# Math 322 - Mathematical Analysis for Engineers

### Sturm-Liouville Eigenfunctions

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.

### Vibrating String

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*= 0.

_{xx}### Circular Elastic Membrane

This MATLAB GUI illustrates 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}* = ∇

^{2}

*u*.

### Rectangular Elastic Membrane

This MATLAB GUI illustrates 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}

### One-dimensional Heat Equation

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*= 0, and may satisfy Dirichlet, Neumann, or mixed boundary conditions.

_{xx}### Heat Equation on the Whole Line

This MATLAB GUI plots the solution to the one-dimensional heat equation, *u _{t}* =

*c*

^{2}

*u*, as a function of time and for "top hat" initial conditions.

_{xx}### Mass-Spring System

This MATLAB 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.