EE575: HEOEM57E5W: OHROKME2W(DOUREK02/13/2020)
Problem 1. Simulate 1D metal rod of unit length. Use 1D assumptions (constant temperature across crossection etc). Initial temperature f(x,0) = 1 sin(2πx),
x+0.1 boundary conditions f (0, t) = f (L, t) = 0. Run the heat equation and plot the tem-
perature as a function of time. (Hint: Refer to the handout posted on blackboard for some useful source code). (10pt)
Problem 2. Simulate 2D unit square plate using a 256×256 regular grid, with ini- tial temperature f (x, y, t = 0) = sin(2πx)cos(2πy). Assume the Dirichlet boundary conditions of 0 temperature on the boundary. Run the heat equation and plot the temperature distribution as a function of time. Take any black and white 256×256 image and run the heat equation on the intensity values. Show the smoothing behavior of the heat equation. (Extra credit (2pt): Create a movie showing heat change over a period of time). Hint: Check delsq function in matlab. (10pt)
Problem 3. (4pt) (Straight Lines as Shortest) Let α : I → R3 be a parameterized curve. Let [a, b] ⊂ I and set α(a) = p, α(b) = q.
a. (2pt) Show that, for any constant vector v, |v| = 1, b b
b. (2pt) Set
and show that
(q − p) · v =
α′(t) · vdt ≤ aa
v=q−p |q − p|
b
|α′(t)|dt
|α(b) − α(a)| ≤
that is, the curve of shortest length from α(a) to α(b) is the straight line joining
these points.
Problem 4. (2pt) The trace of the parameterized curve (arbitrary parameter)
α(t) = (t,cosht), t ∈ R,
is called catenary. Show that the signed curvature of the catenary is
|α′(t)|dt;
.
κ(t) = 1 cosh2 t
Problem 5. (4pt) Plot the Frenet frame for a regular parameterized curve for (tcos(6t), tsin(6t), t). You can write your own code or use the Matlab code that I wrote as a starting point available on blackboard.
1
a