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

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

Separation of Variables Separated solution:

Separation of Variables Separated solution:

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

Separation of Variables Solution for Eq. (1):

Separation of Variables Solution for Eq. (1):

Separation of Variables Solution for Eq. (1):

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