1. Boundary Value Problems
Differential equations
Initial conditions
Prescribed value at t = 0 for the entire region of interest
Boundary conditions
Prescribed value on the boundary for the entire time period of interest

2. Separation of Variables
BVP for an E-field component
One-dimensional case: An example
Differential equation
Boundary conditions
Initial conditions
Boundary Value Problem (BVP)

3. Separation of Variables
Separated solution:

4. Separation of Variables
Separated solution:

5. Separation of Variables
Separated solution:
(1)
(2)
(3)

6. Separation of Variables
Solution for Eq. (1):

7. Separation of Variables
Solution for Eq. (1):

8. Separation of Variables
Solution for Eq. (1):

9. Separation of Variables
Solution for Eq. (1):
Eigenvalue problem
Eigenfunction
Eigenvalue