# MATH 322 - MATLAB GUIs

This MATLAB GUI explores the orthogonality
properties of several families of Sturm-Liouville
eigenfunctions and emphasizes the geometric
significance of the orthogonality relationship
between two (different) eigenfunctions in the same
family.

This MATLAB GUI illustrates the use of Fourier
series to simulate the dynamics of a vibrating
string. The string is clamped at its end points and
its deflection from the horizontal, *u*,
evolves according to the wave equation,
*u*_{tt} - *u*_{xx} =
0.

This MATLAB GUI illustrates graphically how the
vibrating modes of a rectangular membrane, evolve
in time and interact with one another. The membrane
is clamped at its boundary and its deflection from
the horizontal, *u*, evolves according to the
two-dimensional wave equation,
*u*_{tt} =
*c*^{2}(*u*_{xx} +
*u*_{yy}).

This MATLAB GUI illustrates graphically how the
vibrating modes of a circular membrane evolve in
time and interact with one another. The membrane is
clamped at its boundary and its deflection from the
horizontal, *u*, evolves according to the
two-dimensional wave equation,
*u*_{tt} = Ñ^{2}*u*.

This MATLAB GUI illustrates the use of Fourier
series to simulate the diffusion of heat in a
domain of finite size. The quantity *u*
evolves according to the heat equation,
*u*_{t} - *u*_{xx} = 0,
and may satisfy Dirichlet, Neumann, or mixed
boundary conditions.

This MATLAB GUI plots the solution to the
one-dimensional heat equation, *u*_{t}
= *c*^{2}(*u*_{xx} +
*u*_{yy}), as a function of time and
for "top hat" initial conditions.

This GUI simulates the solution to the ordinary
differential equation *m y*'' + *c y*' +
*k y* = *F*(*t*), describing the
response of a one-dimensional mass spring system
with forcing function *F*(*t*) given by
(i) a unit square wave or (ii) a Dirac delta
function (e.g. "hammerblow'). Without loss of
generality, *m* is set to 1.