Definitions, examples, and where they arise
Linearity and building solutions from modes
Derivation, separation of variables, and eigenmodes
Steady–transient splits and boundary forcing
Source terms, eigenfunction expansions, and Green ideas
Solving on infinite/half lines and in time
D’Alembert, energy methods, and standing waves
Harmonic functions, BCs, and maximum principles
Transport-type PDEs and characteristic curves
Conservation laws, shocks, and rarefactions
Euler–Lagrange equations and functionals
Regular vs. singular methods and matched expansions