In your answers below,
for the variable type the word
lambda,
for type the word
gamma;
otherwise treat these as you would any other variable.
We will solve the heat equation
with boundary/initial conditions:
This models temperature in a thin rod of length with
thermal diffusivity
where one end is insulated and the other end has fixed
temperature and the initial temperature distribution is .
For extra practice we will solve this problem from scratch.
Separate variables.
Assume and split the PDE into two
differential equations, one with and one with .
=
=
(
Notation: Write
X'' and
T ' for derivatives.
Place all constants in the differential equation with
T).
The problem splits into cases based on the sign of .
(
Notation: For the cases below, use constants
a and
b)
Plug the eigenvalues from
Case 3 into the
differential equation for and solve:
(
Notation: use
c for the unknown constant.)
Combining all of the and we get that
where are unknown constants.
You can earn partial credit on this problem.