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. 17

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.