1st Edition
Green's Functions and Linear Differential Equations Theory, Applications, and Computation
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 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.
