Generalized Functions

Name: Generalized Functions | Theory and Technique
Brand: Academic Press
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Theory and Technique

Kanwal(Author)

Kanwal(Editor)

Academic Press

Published on 14. May 2014

443 pages

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978-0-08-095676-3 (ISBN)

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Front Cover
Generalized Functions: Theory and Technique
Copyright Page
Contents
PREFACE
CHAPTER 1. THE DIRAC DELTA FUNCTION AND DELTA SEQUENCES
1.1 The Heaviside Function
1.2 The Dirac Delta Function
1.3 The Delta Sequences
1.4 A Unit Dipole
1.5 The Heaviside Sequences
Exercises
CHAPTER 2. THE SCHWRTZ-SOBOLEV THEORY OF DISTRIBUTIONS
2.1 Some Introductory Definitions
2.2 Test Functions
2.3 Linear Functionals and the Schwartz-Sobolev Theory of Distributions
2.4 Examples
2.5 Algebraic Operations on Distributions
2.6 Analytic Operations on Distributions
2.7 Examples
2.8 The Support and Singular Support of a Distribution
Exercises
CHAPTER 3. ADDITIONAL PROPERTIES OF DISTRIBUTIONS
3.1 Transformation Properties of the Delta Distribution
3.2 Convergence of Distributions
3.3 Delta Sequences with Parametric Dependence
3.4 Fourier Series
3.5 Examples
3.6 The Delta Function as a Stieltjes Integral
Exercises
CHAPTER 4. DISTRIBUTIONS DEFINED BY DIVERGENT INTEGRALS
4.1 Introduction
4.2 The Psuedofunction H(x)/xn, n = 1, 2, 3, . . .
4.3 Functions with Algebraic Singularity of Order m
4.4 Examples
Exercises
CHAPTER 5. DISTRIBUTIONAL DERIVATIVES OF FUNCTIONS WITH JUMP DISCONTINUITIES
5.1 Distributional Derivatives in R1
5.2 Rn, n = 2
Moving Surfaces of Discontinuity
5.3 Surface Distributions
5.4 Various Other Representations
5.5 First-Order Distributional Derivatives
5.6 Second-Order Distributional Derivatives
5.7 Higher-Order Distributional Derivatives
5.8 The Two-Dimensional Case
5.9 Examples
CHAPTER 6. TEMPERED DISTRIBUTIONS AND THE FOURIER TRANSFORMS
6.1 Preliminary Concepts
6.2 Distributions of Slow Growth (Tempered Distributions)
6.3 The Fourier Transform
6.4 Examples
Exercises
CHAPTER 7. DIRECT PRODUCTS AND CONVOLUTIONS OF DISTRIBUTIONS
7.1 Definition of the Direct Product
7.2 The Direct Product of Tempered Distributions
7.3 The Fourier Transform of the Direct Product of Tempered Distributions
7.4 The Convolution
7.5 The Role of Convolution in the Regularization of the Distributions
7.6 Examples
7.7 The Fourier Transform of the Convolution
Exercises
CHAPTER 8. THE LAPLACE TRANSFORM
8.1 A Brief Discussion of the Classical Results
8.2 The Laplace Transform of Distributions
8.3 The Laplace Transform of the Distributional Derivatives and Vice Versa
8.4 Examples
Exercises
CHAPTER 9. APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS
9.1 Ordinary Differential Operators
9.2 Homogeneous Differential Equations
9.3 Inhomogeneous Differential Equations: The Integral of a Distribution
9.4 Examples
9.5 Fundamental Solutions and Green's Functions
9.6 Second-Order Differential Equations with Constant Coefficients
9.7 Eigenvalue Problems
9.8 Second-Order Differential Equations with Variable Coefficients
9.9 Fourth-Order Differential Equations
9.10 Differential Equations of the nth Order
9.11 Ordinary Differential Equations with Singular Coefficients
Exercises
CHAPTER 10. APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS
10.1 Introduction
10.2 Classical and Generalized Solutions
10.3 Fundamental Solutions
10.4 The Cauchy-Riemann Operator
10.5 The Transport Operator
10.6 The Laplace Operator
10.7 The Heat Operator
10.8 The Schrödinger Operator
10.9 The Helmholtz Operator
10.10 The Wave Operator
10.11 The Inhomogeneous Wave Equation
10.12 The Klein-Gordon Operator
Exercises
CHAPTER 11. APPLICATIONS TO BOUNDARY VALUE PROBLEMS
11.1 Poisson's Equation
11.2 Dumbbell-Shaped Bodies
11.3 Uniform Axial Distributions
11.4 Linear Axial Distributions
11.5 Parabolic Axial Distributions, n = 5
11.6 The Fourth-Order Polynomial Distribution, n = 7
Spheroidal Cavities
11.7 The Polarization Tensor for a Spheroid
11.8 The Virtual Mass Tensor for a Spheroid
11.9 The Electric and Magnetic Polarizability Tensors
11.10 The Distributional Approach to Scattering Theory
11.11 Stokes Flow
11.12 Displacement-Type Boundary Value Problems in Elastostatics
11.13 The Extension to Elastodynamics
11.14 Distributions on Arbitrary Lines
11.15 Distributions on Plane Curves
11.16 Distributions on a Circular Disk
CHAPTER 12. APPLICATIONS TO WAVE PROPAGATION
12.1 Introduction
12.2 The Wave Equation
12.3 First-Order Hyperbolic Systems
12.4 Aerodynamic Sound Generation
12.5 The Rankine-Hugoniot Conditions
CHAPTER 13. FUNCTIONS THAT HAVE INFINITE SlNGULARlTlES AT AN INTERFACE
13.1 Introduction
13.2 Distributional Field Equations of the First Order
13.3 Applications to Electrodynamics
13.4 Magnetohydrodynamic Waves in a Compressible Perfectly Conducting Fluid
13.5 Second-Order Differential Equations
CHAPTER 14. LINEAR SYSTEMS
14.1 Operators
14.2 The Step Response
14.3 The Impulse Response
14.4 The Response to an Arbitrary Input
14.5 Generalized Functions as Impulse Response Functions
14.6 The Transfer Function
14.7 Discrete-Time Systems
CHAPTER 15. MISCELLANEOUS TOPICS
15.1 The Cauchy Representation of Distributions
15.2 Distributional Weight Functions for Orthogonal Polynomials
15.3 Applications to Probability and Statistics
15.4 Applications of Generalized Functions in Economics
15.5 Distributional Solutions of Integral Equations
References
Additional Reading
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Generalized Functions

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