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

    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
    Linear Transformations
    Cramer’s Rule
    Green’s Identities
    Differentiation and Integration

    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

    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 Solutions
    Appendix C: List of Spherical Harmonics

    Appendix D: Tables of Integral Transforms
    Appendix E: Fractional Derivatives
    Appendix F: Systems of Ordinary Differential Equations



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


    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.