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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. Reade...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge, UK ; New York :
Cambridge University Press,
2010.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.