Linear Partial Differential Equations and Fourier Theory

AUTHOR: Marcus Pivato



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.

Online resources include full-colour and three-dimensional illustrations, practice problems and complete solutions for instructors
Includes a suggested twelve-week syllabus and lists recommended prerequisites for each section
Contains nearly 400 challenging theoretical exercises

Table of Contents

Preface
Notation
What's 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.



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