Linear Partial Differential Equations and Fourier Theory

Do you want a rigorous book that remembers where PDEs come from and what they look like? This highly visual introduction to linear PDEs and initial/boundary value problems connects the math to physical reality, all the time providing a rigorous mathematical foundation for all solution methods. Readers are gradually introduced to abstraction – the most powerful tool for solving problems – rather than simply drilled in the practice of imitating solutions to given examples. The book is therefore ideal for students in mathematics and physics who require a more theoretical treatment than given in most introductory texts. Also designed with lecturers in mind, the fully modular presentation is easily adapted to a course of one-hour lectures, and a suggested 12-week syllabus is included to aid planning. Downloadable files for the hundreds of figures, hundreds of challenging exercises, and practice problems that appear in the book are available online, as are solutions.

Spis treści:
Preface
Notation
Whats good about this book?
Suggested twelve-week syllabus
Part I. Motivating Examples and Major Applications
1. Heat and diffusion
2. Waves and signals
3. Quantum mechanics
Part II. General Theory
4. Linear partial differential equations
5. Classification of PDEs and problem types
Part III. Fourier Series on Bounded Domains
6. Some functional analysis
7. Fourier sine series and cosine series
8. Real Fourier series and complex Fourier series
9. Mulitdimensional Fourier series
10. Proofs of the Fourier convergence theorems
Part IV. BVP Solutions Via Eigenfunction Expansions
11. Boundary value problems on a line segment
12. Boundary value problems on a square
13. Boundary value problems on a cube
14. Boundary value problems in polar coordinates
15. Eigenfunction methods on arbitrary domains
Part V. Miscellaneous Solution Methods
16. Separation of variables
17. Impulse-response methods
18. Applications of complex analysis
Part VI. Fourier Transforms on Unbounded Domains
19. Fourier transforms
20. Fourier transform solutions to PDEs
Appendices
References
Index.