Green's Function in Some Contributions of 19th Century Mathematicians

Abstract

Many questions in mathematical physics lead to a solution in terms of a harmonic function in a closed region with given continuous boundary values. This problem is known as Dirichlet's problem, whose solution is based on an existence principle-the so-called Dirichlet's principle. However, in the second half of the 19th century many mathematicians doubted the validity of Dirichlet's principle. They used direct methods in order to overcome the difficulties arising from this principle and also to find an explicit solution of the Dirichlet problem at issue. Many years before, one of these methods had been developed by Green in 1828, which consists in finding a function-called a Green's function-satisfying certain conditions and appearing in the analytical expression of the solution of the given Dirichlet problem. Helmholtz, Riemann, Lipschitz, Carl and Franz Neumann, and Betti deduced functions similar to Green's function in order to solve problems in acoustics, electrodynamics, magnetism, theory of heat, and elasticity. C 2001 Academic Press Molte questioni fisico matematiche conducono a una soluzione in termini di una funzione armonica in una regione chiusa con dati valori continui al contorno. Questo problemaè noto come problema di Dirichlet, la cui soluzione si basa su un principio di esistenza, il cosiddetto principio di Dirichlet. Tuttavia, nella seconda metà del diciannovesimo secolo, molti matematici cominciarono a mettere in dubbio la validità del principio di Dirichlet. Sia per superare le difficoltà sorte da tale principio, sia per trovare una soluzione esplicita del problema di Dirichlet dato, essi presero ad adoperare metodi diretti. Molti anni prima, uno di questi metodi era stato sviluppato da Green nel 1828 e consiste nel trovare una funzione, detta funzione di Green, che soddisfa certe condizioni e mediante la quale si rappresenta analiticamente la soluzione del problema di Dirichlet in questione. Helmholtz, Riemann, Lipschitz, Carl e Franz Neumann, e Betti dedussero delle funzioni simili alla funzione di Green allo scopo di risolvere problemi di acustica, elettrodinamica, magnetismo, teoria del calore ed elasticità. C 2001 Academic Press Nombreuses questions de physique mathématique mènentà une solution en termes d'une fonction harmonique dans une région fermée avec des valeurs continus donnés sur la frontière. Ce problème est connu comme problème de Dirichlet, la solution duquel est fondée sur un principe d'existence, le principe de Dirichlet. Cependant dans la seconde moitié du dix-neuvième siècle plusieurs mathémati-ciens mirent en doute la validité du principe de Dirichlet. Alors ils employèrent des méthodes directes soit pour surmonter le difficultés nées de ce principe, soit pour déduire une solution explicite du problème de Dirichlet en question. Avant plusieurs années une de ces méthodes aété développée par Green en 1828 et consisteà trouver une fonction, dite fonction de Green, qui satisfait certaines conditions et moyennant laquelle on représente analytiquement la solution du problème de D...

Heritage and early history of the boundary element method

“…Green's formula (25), originally designed to solve electrostatic problems, was such a success that the idea was followed to solve many other physical problems [166]. For example, Hermann Ludwig Ferdinand von Helmholtz (1821-1894) in his study of acoustic problems presented the following equation in 1860 [87], known as the Helmholtz equation…”

“…Furthermore, G takes the null value at the boundary G. G is known as the Green's function. Green went on to prove that for a harmonic function f, whose boundary value is given by a continuous function f(x), x2G, its solution is represented by the boundary integral equation [100,166] fðxÞ Z K 1 4p…”

Historical and epistemological point of view of mathematical physics

“…Their goal was to develop procedures and methods of potential theory to determine the solution of problems similar to that of Dirichlet [29]. Among these, the so-called Neumann problem, which aims to determine a function V harmonic in a domain with assigned values for the normal derivative of V on the boundary, must be mentioned.…”

“…In the period between 1860 and 1870 some mathematicians, including Enrico Betti, Rudolf Otto Sigismund Lipschitz (1832–1903), Franz Ernst Neumann (1798–1895), and Carl Gottfried Neumann, tried to deduce the functions that held the same role as the Green’s function in the theories of heat, magnetism, and electrodynamics of elasticity. Their goal was to develop procedures and methods of potential theory to determine the solution of problems similar to that of Dirichlet [29]. Among these, the so-called Neumann problem, which aims to determine a function V harmonic in a domain with assigned values for the normal derivative of V on the boundary, must be mentioned.…”

Interplay between local and international journals: The case of Sicily, 1880–1920

“…This memoir was published in Acta Mathematica in 1909[Lauricella, 1909]. For details on the Green method in the 19th century, see[Tazzioli, 2001].…”