You are previewing Advanced Mathematics for Applications.
O'Reilly logo
Advanced Mathematics for Applications

Book Description

The partial differential equations that govern scalar and vector fields are the very language used to model a variety of phenomena in solid mechanics, fluid flow, acoustics, heat transfer, electromagnetism and many others. A knowledge of the main equations and of the methods for analyzing them is therefore essential to every working physical scientist and engineer. Andrea Prosperetti draws on many years' research experience to produce a guide to a wide variety of methods, ranging from classical Fourier-type series through to the theory of distributions and basic functional analysis. Theorems are stated precisely and their meaning explained, though proofs are mostly only sketched, with comments and examples being given more prominence. The book structure does not require sequential reading: each chapter is self-contained and users can fashion their own path through the material. Topics are first introduced in the context of applications, and later complemented by a more thorough presentation.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. To the reader
  9. List of Tables
  10. Part 0 General Remarks and Basic Concepts
    1. 1. The Classical Field Equations
      1. 1.1 Vector fields
      2. 1.2 The fundamental equations
      3. 1.3 Diffusion
      4. 1.4 Fluid mechanics
      5. 1.5 Electromagnetism
      6. 1.6 Linear elasticity
      7. 1.7 Quantum mechanics
      8. 1.8 Eigenfunction expansions: introduction
    2. 2. Some Simple Preliminaries
      1. 2.1 The method of eigenfunction expansions
      2. 2.2 Variation of parameters
      3. 2.3 The “δ-function”
      4. 2.4 The idea of Green’s functions
      5. 2.5 Power series solution
  11. Part I Applications
    1. 3. Fourier Series: Applications
      1. 3.1 Summary of useful relations
      2. 3.2 The diffusion equation
      3. 3.3 Waves on a string
      4. 3.4 Poisson equation in a square
      5. 3.5 Helmholtz equation in a semi-infinite strip
      6. 3.6 Laplace equation in a disk
      7. 3.7 The Poisson equation in a sector
      8. 3.8 The quantum particle in a box: eigenvalues of the Laplacian
      9. 3.9 Elastic vibrations
      10. 3.10 A comment on the method of separation of variables
      11. 3.11 Other applications
    2. 4. Fourier Transform: Applications
      1. 4.1 Useful formulae for the exponential transform
      2. 4.2 One-dimensional Helmholtz equation
      3. 4.3 Schrödinger equation with a constant force
      4. 4.4 Diffusion in an infinite medium
      5. 4.5 Wave equation
      6. 4.6 Laplace equation in a strip and in a half-plane
      7. 4.7 Example of an ill-posed problem
      8. 4.8 The Hilbert transform and dispersion relations
      9. 4.9 Fredholm integral equation of the convolution type
      10. 4.10 Useful formulae for the sine and cosine transforms
      11. 4.11 Diffusion in a semi-infinite medium
      12. 4.12 Laplace equation in a quadrant
      13. 4.13 Laplace equation in a semi-infinite strip
      14. 4.14 One-sided transform
      15. 4.15 Other applications
    3. 5. Laplace Transform: Applications
      1. 5.1 Summary of useful relations
      2. 5.2 Ordinary differential equations
      3. 5.3 Difference equations
      4. 5.4 Differential-difference equations
      5. 5.5 Diffusion equation
      6. 5.6 Integral equations
      7. 5.7 Other applications
    4. 6. Cylindrical Systems
      1. 6.1 Cylindrical coordinates
      2. 6.2 Summary of useful relations
      3. 6.3 Laplace equation in a cylinder
      4. 6.4 Fundamental solution of the Poisson equation
      5. 6.5 Transient diffusion in a cylinder
      6. 6.6 Formulae for the Hankel transform
      7. 6.7 Laplace equation in a half-space
      8. 6.8 Axisymmetric waves on a liquid surface
      9. 6.9 Dual integral equations
    5. 7. Spherical Systems
      1. 7.1 Spherical polar coordinates
      2. 7.2 Spherical harmonics: useful relations
      3. 7.3 General solution of the Laplace and Poisson equations
      4. 7.4 A sphere in a uniform field
      5. 7.5 Half-sphere on a plane
      6. 7.6 The Poisson equation in free space
      7. 7.7 General solution of the biharmonic equation
      8. 7.8 Exterior Poisson formula for the Dirichlet problem
      9. 7.9 Point source near a sphere
      10. 7.10 A domain perturbation problem
      11. 7.11 Conical boundaries
      12. 7.12 Spherical Bessel functions: useful relations
      13. 7.13 Fundamental solution of the Helmholtz equation
      14. 7.14 Scattering of scalar waves
      15. 7.15 Expansion of a plane vector wave
      16. 7.16 Scattering of electromagnetic waves
      17. 7.17 The elastic sphere
      18. 7.18 Toroidal–poloidal decomposition and viscous flow
  12. Part II Essential Tools
    1. 8. Sequences and Series
      1. 8.1 Numerical sequences and series
      2. 8.2 Sequences of functions
      3. 8.3 Series of functions
      4. 8.4 Power series
      5. 8.5 Other definitions of the sum of a series
      6. 8.6 Double series
      7. 8.7 Practical summation methods
    2. 9. Fourier Series: Theory
      1. 9.1 The Fourier exponential basis functions
      2. 9.2 Fourier series in exponential form
      3. 9.3 Point-wise convergence of the Fourier series
      4. 9.4 Uniform and absolute convergence
      5. 9.5 Behavior of the coefficients
      6. 9.6 Trigonometric form
      7. 9.7 Sine and cosine series
      8. 9.8 Term-by-term integration and differentiation
      9. 9.9 Change of scale
      10. 9.10 The Gibbs phenomenon
      11. 9.11 Other modes of convergence
      12. 9.12 The conjugate series
    3. 10. The Fourier and Hankel Transforms
      1. 10.1 Heuristic motivation
      2. 10.2 The exponential Fourier transform
      3. 10.3 Operational formulae
      4. 10.4 Uncertainty relation
      5. 10.5 Sine and cosine transforms
      6. 10.6 One-sided and complex Fourier transform
      7. 10.7 Integral asymptotics
      8. 10.8 Asymptotic behavior
      9. 10.9 The Hankel transform
    4. 11. The Laplace Transform
      1. 11.1 Direct and inverse Laplace transform
      2. 11.2 Operational formulae
      3. 11.3 Inversion of the Laplace transform
      4. 11.4 Behavior for small and large t
    5. 12. The Bessel Equation
      1. 12.1 Introduction
      2. 12.2 The Bessel functions
      3. 12.3 Spherical Bessel functions
      4. 12.4 Modified Bessel functions
      5. 12.5 The Fourier–Bessel and Dini series
      6. 12.6 Other series expansions
    6. 13. The Legendre Equation
      1. 13.1 Introduction
      2. 13.2 The Legendre equation
      3. 13.3 Legendre polynomials
      4. 13.4 Expansion in series of Legendre polynomials
      5. 13.5 Legendre functions
      6. 13.6 Associated Legendre functions
      7. 13.7 Conical boundaries
      8. 13.8 Extensions
      9. 13.9 Orthogonal polynomials
    7. 14. Spherical Harmonics
      1. 14.1 Introduction
      2. 14.2 Spherical harmonics
      3. 14.3 Expansion in series of spherical harmonics
      4. 14.4 Vector harmonics
    8. 15. Green’s Functions: Ordinary Differential Equations
      1. 15.1 Two-point boundary value problems
      2. 15.2 The regular eigenvalue Sturm–Liouville problem
      3. 15.3 The eigenfunction expansion
      4. 15.4 Singular Sturm–Liouville problems
      5. 15.5 Initial-value problem
      6. 15.6 A broader perspective on Green’s functions
      7. 15.7 Modified Green’s function
    9. 16 Green’s Functions: Partial Differential Equations
      1. 16.1 Poisson equation
      2. 16.2 The diffusion equation
      3. 16.3 Wave equation
    10. 17. Analytic Functions
      1. 17.1 Complex algebra
      2. 17.2 Analytic functions
      3. 17.3 Integral of an analytic function
      4. 17.4 The Taylor and Laurent series
      5. 17.5 Analytic continuation I
      6. 17.6 Multi-valued functions
      7. 17.7 Riemann surfaces
      8. 17.8 Analytic continuation II
      9. 17.9 Residues and applications
      10. 17.10 Conformal mapping
    11. 18. Matrices and Finite-Dimensional Linear Spaces
      1. 18.1 Basic definitions and properties
      2. 18.2 Determinants
      3. 18.3 Matrices and linear operators
      4. 18.4 Change of basis
      5. 18.5 Scalar product
      6. 18.6 Eigenvalues and eigenvectors
      7. 18.7 Simple, normal and Hermitian matrices
      8. 18.8 Spectrum and singular values
      9. 18.9 Projections
      10. 18.10 Defective matrices
      11. 18.11 Functions of matrices
      12. 18.12 Systems of ordinary differential equations
  13. Part III Some Advanced Tools
    1. 19. Infinite-Dimensional Spaces
      1. 19.1 Linear vector spaces
      2. 19.2 Normed spaces
      3. 19.3 Hilbert spaces
      4. 19.4 Orthogonality
      5. 19.5 Sobolev spaces
      6. 19.6 Linear functionals
    2. 20. Theory of Distributions
      1. 20.1 Introduction
      2. 20.2 Test functions and distributions
      3. 20.3 Examples
      4. 20.4 Operations on distributions
      5. 20.5 The distributional derivative
      6. 20.6 Sequences of distributions
      7. 20.7 Multi-dimensional distributions
      8. 20.8 Distributions and analytic functions
      9. 20.9 Convolution
      10. 20.10 The Fourier transform of distributions
      11. 20.11 The Fourier series of distributions
      12. 20.12 Laplace transform of distributions
      13. 20.13 Miscellaneous applications
      14. 20.14 The Hilbert transform
    3. 21. Linear Operators in Infinite-Dimensional Spaces
      1. 21.1 Linear operators
      2. 21.2 Bounded operators
      3. 21.3 Compact operators
      4. 21.4 Unitary operators
      5. 21.5 The inverse of an operator
      6. 21.6 Closed operators and their adjoint
      7. 21.7 Solvability conditions and the Fredholm alternative
      8. 21.8 Invariant subspaces and reduction
      9. 21.9 Resolvent and spectrum
      10. 21.10 Analytic properties of the resolvent
      11. 21.11 Spectral theorems
  14. Appendix
    1. A.1 Sets
    2. A.2 Measure
    3. A.3 Functions
    4. A.4 Integration
    5. A.5 Curves
    6. A.6 Bounds and limits
  15. References
  16. Index