Section 26.9

Summary of Separation of Variables

A quick reference guide for solving PDEs.

1

The Algorithm

  1. Separate: Assume or .
  2. BVP: Solve the spatial BVP (Sturm-Liouville problem) to find Eigenvalues .
  3. ODE: Solve the remaining ODE (Time or Y) using .
  4. Superposition: Sum all product solutions .
  5. Fourier: Use orthogonality to match Initial Conditions or Boundary Conditions.
2

Common Boundary Conditions

BC TypeConditionEigenfunctionsEigenvalues
Dirichlet
Neumann
Mixed/RobinComplexRoots of
PeriodicFull Fourier
3

Equations Review

Heat Eq

Parabolic. Diffusion. Solutions decay exponentially in time.

Wave Eq

Hyperbolic. Propagation. Solutions oscillate forever (ideally).

Laplace Eq

Elliptic. Potential/Steady State. No time dependence.

4

Practice Quiz

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