Chapter 2

FOUR IMPORTANT

LINEAR PARTIAL

DIFFERENTIAL

EQUATIONS

2.1 Transport equation

2.2 Laplace’s equation

2.3 Heat equation

2.4 Wave equation

2.5 Problems

2.6 References

In this chapter we introduce four fundamental linear partial diﬀeren-

tial equations for which various explicit formulas for solutions are available.

These are

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.

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