An integral representation of a Green’s function of harmonic oscillators

1972 ◽  
Vol 5 (8) ◽  
pp. 613-619 ◽  
Author(s):  
G. Berendt ◽  
E. Weimar
Keyword(s):  
Integral Representation ◽  
Green’S Function ◽  
Green's Function ◽  
Harmonic Oscillators

scholarly journals SOME BOUNDARY VALUE PROBLEMS WITH INVOLUTION FOR THE NONLOCAL POISSON EQUATION

2020 ◽  
Vol 71 (3) ◽  
pp. 74-83
Author(s):  
M. Koshanova ◽  
◽  
М. Muratbekova ◽  
B. Turmetov ◽  
◽  
...  
Keyword(s):  
Integral Representation ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Poisson Equation ◽  
Green's Function ◽  
Boundary Value ◽  
Classical Case ◽  
Exact Solvability ◽  
Generalized Poisson ◽  
Uniqueness Of The Solution

In this paper, we study new classes of boundary value problems for a nonlocal analogue of the Poisson equation. The boundary conditions, as well as the nonlocal Poisson operator, are specified using transformation operators with orthogonal matrices. The paper investigates the questions of solvability of analogues of boundary value problems of the Dirichlet and Neumann type. It is proved that, as in the classical case, the analogue of the Dirichlet problem is unconditionally solvable. For it, theorems on the existence and uniqueness of the solution to the problem are proved. An explicit form of the Green's function, a generalized Poisson kernel, and an integral representation of the solution are found. For an analogue of the Neumann problem, an exact solvability condition is found in the form of a connection between integrals of given functions. The Green's function and an integral representation of the solution of the problem under study are also constructed.

scholarly journals Green's function for the lossy wave equation

2008 ◽  
Vol 30 (1) ◽  
pp. 1302.1-1302.5 ◽  
Author(s):  
R. Aleixo ◽  
E. Capelas de Oliveira
Keyword(s):  
Wave Equation ◽  
Integral Representation ◽  
Green’S Function ◽  
Bessel Function ◽  
Green's Function ◽  
Integral Transform ◽  
Dispersive Media ◽  
Order Partial Differential Equation ◽  
Fourier Cosine ◽  
Transient Electromagnetic

Using an integral representation for the first kind Hankel (Hankel-Bessel Integral Representation) function we obtain the so-called Basset formula, an integral representation for the second kind modified Bessel function. Using the Sonine-Bessel integral representation we obtain the Fourier cosine integral transform of the zero order Bessel function. As an application we present the calculation of the Green's function associated with a second-order partial differential equation, particularly a wave equation for a lossy two-dimensional medium. This application is associated with the transient electromagnetic field radiated by a pulsed source in the presence of dispersive media, which is of great importance in the theory of geophysical prospecting, lightning studies and development of pulsed antenna systems.

scholarly journals On a Radially Symmetrical Green’s Function

2008 ◽  
Vol Volume 9, 2007 Conference in... ◽  
Author(s):  
Ould Ahmed Izid Bih Isselkou
Keyword(s):  
Fixed Point ◽  
Integral Representation ◽  
Green’S Function ◽  
Green's Function ◽  
Representation Formula ◽  
Laplacian Operator ◽  
Elliptic Pde ◽  
Integral Representation Formula ◽  
Nous Montrons ◽  
International Audience

International audience It is quite usual to transform elliptic PDE problems of second order into fixed point integral problems, via the Green’s function. But it is not easy, in general, to handle integrals involved in such a formulation. When it comes to the Laplacian operator on balls of Rn, we give here a radially symmetrical Green’s function which, under some nonlinearity assumptions, makes the Green’s Integral representation formula easier to use; we give three examples of application. Il est courant de transformer un problème, donné sous forme d’EDP elliptique de second ordre, en un problème intégral de point fixe, et ce en utilisant la fonction de Green. En général, les intégrales intervenant dans une telle formulation, sont de maniement difficile. Lorsqu’il s’agit de l’opérateur du Laplacien sur des boules de Rn, nous montrons l’existence d’une fonction de Green à symétrie radiale; elle permet, moyennant des hypothèses adéquates sur la non linéarité, de faciliter l’usage de la Formule de représentation de Green; nous donnons trois exemples d’application.

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