# Quiz 2

Math 421:03, Fall 2002

Date: 30 October 2002

There are two (2) problems on this quiz. The other one is on the back of this page.

1.Formulate a boundary value problem modeling heat conduction in a thin bar of length if the left end is insulated and the right end is kept at a fixed temperate . The initial temperature distribution in the bar at is . (Just set up the problem and stop!)

Proof. [Solution]The heat equation is for , . The boundary conditions are for all (because the left side is insulated), and for all (because the right side is held to a constant temperature). The initial conditions are for .

2.Solve the boundary-value problem

Proof. [Solution]The general solution to this problem is

where

In our case we have

This was enough as far as I was concerned. The simplification is tricker than on the homework, but can be done like this:

 difference 1 0 -1 1 2 -1 1 -2 3 0 -1 1 4 1 1 0 5 0 -1 1 6 -1 1 -2 7 0 -1 1 8 1 1 0

So clearly if is odd. If is even, can be written as , but depends on whether is even or odd! The numbers that can be written as 2 times an odd number look like . Therefore we have

Matthew Leingang 2002-11-12