This is a text for a two-semester or three-quarter sequence of courses in partial differential equations. It is assumed that the student has a good background in vector calculus and ordinary differential equations and has been introduced to such elementary aspects of partial differential equations as separation of variables, Fourier series, and eigenfunction expansions. Some familiarity is also assumed with the application of complex variable techniques, including conformal mapping, integration in the complex plane, and the use of integral transforms.
Linear theory is developed in the first half of the book and quasilinear and nonlinear problems are covered in the second half, but the material is presented in a manner that allows flexibility in selecting and ordering topics. For example, it is possible to start with the scalar first-order equation in Chapter 5, to include or delete the nonlinear equation in Chapter 6, and then to move on to the second-order equations, selecting and omitting topics as dictated by the course. At the University of Washington, the material in Chapters 1-4 is covered during the third quarter of a three-quarter sequence that is part of the required program for first-year graduate students in Applied Mathematics. We offer the material in Chapters 5-8 to more advanced students in a two-quarter sequence.
The primary purpose of this book is to analyze the formulation and solution of representative problems that arise in the physical sciences and engineering and are modeled by partial differential equations. To achieve this goal, all the basic physical principles of a given subject are first discussed in detail and then incorporated into the analysis. Although proofs are often omitted, the underlying mathematical concepts are carefully explained. The emphasis throughout is on deriving explicit analytical results, rather than on the abstract properties of solutions. Whenever a new idea is introduced, it is illustrated with an example from an appropriate area of application. There is a selection of additional problems at the end of each chapter, ranging in difficulty from straightforward extensions of the textual material to rather challenging departures, testing the student's skill at application. In addition, there are selected review exercises for Chapters 1 and 2 that illustrate some of the prerequisite ideas that are not explicit in the text.
The numerical solution of partial differential equations is a vast topic requiring a separate volume; here, the emphasis is on analytical techniques. Numerical methods of solution are mentioned only in connection with particular examples and, more generally, to illustrate the solution of hyperbolic problems in terms of characteristic variables. Certain analytical techniques covered in specialized texts have also been left out. The notable omissions concern the asymptotic expansion of solutions obtained by integral transforms, integral equation methods, the Wiener-Hopf method, and inverse scattering theory.
I dedicate this book with admiration and gratitude to Paco A. Lagerstrom and Julian D. Cole, who got me started in this field. I thank my wife, Seta, without whose patience and support this book would have remained just a plan. I appreciate the help of Radhakrishnan Srinivasan and David L. Bosley, who read the manuscript and made valuable comments and corrections. I applaud the skill and efficiency demonstrated by Lilly Harper, who transformed barely legible handwriting into flawless copy throughout the manuscript.