Imagine that we took a uniform, one-dimensional rod of length 2L with source Q=0 and cp=1=Ko and bent the rod into a circle with the ends touching - a perfectly round ring. The temperature ur of this ring satisfies the heat equation.
This ring is so perfectly formed that we have determined there is perfect thermal contact at the joint and we have determined that there is perfect heat flux at the joint.
a. Determine appropriate boundary conditions for this partial differential equation.
b. Given an initial condition of urt=0=f, solve for the temperature of this partial differential equation, using the boundary conditions determined above. Apply the method of separation of variables. Determine appropriate values of a separation constant. (Example: Is this separation constant greater than zero? Less than zero? Equal to zero?)
c. Describe why separation of variables is an appropriate method to use with this partial differential equation given these boundary conditions.
d. Define boundary conditions to the heat equation =g for a rod of length L that would prevent the use of the method of separation of variables. Do not attempt to solve this partial differential equation.