2.1 Transport equation
2.2 Laplace’s equation
2.3 Heat equation
2.4 Wave equation
In this chapter we introduce four fundamental linear partial diﬀeren-
tial equations for which various explicit formulas for solutions are available.
the transport equation ut + b · Du = 0 (§2.1),
Laplace’s equation Δu = 0 (§2.2),
the heat equation ut − Δu = 0 (§2.3),
the wave equation utt − Δu = 0 (§2.4).
Before going further, the reader should review the discussions of inequal-
ities, integration by parts, Green’s formulas, convolutions, etc., in Appen-
dices B and C and later refer back to these as necessary.