The pictures in each column are *not* the same! Things differ in
subtle ways. You need to look closely. Material similar to the first
example was handed out in class. All of the pictures used similar
`Maple` commands, also available in the class handout.

The boundary value problem | Its eigenvalues and associated eigenfunctions | Graph of the 100^{th} partial sum of the
Fourier-like series associated with this eigenfunction
expansion |
Behavior at t=.001 | Behavior at t=.01 | Behavior at t=.1 | Behavior at t=1 | Behavior at t=10 |
---|---|---|---|---|---|---|---|

Temperature 0
at 0 and Pi u(0,t)=0 and u(Pi,t)=0 |
Positive integers, n. sin(n x) |
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When t gets large, the heat oozes out the ends, and the temperature drops to 0 in the whole interval. | |||||||

The boundary value problem | Its eigenvalues and associated eigenfunctions | Graph of the 100^{th} partial sum of the
Fourier-like series associated with this eigenfunction
expansion |
Behavior at t=.001 | Behavior at t=.01 | Behavior at t=.1 | Behavior at t=1 | Behavior at t=10 |

Insulated ends Flux 0 at 0 and Pi
u |
Nonnegative integers, n. cos(n x) For n=0, this is the function 1. |
||||||

When t gets large, the heat tends to even out over the whole interval, and the temperature approaches a constant (determined by the total area). | |||||||

The boundary value problem | Its eigenvalues and associated eigenfunctions | Graph of the 100^{th} partial sum of the
Fourier-like series associated with this eigenfunction
expansion |
Behavior at t=.001 | Behavior at t=.01 | Behavior at t=.1 | Behavior at t=1 | Behavior at t=10 |

Temperature 0 at 0 and
insulated end at Pi
u(0,t)=0 and u |
Half of odd positive integers, (1/2)(2n+1). (n at least 0) sin((1/2)(2n+1)x) |
||||||

When t gets large, the heat oozes out the left end, while the temperature curve always has a horizontal tangent at the right end because there is no heat flow there. The temperature eventually drops to 0 throughout the interval. |

**
Maintained by
greenfie@math.rutgers.edu and last modified 8/27/2004.
**