The final exam will take place 8:30-10:20 Wed. March 18.
The exam will cover the material on complex analysis (i.e., it is NOT a comprehensive exam on the entire course).
You can use your textbook and 1 page of notes duing the exam.
EDGE STUDENTS: PLEASE make sure that your exam gets scanned and e-mailed directly to me on Wednesday morning. (Otherwise, I cannot get your grade submitted on time!) You might also want to submit a copy to the EDGEoffice so that they have a record of your submission.

The final homework assignment is now posted on the assignments page.

A number of people have hit a snag on problem #3 on the take-home exam. Click here to download an alternative version that you can do instead.

If you have already made good progress on (or sunk too much time into) #3, write up what you have done on the original version and hand it in. Please state which version you are doing!

The take-home exam on PDEs has been sent to the class e-mail lists, so all students should have received it by now. If not, it is also available via this link to download Exam 2.

New notebooks have now been posted for vibrations of a membrane/annulus using Bessel functions (discussed in class Friday 20) and for solution of PDEs by integral transforms (to be discussed next week).

Notebooks are now available with info on eigenfunctions for problem 12.3.9 and solution of Laplace's equation in spherical coordinates using Legendre polynomials.

Please download HW#4 in PDF format if you have not already received it by e-mail.

The Mathematica notebook (discussed in class on 2/9) using Bessel functions to solve a heat equation in polar coordinates is now available.

Also, in case you did not already receive the e-mail, here are some hints on my embellishment of Problem #12.3.19. From Problem 15, you know that the x-dependence looks like:
f(x) = A cos(bx) + B sin(bx) + C cosh(bx) + D sinh(bx)

Applying the boundary conditions at the left end (f(x)=0, f'(x)=0), you should be able to rewrite this as:
f(x) = A (cos(bx) - cosh(bx) ) + B (sin(bx) - sinh(bx) )

Applying the boundary conditions at the right end ( f(L)=0, f'(L)=0 ) gives two conditions on A,B:
A ( cos(bL) - cosh(bL) ) + B (sin(bL) - sinh(bL) ) = 0
A (-sin(bL) - sinh(bL) ) + B (cos(bL) - cosh(bL) ) = 0

My advice from here is to write this as a matrix of coefficients multiplying the vector Transpose( A, B ).

When do non-trivial solutions exist or the coefficients A and B? (Relate this to the Determinant of the coefficient matrix.) Find the first few roots of the characteristic function.

What do the non-trivial solutions for A and B look like? (Find the eigenvector associated with the zero eigenvalue.)
Plug the eigenvector and a root of the characteristic function into your generic expression for f(x) to produce the first few f_n(x) functions. (A number of terms in the single digits should suffice.)

Plot the functions and compute the inner products f_m(x) . f_ n(x). Show that these functions are orthogonal.

Once the orthogonality relation is established, project the initial conditions onto the f_n(x) functions to determine the constants in the solution. Plot your result and quantify how well your solution does (or does not) satisfy the initial conditions, boundary conditions, and the PDE.

I hope that helps. -- DS

The Mathematica notebooks discussed in class on Wed. Feb. 04 (visualizing the solutions of the 1D wave equation and heat equation) are not available on the downloads page.

HW#3 is now posted on the assignments page.

The Fourier Series notebook discussed in class on Jan. 7 is now posted on the downloads page.

Click here to download the course information sheet discussed in class on 1/5.

The solution of HW5 is posted on the download page