Sample Quiz 3 problems: BVPs#
Problem 1: Finite difference method#
The temperature distribution
where

Fig. 1 Annular fin#
a.) Write the finite-difference representation of the ODE (that applies at a location
b.) Based on the last part, write the recursion formula.
c.) The boundary condition at the outer radius
Write the boundary condition at
Solution#
a.) Replace the derivatives in the given ODE with finite differences, and replace any locations with the
or
b.) Rearrange and combine terms:
c.) We can use a backward difference to approximate the
Problem 2: eigenvalue#
Given the equation
a.) Find the expression that gives all eigenvalues (
b.) Calculate the principal eigenvalue.
Solution#
a.)
The eigenfunction is then the solution function associated with an eigenvalue:
b.) The principal eigenvalue is just that associated with
Problem 3: shooting method#
Use the shooting method to solve the boundary value problem
where
Solution#
First decompose into two 1st-order ODEs:
with BCs
Try some guess #1:
so for solution 1:
For guess #2:
so for solution 1:
For guess #3, we can interpolate:
then, use this guess:
so for solution 3:
So our answer is