Preface ix Sections 7.9 and 7.10 are new.They contain Duhamel's principle,with appli- cations to the nonhomogeneous heat and wave equations.They also introduce the notions of fundamental solutions and weak solutions of partial differen- tial equations.Even though the topics are theoretical in nature,the sections contain many interesting applications and exercises that should appeal to a student seeing this material for the first time. Chapter 12 on Green's functions and conformal mappings is new.This is a self-contained treatment of Green's functions and conformal mappings that is written in the same easy-to-follow style as the rest of the book.The chapter is packed with interesting exercises and applications.Section 12.1 starts with the basic properties of line integrals,proves Green's theorem, derives Green's formulas,and then proves fundamental results about har- monic functions.In Sections 12.2 and 12.3,we prove Gauss's mean value property of harmonic functions and the maximum modulus principle,derive Green's functions and discuss their theoretical and physical significance.Sec- tion 12.3 also treats the eigenfunction method for finding Green's functions. In Section 12.4,the method of images is used to derive Green's functions. Section 12.5 is a resourceful introduction to the theory of analytic functions and their applications to partial differential equations.This section includes the Cauchy-Riemann equations and many interesting applications of analytic functions to the solution of Dirichlet problems.Section 12.6 presents the method of conformal mappings and its applications to boundary value prob- lems.In Sections 12.7 and 12.8 we use conformal mappings to derive Green's functions and Neumann functions. .Appendix A.4 is new.It provides a review of power series that is suitable for the power series methods of the following sections. Possible Course Outlines Basic Course in Partial Differential Equations.This course consists of the core material. Chapter 1. -ections2.1-2.4.2.6 Chapter 3(Sections 3.9-3.11 are optional). -Sections 4.1,4.2,4.4 (appealing to Sections 4.7 and 4.8 as needed) -Sections 5.1,5.2 (appealing to Sections 5.5 and 5.6 as needed). Sections 6.1,6.2 一 Sections 7.1-7.7. Section 8.3 (appealing to Sections 8.1 and 8.2 as needed) A Course with an Emphasis on Engineering Applications.Substitute the material from Chapter 5 by any of the following sections:Sections 2.7, 6.3,6.5,6.6and6.7. A Course with an Emphasis on Physics.Substitute sections from Chap- ters 7 and 8 by sections from Chapters 5 and 11