Partial Differential Equations

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