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Green's Functions and Linear Differential Equations

Theory, Applications, and Computation

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## Book Description

**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

**Some Basic Results**Euclidean Space

Classes of Continuous Functions

Convergence

Functionals

Linear Transformations

Cramer’s Rule

Green’s Identities

Differentiation and Integration

Inequalities

**The Concept of Green’s Functions**Generalized Functions

Singular Distributions

The Concept of Green’s Functions

Linear Operators and Inverse Operators

Fundamental Solutions

**Sturm–Liouville Systems**Ordinary Differential Equations

Initial Value Problems

Boundary Value Problems

Eigenvalue Problem for Sturm–Liouville Systems

Periodic Sturm–Liouville Systems

Singular Sturm–Liouville Systems

**Bernoulli’s Separation Method **Coordinate Systems

Partial Differential Equations

Bernoulli’s Separation Method

Examples

**Integral Transforms **Integral Transform Pairs

Laplace Transform

Fourier Integral Theorems

Fourier Sine and Cosine Transforms

Finite Fourier Transforms

Multiple Transforms

Hankel Transforms

Summary: Variables of Transforms

**Parabolic Equations**1-D Diffusion Equation

2-D Diffusion Equation

3-D Diffusion Equation

Schrödinger Diffusion Operator

Min-Max Principle

Diffusion Equation in a Finite Medium

Axisymmetric Diffusion Equation

1-D Heat Conduction Problem

Stefan Problem

1-D Fractional Diffusion Equation

1-D Fractional Schrödinger Diffusion Equation

Eigenpairs and Dirac Delta Function

**Hyperbolic Equations**1-D Wave Equation

2-D Wave Equation

3-D Wave Equation

2-D Axisymmetric Wave Equation

Vibrations of a Circular Membrane

3-D Wave Equation in a Cube

Schrödinger Wave Equation

Hydrogen Atom

1-D Fractional Nonhomogeneous Wave Equation

Applications of the Wave Operator

Laplace Transform Method

Quasioptics and Diffraction

**Elliptic Equations**Green’s Function for 2-D Laplace’s Equation

2-D Laplace’s Equation in a Rectangle

Green’s Function for 3-D Laplace’s Equation

Harmonic Functions

2-D Helmholtz’s Equation

Green’s Function for 3-D Helmholtz’s Equation

2-D Poisson’s Equation in a Circle

Method for Green’s Function in a Rectangle

Poisson’s Equation in a Cube

Laplace’s Equation in a Sphere

Poisson’s Equation and Green’s Function in a Sphere

Applications of Elliptic Equations

**Spherical Harmonics **Historical Sketch

Laplace’s Solid Spherical Harmonics

Surface Spherical Harmonics

**Conformal Mapping Method**Definitions and Theorems

Dirichlet Problem

Neumann Problem

Green’s and Neumann’s Functions

Computation of Green’s Functions

**Appendix A: Adjoint Operators****Appendix B: List of Fundamental SolutionsAppendix C: List of Spherical Harmonics**

**Appendix D: Tables of Integral Transforms**

**Appendix E: Fractional Derivatives**

**Appendix F: Systems of Ordinary Differential Equations**

**Bibliography**

**Index**

*Exercises appear at the end of each chapter, with hints, answers, and, sometimes, complete solutions.*

## Author(s)

### Biography

**Prem K. Kythe** is a professor emeritus of mathematics at the University of New Orleans. Dr. Kythe is the co-author of *Handbook of Computational Methods for Integration* (CRC Press, December 2004) and *Partial Differential Equations and Boundary Value Problems with Mathematica, Second Edition* (CRC Press, November 2002). His research encompasses complex function theory, boundary value problems, wave structure, and integral transforms.