Green's Functions: Construction and Applications

✍ Scribed by Yuri A. Melnikov; Max Y. Melnikov

• Publisher: De Gruyter
• Year: 2012
• Tongue: English
• Leaves: 448
• Series: De Gruyter Studies in Mathematics; 42
• Category: Library

✦ Synopsis

Green's functions represent one of the classical and widely used issues in the area of differential equations.

This monograph is looking at applied elliptic and parabolic type partial differential equations in two variables. The elliptic type includes the Laplace, static Klein-Gordon and biharmonic equation. The parabolic type is represented by the classical heat equation and the Black-Scholes equation which has emerged as a mathematical model in financial mathematics. The book is attractive for practical needs: It contains many easily computable or computer friendly representations of Green's functions, includes all the standard Green's functions and many novel ones, and provides innovative and new approaches that might lead to Green's functions.

The book is a useful source for everyone who is studying or working in the fields of science, finance, or engineering that involve practical solution of partial differential equations.

✦ Table of Contents

Preface
0 Introduction
1 Green’s Functions for ODE
1.1 Standard Procedure for Construction
1.2 Symmetry of Green’s Functions
1.3 Alternative Construction Procedure
1.4 Chapter Exercises
2 The Laplace Equation
2.1 Method of Images
2.2 Conformal Mapping
2.3 Method of Eigenfunction Expansion
2.4 Three-Dimensional Problems
2.5 Chapter Exercises
3. The Static Klein-Gordon Equation
3.1 Definition of Green’s Function
3.2 Method of Images
3.3 Method of Eigenfunction Expansion
3.4 Three-Dimensional Problems
3.5 Chapter Exercises
4 Higher Order Equations
4.1 Definition of Green’s Function
4.2 Rectangular-Shaped Regions
4.3 Circular-Shaped Regions
4.4 The equation ∇2∇2w(P) + λ4w(P) = 0
4.5 Elliptic Systems
4.6 Chapter Exercises
5 Multi-Point-Posed Problems
5.1 Matrix of Green’s Type
5.2 Influence Function of a Multi-Span Beam
5.3 Further Extension of the Formalism
5.4 Chapter Exercises
6 PDE Matrices of Green’s type
6.1 Introductory Comments
6.2 Construction of Matrices of Green’s Type
6.3 Fields of Potential on Surfaces of Revolution
6.4 Chapter Exercises
7 Diffusion Equation
7.1 Basics of the Laplace Transform
7.2 Green’s Functions
7.3 Matrices of Green’s Type
7.4 Chapter Exercises
8 Black-Scholes Equation
8.1 The Fundamental Solution
8.2 Other Green’s Functions
8.3 A Methodologically Valuable Example
8.4 Numerical Implementations
8.5 Chapter Exercises
Appendix Answers to Chapter Exercises
Bibliography
Index