1. §3.3.1 Separation of Cartesian variables: Orthogonal functions
Christopher Crawford
PHY 416

2. Outline
Analogies from Chapter 1 Inner, outer products Projections, eigenstuff
Example boundary value problem – square box Separation of variables Solution of boundary conditions
Numerical methods for BVPs Relaxation method Finite difference method Finite element method

3. Analogies from Chapter 1
Inner vs. outer product – projections & components
Linear operators – stretches & rotations

4. Separation of variables technique
Goal: solve Laplace’s equation (PDE) by converting it into one ODE per variable
Method: separate the equation into separate terms in x, y, z start with factored solution V(x,y,z) = X(x) Y(y) Z(z)
Trick: if f (x) = g (y) then they both must be constant
Endgame: form the most general solution possible as a linear combination of single-variable solutions. Solve for coefficients using the boundary conditions.
Trick 2: use orthogonal functions to project out coefficients
Analogy: the set of all solutions forms a vector space the basis vectors are independent individual solutions

5. Example: rectangular box
Boundary value problem: Laplace equation

6. Example: rectangular box
Boundary value problem: Boundary conditions 6