Green's Functions (eBook)

lt;!doctype html public "-//w3c//dtd html 4.0 transitional//en">

Yuri A. Melnikov,Middle Tennessee State University, Murfreesboro, Tennessee, USA; Max Y.Melnikov, Cumberland University, Lebanon, Tennessee, USA.

Preface 7
0 Introduction 13
1 Green’s Functions for ODE 21
1.1 Standard Procedure for Construction 21
1.2 Symmetry of Green’s Functions 37
1.3 Alternative Construction Procedure 45
1.4 Chapter Exercises 63
2 The Laplace Equation 67
2.1 Method of Images 67
2.2 Conformal Mapping 86
2.3 Method of Eigenfunction Expansion 92
2.4 Three-Dimensional Problems 134
2.5 Chapter Exercises 140
3. The Static Klein-Gordon Equation 142
3.1 Definition of Green’s Function 142
3.2 Method of Images 145
3.3 Method of Eigenfunction Expansion 157
3.4 Three-Dimensional Problems 172
3.5 Chapter Exercises 177
4 Higher Order Equations 179
4.1 Definition of Green’s Function 180
4.2 Rectangular-Shaped Regions 181
4.3 Circular-Shaped Regions 192
4.4 The equation .2.2w(P) + .4w(P) = 0 211
4.5 Elliptic Systems 218
4.6 Chapter Exercises 235
5 Multi-Point-Posed Problems 238
5.1 Matrix of Green’s Type 239
5.2 Influence Function of a Multi-Span Beam 250
5.3 Further Extension of the Formalism 262
5.4 Chapter Exercises 275
6 PDE Matrices of Green’s type 277
6.1 Introductory Comments 277
6.2 Construction of Matrices of Green’s Type 280
6.3 Fields of Potential on Surfaces of Revolution 305
6.4 Chapter Exercises 327
7 Diffusion Equation 329
7.1 Basics of the Laplace Transform 330
7.2 Green’s Functions 334
7.3 Matrices of Green’s Type 362
7.4 Chapter Exercises 372
8 Black-Scholes Equation 374
8.1 The Fundamental Solution 375
8.2 Other Green’s Functions 382
8.3 A Methodologically Valuable Example 400
8.4 Numerical Implementations 405
8.5 Chapter Exercises 413
Appendix Answers to Chapter Exercises 414
Bibliography 433
Index 439

lt;P>"[...] the book is a useful source for everyone who is studying or working in fields of science, finance, or engineering that involve practical solutions of partial differential equations."
L'Enseignement Mathématique 2/2012

"The monograph is a valuable source to any researcher or postgraduate student working in applied sciences or engineering that concern practical solutions of partial or ordinary differential equations."
Marius Ghergu, Zentralblatt für Mathematik