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

Prem K. Kythe
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series
We do not plan to review this book.

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