Linear Partial Differential Equations and Fourier Theory

 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...