Chapter 1 INTRODUCTION 1.1 Partial differential equations 1.2 Examples 1.3 Strategies for studying PDE 1.4 Overview 1.5 Problems 1.6 References This chapter surveys the principal theoretical issues concerning the solv- ing of partial differential equations. To follow the subsequent discussion, the reader should first of all turn to Appendix A and look over the notation presented there, particularly the multiindex notation for partial derivatives. 1.1. PARTIAL DIFFERENTIAL EQUATIONS A partial differential equation (PDE) is an equation involving an unknown function of two or more variables and certain of its partial derivatives. Using the notation explained in Appendix A, we can write out symbol- ically a typical PDE, as follows. Fix an integer k 1 and let U denote an open subset of Rn. DEFINITION. An expression of the form (1) F (Dku(x), Dk−1u(x), . . . , Du(x), u(x), x) = 0 (x U) is called a kth-order partial differential equation, where F : Rnk × Rnk−1 × · · · × Rn × R × U R 1
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