Green s Functions and Linear Differential Equations Theory Applications and Computation 1st Edition by Prem Kythe ISBN 978-1439840085 1439840085
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Product details:
ISBN 10: 1439840085
ISBN 13: 978-1439840085
Author: Prem K. Kythe
Green’s Functions and Linear Differential Equations: Theory, Applications, and Computation presents a variety of methods to solve linear ordinary differential equations (ODEs) and partial differential equations (PDEs). The text provides a sufficient theoretical basis to understand Green’s function method, which is used to solve initial and boundary value problems involving linear ODEs and PDEs. It also contains a large number of examples and exercises from diverse areas of mathematics, applied science, and engineering.
Taking a direct approach, the book first unravels the mystery of the Dirac delta function and then explains its relationship to Green’s functions. The remainder of the text explores the development of Green’s functions and their use in solving linear ODEs and PDEs. The author discusses how to apply various approaches to solve initial and boundary value problems, including classical and general variations of parameters, Wronskian method, Bernoulli’s separation method, integral transform method, method of images, conformal mapping method, and interpolation method. He also covers applications of Green’s functions, including spherical and surface harmonics.
Filled with worked examples and exercises, this robust, self-contained text fully explains the differential equation problems, includes graphical representations where necessary, and provides relevant background material. It is mathematically rigorous yet accessible enough for readers to grasp the beauty and power of the subject.
Table of contents:
1. Some Basic Results
1.1. Euclidean Space
1.1.1. Metric Space
1.1.2. Inner Product
1.2. Classes of Continuous Functions
1.3. Convergence
1.3.1. Convergence of Sequences
1.3.2. Weak Convergence
1.3.3. Metric
1.3.4. Convergence of Infinite Series
1.3.5. Tests for Convergence of Positive Series
1.4. Functionals
1.4.1. Examples of Linear Functionals
1.5. Linear Transformations
1.6. Cramer’s Rule
1.7. Green’s Identities
1.8. Differentiation and Integration
1.8.1. Leibniz’s Rules
1.8.2. Integration by Parts
1.9. Inequalities
1.9.1. Bessel’s Inequality for Fourier Series
1.9.2. Bessel’s Inequality for Square-Integrable Functions
1.9.3. Schwarz’s Inequality for Infinite Sequences
1.10. Exercises
2. The Concept of Green’s Functions
2.1. Generalized Functions
2.1.1. Heaviside Function
2.1.2. Delta Function in Curvilinear Coordinates
2.2. Singular Distributions
2.3. The Concept of Green’s Functions
2.4. Linear Operators and Inverse Operators
2.4.1. Linear Operators and Inverse Operators
2.4.2. Adjoint Operators
2.5. Fundamental Solutions
2.6. Exercises
3. Sturm-Liouville Systems
3.1. Ordinary Differential Equations
3.1.1. Initial and Boundary Conditions
3.1.2. General Solution
3.1.3. Method of Variation of Parameters
3.2. Initial Value Problems
3.2.1. One-Sided Green’s Functions
3.2.2. Wronskian Method
3.2.3. Systems of First-Order Differential Equations
3.3. Boundary Value Problems
3.3.1. Sturm-Liouville Boundary Value Problems
3.3.2. Properties of Green’s Functions
3.3.3. Green’s Function Method
3.4. Eigenvalue Problem for Sturm-Liouville Systems
3.4.1. Eigenpairs
3.4.2. Orthonormal Systems
3.4.3. Eigenfunction Expansion
3.4.4. Data for Eigenvalue Problems
3.5. Periodic Sturm-Liouville Systems
3.6. Singular Sturm-Liouville Systems
3.7. Exercises
4. Bernoulli’s Separation Method
4.1. Coordinate Systems
4.2. Partial Differential Equations
4.3. Bernoulli’s Separation Method
4.3.1. Laplace’s Equation in a Cube
4.3.2. Laplace’s Equation in a Cylinder
4.3.3. Laplace’s Equation in a Sphere
4.3.4. Helmholtz’s Equation in Cartesian Coordinates
4.3.5. Helmholtz’s Equation in Spherical Coordinates
4.3.6. Wave Equation
4.4. Examples
4.5. Exercises
5. Integral Transforms
5.1. Integral Transform Pairs
5.2. Laplace Transform
5.2.1. Definition of Dirac Delta Function
5.3. Fourier Integral Theorems
5.3.1. Properties of Fourier Transforms
5.3.2. Fourier Transforms of Derivatives of a Function
5.3.3. Convolution Theorems for Fourier Transform
5.4. Fourier Sine and Cosine Transforms
5.4.1. Properties of Fourier Sine and Cosine Transforms
5.4.2. Convolution Theorems for Fourier Sine and Cosine Transforms
5.5. Finite Fourier Transforms
5.5.1. Properties
5.5.2. Periodic Extensions
5.5.3. Convolution
5.6. Multiple Transforms
5.7. Hankel Transforms
5.8. Summary: Variables of Transforms
5.9. Exercises
6. Parabolic Equations
6.1. 1-D Diffusion Equation
6.1.1. Sturm-Liouville System for 1-D Diffusion Equation
6.1.2. Green’s Function for 1-D Diffusion Equation
6.2. 2-D Diffusion Equation
6.2.1. Dirichlet Problem for the General Parabolic Equation in a Square
6.3. 3-D Diffusion Equation
6.3.1. Electrostatic Analog
6.4. Schrödinger Diffusion Operator
6.5. Min-Max Principle
6.6. Diffusion Equation in a Finite Medium
6.7. Axisymmetric Diffusion Equation
6.8. 1-D Heat Conduction Problem
6.9. Stefan Problem
6.10. 1-D Fractional Diffusion Equation
6.10.1. 1-D Fractional Diffusion Equation in Semi-Infinite Medium
6.11. 1-D Fractional Schrödinger Diffusion Equation
6.12. Eigenpairs and Dirac Delta Function
6.13. Exercises
7. Hyperbolic Equations
7.1. 1-D Wave Equation
7.1.1. Sturm-Liouville System for 1-D Wave Equation
7.1.2. Vibrations of a Variable String
7.1.3. Green’s Function for 1-D Wave Equation
7.2. 2-D Wave Equation
7.3.3-D Wave Equation
7.4. 2-D Axisymmetric Wave Equation
7.5. Vibrations of a Circular Membrane
7.6. 3-D Wave Equation in a Cube
7.7. Schrödinger Wave Equation
7.8. Hydrogen Atom
7.8.1. Harmonic Oscillator
7.9. 1-D Fractional Nonhomogeneous Wave Equation
7.10. Applications of the Wave Operator
7.10.1. Cauchy Problem for 2-D and 3-D Wave Equation
7.10.2. d’Alembert Solution of the Cauchy Problem for Wave Equation
7.10.3. Free Vibration of a Large Circular Membrane
7.10.4. Hyperbolic or Parabolic Equations in Terms of Green’s Functions
7.11. Laplace Transform Method
7.12. Quasioptics and Diffraction
7.12.1. Diffraction of Monochromatic Waves
(a) Fraunhofer Approximation
(b) Fresnel Approximation
7.13. Exercises
8. Elliptic Equations
8.1. Green’s Function for 2-D Laplace’s Equation
8.2. 2-D Laplace’s Equation in a Rectangle
8.3. Green’s Function for 3-D Laplace’s Equation
8.3.1. Laplace’s Equation in a Rectangular Parallelopiped
8.4. Harmonic Functions
8.5. 2-D Helmholtz’s Equation
8.5.1. Closed-Form Green’s Function for Helmholtz’s Equation
8.6. Green’s Function for 3-D Helmholtz’s Equation
8.7. 2-D Poisson’s Equation in a Circle
8.8. Method for Green’s Function in a Rectangle
8.9. Poisson’s Equation in a Cube
8.10. Laplace’s Equation in a Sphere
8.11. Poisson’s Equation and Green’s Function in a Sphere
8.12. Applications of Elliptic Equations
8.12.1. Dirichlet Problem for Laplace’s Equation
8.12.2. Neumann Problem for Laplace’s Equation
8.12.3. Robin Problem for Laplace’s Equation
8.12.4. Dirichlet Problem for Helmholtz’s Equation
8.12.5. Dirichlet Problem for Laplace’s Equation in the Half-Plane
8.12.6. Dirichlet Problem for Laplace’s Equation in a Circle
8.12.7. Dirichlet Problem for Laplace’s Equation in the Quarter Plane
8.12.8. Vibration Equation for the Unit Sphere
8.13. Exercises
9. Spherical Harmonics
9.1. Historical Sketch
9.2. Laplace’s Solid Spherical Harmonics
9.2.1. Orthonormalization
9.2.2. Condon-Shortley Phase Factor
9.2.3. Spherical Harmonics Expansion
9.2.4. Addition Theorem
9.2.5. Laplace’s Coefficients
9.3. Surface Spherical Harmonics
9.3.1. Poisson Integral Representation
9.3.2. Representation of a Function f(0, $)
9.3.3. Addition Theorem for Spherical Harmonics
9.3.4. Discrete Energy Spectrum
9.3.5. Further Developments
9.4. Exercises
10. Conformal Mapping Method
10.1. Definitions and Theorems
10.1.1. Cauchy-Riemann Equations
10.1.2. Conformal Mapping
10.1.3. Symmetric Points
10.1.4. Cauchy’s Integral Formula
10.1.5. Mean-Value Theorem
10.2. Dirichlet Problem
10.2.1. Dirichlet Problem for a Circle in the (x, y)-Plane
10.3. Neumann Problem
10.4. Green’s and Neumann’s Functions
10.4.1. Laplacian
10.4.2. Green’s Function for a Circle
10.4.3. Green’s Function for an Ellipse
10.4.4. Green’s Function for an Infinite Strip
10.4.5. Green’s Function for an Annulus
10.5. Computation of Green’s Functions
10.5.1. Interpolation Method
10.6. Exercises
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Tags: Prem Kythe, Green s Functions, Linear Differential, Theory Applications