Linear partial differential equations and Fourier theory

✍ Scribed by Marcus Pivato

• Publisher: Cambridge University Press
• Year: 2010
• Tongue: English
• Leaves: 631
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Content: Cover --
Half title --
Title --
Copyright --
Dedication --
Contents --
Preface --
Prerequisites and intended audience --
Conventions in the text --
Acknowledgements --
What's good about this book? --
Illustrations --
Physical motivation --
Detailed syllabus --
Explicit prerequisites for each chapter and section --
Flat dependency lattice --
Highly structured exposition, with clear motivation up front --
Many 8216;practice problems (with complete solutions and source code available online) --
Challenging exercises without solutions --
Appropriate rigour --
Appropriate abstraction --
Gradual abstraction --
Expositional clarity --
Clear and explicit statements of solution techniques --
Suggested 12-week syllabus --
Part I Motivating examples and major applications --
1 Heat and diffusion --
1A Fouriers law --
1B The heat equation --
1C The Laplace equation --
1D The Poisson equation --
1E Properties of harmonic functions --
1F8727; Transport and diffusion --
1G8727; Reaction and diffusion --
1H Further reading --
1I Practice problems --
2 Waves and signals --
2A The Laplacian and spherical means --
2B The wave equation --
2C The telegraph equation --
2D Practice problems --
3 Quantum mechanics --
3A Basic framework --
3B The Schr168;odinger equation --
3C Stationary Schr168;odinger equation --
3D Further reading --
3E Practice problems --
Part II General theory --
4 Linear partial differential equations --
4A Functions and vectors --
4B Linear operators --
4C Homogeneous vs. nonhomogeneous --
4D Practice problems --
5 Classification of PDEs and problem types --
5A Evolution vs. nonevolution equations --
5B Initial value problems --
5C Boundary value problems --
5D Uniqueness of solutions --
5E8727; Classification of second-order linear PDEs --
5F Practice problems --
Part III Fourier series on bounded domains --
6 Some functional analysis --
6A Inner products --
6B L2-space --
6C8727; More about L2-space --
6D Orthogonality --
6E Convergence concepts --
6F Orthogonal and orthonormal bases --
6G Further reading --
6H Practice problems --
7 Fourier sine series and cosine series --
7A Fourier (co)sine series on [0, 960;] --
7B Fourier (co)sine series on [0,L] --
7C Computing Fourier (co)sine coefficients --
7D Practice problems --
8 Real Fourier series and complex Fourier series --
8A Real Fourier series on [8722;960;, 960;] --
8B Computing real Fourier coefficients --
8C Relation between (co)sine series and real series --
8D Complex Fourier series --
9 Multidimensional Fourier series --
9A --
9B --
9C Practice problems --
10 Proofs of the Fourier convergence theorems --
10A Bessel, Riemann, and Lebesgue --
10B Pointwise convergence --
10C Uniform convergence --
10D L2-convergence --
10D(i) Integrable functions and step functions in L2[8722;960;, 960;] --
10D(ii) Convolutions and mollifiers --
10D(iii) Proofs of Theorems 8A.1(a) and 10D.1 --
Part IV BVP solutions via eigenfunction expansions --
11 Boundary value problems on a line segment --
11A The heat equation on a line segment --
11B The wave equation on a line (the vibrating string) --
11C The Poisson problem on a line segment --
11D Practice problems --
1.
Abstract:

This highly visual introductory textbook provides a rigorous mathematical foundation for all solution methods and reinforces ties to physical motivation. Read more...