1st Edition

Green's Functions and Linear Differential Equations Theory, Applications, and Computation

By Prem K. Kythe Copyright 2011
382 Pages 47 B/W Illustrations
by Chapman & Hall

382 Pages
by Chapman & Hall

Also available as eBook on:

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.

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 B: List of Fundamental Solutions
Appendix 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.

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.