GREEN FUNCTION
DOI:
https://doi.org/10.65405/prckhc65Abstract
The green function theory was developed by the scientist George Green 1793-1841 and is a mathematical function. Where this theory was developed to improve and manage ordinary and partial differential equations with different dimensions and for a time-dependent and time-independent problem .The theory was also developed to serve physics and mechanics, largely in quantum field theory and thermodynamics, as well as in statistical field theory. Thus, during the understanding and analysis of the theory of green function, its role and importance in science were shown, and many applications were presented to understand it, such as i.e. Boundary and Initial Value problem, Wave Equation, Kirchhoff Diffusion Equation, Diffraction Theory, Helmholtz Equation and etc.
So the green function have many important roles in many aspect of sciences, so we try to cover the theory from all of it's aspect.
Downloads
References
1- Any theory in which the basic quantities are fields, such as electromagnetic theory.
2- Green, G., 1850, 1852, 1854: An essay on the application of mathematical analysis to the theories of electricity and magnetism. J. Reine Angew. Math.
3- Poincar´e, H., 1894: Sur les ´equations de la physique math´ematique. Rend. Circ. Mat. Palermo.
4- Morse, P. M., and H. Feshbach, 1946: Methods of Theoretical Physics. MIT Technology Press.
5- Mackie, A. G., 1965: Boundary Value Problems. Oliver & Boyd.
6- Neumann, C., 1861: Ueber die Integration der partiellen Differentialgleichung.
7- Meutzner, P., 1875: Untersuchungen im Gebiete des logarithmischen Potentiales.Math. Ann.,. For an alternative derivation, see Sections 15 and 17 in Neumann, C., 1906: Uber das logarithmische Potential. ¨ Ber. Verh. K. Sachs. Ges. Wiss.Leipzig, Math.-Phys. Klasse.
8- See Chapter 2 in Harnack, A., 1887: Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene. Leipzig, B. G. Teubner.
9- Dougall, J., 1900: The determination of Green’s function by means of cylindrical or spherical harmonics. Proc. Edinburgh Math.
10- Greenhill, A. G., 1879: On Green’s function for a rectangular parallelepiped. Proc. Cambridge Philos.
11- Macdonald, H. M., 1895: The electrical distribution on a conductor bounded by two spherical surfaces cutting at any angle. Proc. London Math. Soc., Ser. Macdonald, H. M., 1900: Demonstration of Green’s formula for electric density near the vertex of a right cone. Trans. Cambridge Philos.
12- Sommerfeld, A., 1897: Uber verzweigte Potentiale im Raum. ¨ Proc. London Math.
13- Hobson, E. W., 1900: On Green’s function for a circular disc, with applications to electrostatic problems. Trans. Cambridge Philos.
14- Waldmann, L., 1937: Zwei Anwendungen der Sommerfeld’schen Methode der verzweigten Potentiale. Phys.
15- Kneser, A., 1911: Integralgleichungen und ihre Anwendungen in der mathematischenPhysik. Braunschweig.
16- See, for example, Bouwkamp, C. J., and N. G. de Bruijn, 1947: The electrostatic field of a point charge inside a cylinder, in connection with wave guide theory. J. Appl. Phys, This paper is of particular note because of its use of the modern definition of the delta function.
17- Weber, E., 1939: The electrostatic potential produced by a point charge on the axis of a cylinder. J. Appl. Phys.
18- Thomson, W., 1845: Extrait d’une lettre de M. William Thomson ` a M. Liouville. J. Math. Pures Appl; Thomson, W., 1847: Extraits de deux lettres address´ees a M. Liouville. J. Math. Pures Appl
19- .Thomson, W., 1854/55: On the theory of the electric telegraph. Proc. R. Soc. London,.
20- Hobson, E. W., 1887: Synthetical solutions in the conduction of heat. Proc. London Math.
21- Hobson, E. W., 1888: On a radiation problem. Math. Proc. Cambridge Philos.
22- Bryan, G. H., 1892: Note on a problem in the linear conduction of heat. Math. Proc.Cambridge Philos.
23- Sommerfeld, A., 1894: Zur analytischen Theorie der W¨armeleitung. Math.
24- Solovieff, P. V., 1939: Die Greensche Funktion der W¨ armeleitungsgleichung. Dokl. Acad. Sci. USSR,; Solovieff, P. V., 1939: Fonctions de Green des ´equations paraboliques. Dokl. Acad. Sci. USSR
25- .Dougall, J., 1901: Note on the application of complex integration to the equation of conduction of heat, with special reference to Dr. Peddie’s problem. Proc. Edinburgh Math.
26- Carslaw, H. S., 1902: The use of Green’s functions in the mathematical theory of the conduction of heat. Proc. Edinburgh Math.
27- Bernstein, F., and G. Doetsch, 1925: Probleme aus der Theorie der W¨armeleitung. I. Mitteilung. Eine neue Methode zur Integration partieller Differentialgleichungen. Der lineare W¨armeleiter mit verschwindender Anfangstemperatur. Math.; Doetsch, G., 1925: Probleme aus der Theorie der W¨ armeleitung. II. Mitteilung. Der lineare W¨armeleiter mit verschwindender Anfangstemperatur. Die allgemeinste L¨ osung und die Frage der Eindeutigkeit. Math; Doetsch, G., 1925: Probleme aus der Theorie der W¨armeleitung. III. Mitteilung. Der lineare W¨armeleiter mit beliebiger Anfangstemperatur. Die zeitliche Fortsetzung des W¨ armezustandes. Math; Bernstein, F., and G. Doetsch, 1927: Probleme aus der Theorie der W¨armeleitung. IV. Mitteilung. Die r¨aumliche Fortsetzung des Temperaturablaufs (Bolometerproblem). Math.
28- Goldstein, S., 1932: Some two-dimensional diffusion problems with circular symmetry. Proc. London Math.
29- Lowan, A. N., 1937: On the operational determination of Green’s functions in the theory of heat conduction. Philos. Mag.
30- Lowan, A. N., 1938: On the operational determination of two dimensional Green’s function in the theory of heat conduction. Bull. Amer. Math.
31- Lowan, A. N., 1939: On Green’s functions in the theory of heat conduction in spherical coordinates. Bull. Amer. Math.
32- Carslaw, H. S., and J. C. Jaeger, 1939: On Green’s functions in the theory of heat conduction. Bull. Amer. Math.
33- Carslaw, H. S., 1940: A simple application of the Laplace transformation. Philos. Mag.
34- arslaw, H. S., and J. C. Jaeger, 1940: The determination of Green’s function for the equation of conduction of heat in cylindrical coordinates by the Laplace transformation. J. London Math.
35- Carslaw, H. S., and J. C. Jaeger, 1941: The determination of Green’s function foline sources for the equation of conduction of heat in cylindrical coordinates by the Laplace transformation. Philos. Mag.
36- Helmholtz, H., 1860: Theorie der Luftschwingungen in R¨ohren mit offenen Enden. J. Reine Angew. Math.
37- Pockels, F., 1891: Uber die partielle Di ¨ fferentialgleichung ∆u + k2u = 0 und deren Auftreten in der mathematischer Physik. Leipzig, Teubner.
38- Sommerfeld, A., 1912: Die Greensche Funktion der Schwingungsgleichung. Jahresber. Deutsch. Math.-Verein.
39- Carslaw, H. S., 1912: Integral equations and the determination of Green’s functions in the theory of potential. Proc. Edinburgh Math.; Carslaw, H. S., 1914: The Green’s function for the equation ∇2u + k2u = 0. Proc. London Math. Soc.
40- See Schot, S. M., 1992: Eighty years of Sommerfeld’s radiation condition. Hist. Math.
41- Lyra, G., 1943: Theorie der station¨ aren Leewellenstr¨omung in freier Atmosph¨are. Zeit. Angew. Math. Mech.
42- Foldy, L. L., and H. Primakoff, 1945: A general theory of passive linear electroacoustic transducers and the electroacoustic reciprocity theorem. I. J. Acoust. Soc. Am.
43- Riemann, B., 1860: Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Abh. d. K¨on. Ges. der Wiss. zu G¨ottingen,. An English translation appears in Johnson, J. N., and R. Ch´eret, 1998: Classic Papers in Shock Compression Science. Springer-Verlag.
44- Mackie, A. G., 1964/65: Green’s function and Riemann’s method. Proc. Edinburgh Math.
45- For the derivation, see Section 73 in Webster, A. G., 1966: Partial Differential Equations of Mathematical Physics. Dover.
46- Copson, E. T., 1958: On the Riemann-Green function. Arch. Rat. Mech. Anal.
47- Picard, E., 1894: Sur l’ equation aux d´eriv´ees partielles qui se recontre dans la th´eorie de la propagation de l’´electricit´e. Acad. Sci., Compt. Rend.,; Bois-Reymond, P. du, 1889: Uber lineare partielle Di ¨ fferentialgleichungen zweiter Ordnung. J. Reine Angew. Math.,; Voigt, W., 1899: Ueber die Aenderung der Schwingungsform des Lichtes beim Fortschreiten in einem dispergirenden oder absorbirenden Mittel. Ann. Phys., Gray, M. C., 1923: The equation of telegraphy. Proc. Edinburgh Math. Soc., Rademacker, H., and R. Iglisch, 1961: Randwertprobleme der partiellen Differentialgleichungen zweiter Ordnung, 779–828 in Frank, Ph., and R. von Mises, 1961: Die Differential- und Integralgleichungen der Mechanik und Physik. I. Mathematischer Teil. Dover, A. G., 1966: Partial Differential Equation of Mathematical Physics. Dover, 446 pp.; Wahlberg, C., 1977: Riemann’s function for a Klein-Gordon equation with a non-constant coefficient. J. Phys; Asfar, O. R., 1990: Riemann-Green function solution of transient electromagnetic plane waves in lossy media. IEEE Trans. Electromagn. Compat.
48- Kirchhoff, G., 1882: Zur Theorie der Lichtstrahlen. Sitzber. K. Preuss. Akad. Wiss. Berlin, 641–669; reprinted a year later in Ann. Phys. Chem., Neue Folge,.
49- This appears to have been done by Gutzmer, A., 1895: Uber den analytischen Aus- ¨ druck des Huygens’schen Princips. J. Reine Angew. Math.
50- Poincar´e, H., 1893: Sur la propagation de l’´electricit´e. Acad. Sci., Compt. Rend,Webster, A. G., 1966: Partial Differential Equation of Mathematical Physics. Dover, Section 46.
51- Bromwich, T. J. I’A., 1914: Normal coordinates in dynamical systems. Proc. London Math.
52- Lowan, A. N., 1941: On the problem of wave-motion for the wedge of an angle. Philos. Mag.
53- Lowan, A. N., 1939: On wave motion in an infinite solid bounded internally by a cylinder or a sphere. Bull. Amer. Math.
54- Walters, A. G., 1949: The solution of some transient differential equations by means of Green’s functions. Proc. Cambridge Philos. Soc., 45, 69–80; Walters, A. G., 1951: On the propagation of disturbances from moving sources. Proc. Cambridge Philos.
55- Bhabha, H. J., 1939: Classical theory of mesons. Proc. R. Soc. London, Ser. A, 172.
56- Van der Pol, B., and H. Bremmer, 1964: Operational Calculus Based on the Two-Sided Laplace Transform. Cambridge.
57- Morse, P. M., and H. Feshbach, 1953: Methods of Theoretical Physics. Part I: Chapters 1 to 8. McGraw-Hill
58- .Friedlander, F. G., 1954: Diffraction of pulses by a circular cylinder. Commun. Pure Appl. Math.
59- Cagniard, L., 1935: Diffraction d’une onde progressive par un ´ecran en forme de demiplan. J. Phys. Radium, Ser.; Cagniard, L., 1935: Diffraction d’une onde harmonique par un ´ecran en forme de demi-plan. J. Phys. Radium.
60- Schouten, G., 1999: Two-dimensional effects in the edge sound of vortices and dipoles. J. Acoust. Soc. Am.
61- Turner, R. D., 1956: The diffraction of a cylindrical pulse by a half-plane. Q. Appl. Math.
62- Schouten, op. cit.
63- Burkhardt, M. H., 1894: Sur les fonctions de Green relatives a un domaine d’une dimension. Bull. Soc. Math.
64- Bˆocher, M., 1901: Green’s function in space of one dimension. Bull. Amer. Math.
65- Whyburn, W. M., 1924: An extension of the definition of the Green’s function in one dimension. Ann. Math.
66- Bˆocher, M., 1911/12: Boundary problems and Green’s functions for linear differential and difference equations. Ann. Math.
67- Ince, E. L., 1956: Ordinary Differential Equations. Dover.
68- Birkhoff, G. D., 1908: Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math.
69- Bounitzky, E., 1909: Sur la fonction de Green des ´equations diff´erentielles lin´eaires ordinaires. J. Math. Pures Appl.
70- Hilb, E., 1911: Uber Reihenentwicklungen nach den Eigenfunktionen linearer Diffe- ¨ rentialgleichungen 2ter Ordnung. Math.
71- Sommerfeld, A., 1910: Die Greensche Funktion der Schwingungsgleichungen f¨ ur ein beliebiges Gebiet. Phys. Z.
72- Kneser, op. cit.
73- Sommerfeld, A., 1912: Die Greensche Funktion der Schwingungsgleichung. Jahrber. Deutsch. Math.-Verein.
74- Schot, S. M., 1992: Eighty years of Sommerfeld’s radiation condition. Hist. Math
75- ,See Section 1.22 and 1.23 in Bateman, H., 1959: Partial Differential Equations of Mathematical Physics. Cambridge, See also Von Mises, R., Ph. Frank, H. Weber, and B. Riemann, 1925: Die Differential- und Integralgleichungen der Mechanik und Physik. Vol I. Braunschweig, F. Vieweg.
76- Kneser, A., 1914: Belastete Integralgleichungen. Rend. Circ. Matem. Palermo.
77- Hilbert, D., 1912: Grundz¨ uge einer allgemeiner Theorie der linearen Integralgleichungen. Leipzig, B. G. Teubner.
78- Westfall, W. D. A., 1909: Existence of the generalized Green’s function. Ann. Math.
79- Elliott, W. W., 1928: Generalized Green’s function for compatible differential systems. Amer. J. Math.,; Elliott, W. W., 1929: Green’s function for differential systems containing a parameter. Amer. J. Math.
80- Loud, W. S., 1970: Some examples of generalized Green’s functions and generalized Green’s matrices. SIAM Rev.
81- Locker, J., 1977: The generalized Green’s function for the nth order linear differential operator. Trans. Amer. Math.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Comprehensive Journal of Science

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.








