MAT 461 Project Ideas

The written project is due via email on Tuesday, Nov. 19 (via email). This should consist of either:
Printout of power point slides, or pdf slides, or other projected images during the presentation, and an appendix (which may be handwritten) with details of calculations skipped in the presentation
or
A written report, if you do plan to use the white board during your presentation. This should be approximately 5 typewritten pages. The equations can be written if necessary.

I expect that most of you will want to use computers in some capacity. Equations can be inserted into MS Word documents. You can also do the write-up in Mathematica or PowerPoint. For the actual math part, you will probably want to use Mathematica or Matlab to plot figures. The easiest way to do a project is to use one of these programs, or some other programming language such as C++, to do some "number crunching." If you would rather do a "pencil-and-paper" calculation that is possible too.

I want you to give me a title and a short description of you project on Monday, October 28. Please talk to me before then if you have any questions about what to do, or what would be appropriate.

The project should be about something that you are interested in, and it should involve PDEs (or boundary value problems of ODEs). It would be great if there is some problem with an application in one of the sciences that has captured your interest. On the other hand, I do not want you to duplicate something you have seen elsewhere. Here are some ideas for projects. I have labeled them as pencil and paper (P&P) or computer (C) projects. Of course, the (P&P) projects will probably need computers to produce figures, and the (C) projects will involve a lot of written analysis before you go to the computer for the number crunching.

(P&P) Investigate one or more of the unusual coordinate systems in which the Laplacian is separable.
Use them to solve Laplace's equation, and/or to find eigenvalues of the Laplacian in certain geometries. For example, use bipolar coordinates to solve Laplace's equation on the disk with u = 1 on the upper semi-circle and u = -1 on the lower semi-circle.
Or, use bipolar coordinates to solve Laplace's equation in the region of the plane outside two circle, each held at a different temperature (or electrostatic potential).
Or, use elliptic coordinates to find the eigenvalues of the Laplacian inside and ellipse with u = 0 on the boundary.

(C) Complete problem 2.3.2g, as described in class when I showed this demonstration.

(C) Use the L matrix for approximating the 1-D Laplacian using finite differences to do one of these, numbered L1, L2, etc.

(L1) Solve the wave equation with specific non-constant density rho(x). Pick one, for example rho(x) = 1 + (x/L)^2. Your code will define the function rho(x), so that it can be easily changed and re-run. Use equation (4.2.8) with the boundary conditions(4.4.2). Use the separation of variable like in section 4.4, but solve the eigenvalue problem for phi(x) numerically using the L matrix. Show pictures of the eigenfunctions. If you have time, also solve the PDE with the initial conditions (4.4.3), which some specific choice of f(x) and g(x). This will require you to find the orthogonality condition for the eigenfunctions. Hint: the eigenvalue problem you solve is a regular Sturm-Liouville problem.

(L2) Solve the 1D heat equation with non-constant thermal conductivity and/or other properties. This would involve solving an eigenvalue problem numerically.

(L3) Solve the 1D Schroedinger equation from quantum mechanics numerically (with a potential other than Coulomb's law or Hooke's Law).

(L4) Solve the radial Schroedinger's equation for a central force other than Coulomb's law or Hooke's law. Perhaps the 6-12 rule?

The following few involve eigenvalues and eigenfunctions of the Laplacian in polar coordinates. After assuming cos(m theta) or sin(m theta) dependence, you just need to find a function of r.

(C) Find the eigenvalues of the Laplacian on the disk with fixed temp, zero derivative, and Newton's Law of cooling at the boundary.

(C) Find the eigenvalues of the Laplacian on an annulus, with u = 0 on the boundaries.

(C) Find the eigenvalues of the Laplacian with 0-Dirichlet boundary conditions on some 2-D region using finite difference. For example find the eigenfunction/eigenvalues on a cross using a rectangular grid, or on a hexagon or triangle with a triangular grid.

(C) Solve the 2D Schrödinger equation numerically, with a non-central force.

(C) Implement one or more of the numerical schemes described in Chapter 6 of our textbook.

(P&P) Solve this system of two coupled PDEs for heat flow in a porous medium. (See the work of Peter Vadasz.)
c_u du/dt = K_u d²u/dx² + k(v-u)
c_v dv/dt = K_v d²v/dx² + k(u-v)
u(0, t) = v(0, t) = u(L, t) = v(L, t) = 0 for all t
u(x, 0) = f(x), and v(x, 0) = g(x), where f and g are given functions.

(C) Tone color and Fourier series.
In music, timbre, also known as tone color or tone quality, is the perceived sound quality of a musical note, sound or tone (a quote from the wikipedia page on Timbre). A guitar, or flute, or Frank Sinatra, might all sound the same note (at the same frequency) but it sounds different. Why? It has to do with the "power spectrum", which depends on the coefficients a_n and b_n in the Fourier Series. For example a note in ``concert A" is at a frequency of 440 cycles per second (440 Hz "Hertz"), or a period of 1/440 second. The air pressure p(t) as a function of time t in seconds satisfies p(t+1/440) = p(t). The Fourier Series of this sound is a sum of a<sub>n</sub>cos(n 2 pi 440 t) + b<sub>n</sub> sin(n 2 pi 440 t). (We assume that a<sub>0</sub>=0, since p(t) is the difference of the true air pressure minus the constant average air pressure.) Another way to write this is to use ``polar coordinates'' for the coefficients: a_n = r_ncos(theta_n) and b_n=r_ncos(theta_n). The resulting Fourier series is shown in this Desmos graph FS in Polar Coordinates. (The frequency f is a parameter in that graph, in a range near f = 1 that we couldn't hear.) The power spectrum is sometimes defined r_n² as a function of n, and it describes the loudness of frequency n*440 Hz (the n'th harmonic). The power spectrum is called the Harmonic Spectra in that wikipedia page on timbre.
One could explore many aspects of this as a project. One suggestion I have is to test the assertion that we don't hear the difference caused by the phases theta_n. One could use some software to generate a periodic sound wave with an input set of Fourier amplitudes to test this.

(P&P, C) Investigate the Discrete Time Fourier Transform (DTFT), the Discrete Fourier Transfom (DFT) and the Fast Fourier Transform (FFT), and show how the DTFT is related to the Fourer Series in section 3.6.