Preface I was very pleased with the positive responses that I received from users of the first edition.This edition incorporates many of their suggestions.It contains more engineering and physics applications,and more mathematical proofs and theory. To preserve the original organization of the topics,most changes appear in new sections at the end of chapters. Like the first edition,this book is intended as a modern textbook for an in- troductory course on partial differential equations and boundary value problems, including Fourier series.It is designed for use by students who have completed a first course in ordinary differential equations. The Book's Content and Organization The book is so organized as to provide a smooth transition from a course in ele- mentary ordinary differential equations to more advanced topics in a first course in partial differential equations.Even though the book is intended for students of engineering,mathematics,and physics,with heavy emphasis on applications,the new topics in this edition offer the instructors the option to teach a theory oriented course on partial differential equations or a course with heavier emphasis on Fourier series and boundary value problems.Beyond the core material for a basic course on partial differential equations (see Possible Course Outlines,below),the book includes a number of more specialized topics that the instructor can cover or use as research projects at his or her discretion.These more advanced topics,while building on the core material,are more or less independent of each other. Minimal Prerequisites In Appendix A we review the basics from the theory of linear ordinary differential equations,including series methods.This material may be covered in detail or completely omitted at the discretion of the instructor.It is included as a convenience to the student who wishes to review these topics and to have ready access to them. In particular,Sections A.4-A.6 contain a detailed treatment of the power series method and the Frobenius method that is suitable for students who are seeing this material for the first time.With the recent trend to omit the series methods from a first course in ordinary differential equations,I felt that it was necessary to treat this material in full detail The Exercises and Noteworthy Features for Computer Users Each set of exercises begins with a series of straightforward problems designed to solidify the student's understanding of the basic concepts in that section.The later exercises in each section are usually more involved and lead the student to a deeper understanding of the concepts.They usually include detailed hints to make them accessible to all students at this level.Some sections contain Project Problems which are longer exercises geared toward noteworthy results or interesting applica- tions.They can be used by individual students.or as group projects,or as further illustrations by the instructor