random old notes
For several classes I have written notes on a variety of topics.
Some of these have proven useful to other people.
These notes are informal and may contain errors. The date of last
revision is noted in the description.
Below are a few of my "solutions" to papers/projects which I assigned as part
of a course. While my solutions are far from groundbreaking research, they
included a few interesting results.
- MATLAB Primer. Last edited in 2004. This primer has been
used as an early introduction to programming in the MATLAB environment. The reading
assumes the reader does not know linear algebra and
therefore contains none of the matrix-vector operations that make MATLAB quite useful.
- Numerical solutions to differential equations. Last edited 2003. An
early introduction to using MATLAB to solve differential equations numerically. The focus is
on Euler's method and other simple methods such as midpoint. Introduces Runge-Kutta and
shows how to use ode45, the built in MATLAB function. The reading
assumes student does not know linear algebra and therefore does not solve systems of equations.
- Diffusion. Last edited 2007. An introduction to diffusion processes,
solutions via simple finite difference methods (in MATLAB), RC circuit analogies to 1D conduction,
non-dimensionalization, and an introduction to
separation of variables.
- Fourier series in MATLAB. Last edited 2002. A basic introduction to Fourier series
and how power spectra are computed. Shows the reader how to interpret the data returned by MATLAB's FFT (fast Fourier Transform) command.
- MATLAB's data acquisition toolbox. Last edited 2002. An introduction to using
MATLAB's data acquisition toolbox.
- Ideal gas piston. Last edited 2004. An interesting problem
in thermodynamics is the dynamics of a piston in the middle of a gas filled tube. Such a device was once
used to infer the ratio of specific heats by measuring the system resonance. The piston has the interesting
property that if the dynamics are fast, the gas will be compressed adiabatically and the piston will oscillate
forever, if there were no mechanical friction. If the piston dynamics are slow, then the gas will be compressed
isothermally which will also oscillate forever. Between these two limits, the process
is irreversible and oscillations will be thermodynamically damped. This paper was used very early in an
introductory thermodynamics course.
- Tank draining by a siphon. Last edited 2006. A simple problem in fluid dynamics is the draining
of a tank by a siphon. The paper provides an introduction to making equations dimensionless. This paper
shows a nice transistion from Toricelli's law to an exponential as the length of the tube is increased.
The non-dimensionalization collapses all the data well regardless of tube length. The model
works quite well when compared to experimental data (though I did not include the experiments
in this paper).