**Tags: Matlab simulation, math equation, Dirichlet boundary conditions **

**Problem 1**. Simulate 1D metal rod of unit length. Use 1D assumptions (constant temperature across crossection etc). Initial temperature f(x, 0) = 1x+0.1 sin(2x), boundary conditions f(0, t) = f(L, t) = 0. Run the heat equation and plot the temperature 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 256x256 regular grid, with initial temperature f(x, y, t = 0) = sin(2x)cos(2y). 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 256x256 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 : I ! R3 be a parameterized

curve. Let [a, b] I and set (a) = p, (b) = q.

a. (2pt) Show that, for any constant vector v, |v| = 1,

(q − p) · v =Z ba0(t) · vdt Z ba|0(t)|dt

b. (2pt) Set

v = q − p / |q − p|

and show that

|(b) − (a)| Z ba|0(t)|dt;

that is, the curve of shortest length from (a) to (b) is the straight line joining these points.

**Problem 4.** (2pt) The trace of the parameterized curve (arbitrary parameter)

(t) = (t, cosh t), t 2 R,

is called catenary. Show that the signed curvature of the catenary is

(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 Iwrote as a starting point available on blackboard.