Book contents
- Frontmatter
- Contents
- Preface
- What's good about this book?
- Suggested 12-week syllabus
- Part I Motivating examples and major applications
- Part II General theory
- Part III Fourier series on bounded domains
- Part IV BVP solutions via eigenfunction expansions
- Part V Miscellaneous solution methods
- Part VI Fourier transforms on unbounded domains
- Appendix A Sets and functions
- Appendix B Derivatives – notation
- Appendix C Complex numbers
- Appendix D Coordinate systems and domains
- Appendix E Vector calculus
- Appendix F Differentiation of function series
- Appendix G Differentiation of integrals
- Appendix H Taylor polynomials
- References
- Subject index
- Notation index
Preface
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- What's good about this book?
- Suggested 12-week syllabus
- Part I Motivating examples and major applications
- Part II General theory
- Part III Fourier series on bounded domains
- Part IV BVP solutions via eigenfunction expansions
- Part V Miscellaneous solution methods
- Part VI Fourier transforms on unbounded domains
- Appendix A Sets and functions
- Appendix B Derivatives – notation
- Appendix C Complex numbers
- Appendix D Coordinate systems and domains
- Appendix E Vector calculus
- Appendix F Differentiation of function series
- Appendix G Differentiation of integrals
- Appendix H Taylor polynomials
- References
- Subject index
- Notation index
Summary
This is a textbook for an introductory course on linear partial differential equations (PDEs) and initial/boundary value problems (I/BVPs). It also provides a mathematically rigorous introduction to Fourier analysis (Chapters 7, 8, 9, 10, and 19), which is the main tool used to solve linear PDEs in Cartesian coordinates. Finally, it introduces basic functional analysis (Chapter 6) and complex analysis (Chapter 18). The first is necessary to characterize rigorously the convergence of Fourier series, and also to discuss eigenfunctions for linear differential operators. The second provides powerful techniques to transform domains and compute integrals, and also offers additional insight into Fourier series.
This book is not intended to be comprehensive or encyclopaedic. It is designed for a one-semester course (i.e. 36–40 hours of lectures), and it is therefore strictly limited in scope. First, it deals mainly with linear PDEs with constant coefficients. Thus, there is no discussion of characteristics, conservation laws, shocks, variational techniques, or perturbation methods, which would be germane to other types of PDEs. Second, the book focuses mainly on concrete solution methods to specific PDEs (e.g. the Laplace, Poisson, heat, wave, and Schrödinger equations) on specific domains (e.g. line segments, boxes, disks, annuli, spheres), and spends rather little time on qualitative results about entire classes of PDEs (e.g. elliptic, parabolic, hyperbolic) on general domains.
- Type
- Chapter
- Information
- Linear Partial Differential Equations and Fourier Theory , pp. xv - xviiPublisher: Cambridge University PressPrint publication year: 2010