scholarly journals The properties of Green’s functions for one stationary problem with nonlocal boundary conditions

2019 ◽  
Vol 50 ◽  
Author(s):  
Svetlana Roman ◽  
Artūras Štikonas
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Stationary Problem

In this paper we research Green’s function properties for stationary problem with four-pointnonlocal boundary conditions. Dependence of these functions on values ξ and γ is investigated. Green’sfunctions graphs with various values ξ and γ are presented.

scholarly journals Linear ODE with nonlocal boundary conditions and Green’s functions for such problems

2019 ◽  
Vol 51 ◽  
pp. 379-384
Author(s):  
Svetlana Roman ◽  
Artūras Štikonas
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Linear Ode

In this article we investigate a formula for the Green’s function for the n-orderlinear differential equation with n additional conditions. We use this formula for calculatingthe Green’s function for problems with nonlocal boundary conditions.

Sign Properties of Green's Functions For Two Classes of Boundary Value Problems

1987 ◽  
Vol 30 (1) ◽  
pp. 28-35 ◽  
Author(s):  
P. W. Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Difference Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Value ◽  
Partial Derivatives

AbstractLet G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

One‐way versions of the Kirchhoff Integral

Geophysics ◽  
1989 ◽  
Vol 54 (4) ◽  
pp. 460-467 ◽  
Author(s):  
A. J. Berkhout ◽  
C. P. A. Wapenaar
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Wave Field ◽  
Green's Function ◽  
Green's Functions ◽  
Acoustic Pressure ◽  
Green’S Functions ◽  
Absorbing Boundary Conditions ◽  
Multiple Reflections ◽  
Kirchhoff Integral

The conventional Kirchhoff integral, based on the two‐way wave equation, states how the acoustic pressure at a point A inside a closed surface S can be calculated when the acoustic wave field is known on S. In its general form, the integrand consists of two terms: one term contains the gradient of a Green’s function and the acoustic pressure; the other term contains a Green’s function and the gradient of the acoustic pressure. The integrand can be simplified by choosing reflecting boundary conditions for the two‐way Green’s functions in such a way that either the first term or the second term vanishes on S. This conventional approach to deriving Rayleigh‐type integrals has practical value only for media with small contrasts, so that the two‐way Green’s functions do not contain significant multiple reflections. We present a modified approach for simplifying the integrand of the Kirchhoff integral by choosing absorbing boundary conditions for the one‐way Green’s functions. The resulting Rayleigh‐type integrals are the theoretical basis for true amplitude one‐way wave‐field extrapolation techniques in inhomogeneous media with significant contrasts.

Three-Dimensional Green’s Functions in an Anisotropic Half-Space With General Boundary Conditions

2003 ◽  
Vol 70 (1) ◽  
pp. 101-110 ◽  
Author(s):  
E. Pan
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Three Dimensional ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Different Boundary Conditions

This paper derives, for the first time, the complete set of three-dimensional Green’s functions (displacements, stresses, and derivatives of displacements and stresses with respect to the source point), or the generalized Mindlin solutions, in an anisotropic half-space z>0 with general boundary conditions on the flat surface z=0. Applying the Mindlin’s superposition method, the half-space Green’s function is obtained as a sum of the generalized Kelvin solution (Green’s function in an anisotropic infinite space) and a Mindlin’s complementary solution. While the generalized Kelvin solution is in an explicit form, the Mindlin’s complementary part is expressed in terms of a simple line-integral over [0,π]. By introducing a new matrix K, which is a suitable combination of the eigenmatrices A and B, Green’s functions corresponding to different boundary conditions are concisely expressed in a unified form, including the existing traction-free and rigid boundaries as special cases. The corresponding generalized Boussinesq solutions are investigated in details. In particular, it is proved that under the general boundary conditions studied in this paper, the generalized Boussinesq solution is still well-defined. A physical explanation for this solution is also offered in terms of the equivalent concept of the Green’s functions due to a point force and an infinitesimal dislocation loop. Finally, a new numerical example for the Green’s functions in an orthotropic half-space with different boundary conditions is presented to illustrate the effect of different boundary conditions, as well as material anisotropy, on the half-space Green’s functions.

A Short Survey on Green’s Function for Acoustic Problems

2020 ◽  
Vol 28 (02) ◽  
pp. 1950025
Author(s):  
Augustus R. Okoyenta ◽  
Haijun Wu ◽  
Xueliang Liu ◽  
Weikang Jiang
Keyword(s):  
Integral Equation ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Acoustic Medium ◽  
Acoustic Source ◽  
Impedance Boundary ◽  
Source Method

Green’s functions for acoustic problems is the fundamental solution to the inhomogeneous Helmholtz equation for a point source, which satisfies specific boundary conditions. It is very significant for the integral equation and also serves as the impulse response of an acoustic wave equation. They are important for acoustic problems that involve the propagation of sound from the source point to the observer position. Once the Green’s function, which satisfies the necessary boundary conditions, is obtained, the sound pressure at any point away from the source can be easily calculated by the integral equation. The major problem faced by researchers is in the process of constructing these Green’s functions which satisfy a specific boundary condition. The aim of this work is to review some of these fundamental solutions available in the literature for different boundary conditions for the ease of analyzing acoustics problems. The review covers the free-space Green’s functions for stationary source and rotational source, for both when the observer and the acoustic medium are at rest and when the medium is in uniform flow. The half-space Green’s functions are also summarized for both stationary acoustic source and moving acoustic source, derived using the image source method, equivalent source method and complex equivalent method in both time domain and frequency domain. Each of these methods used depends on the different impedance boundary conditions for which the Green’s function will satisfy. Finally, enclosed spaced Green’s functions for both rectangular duct and cylindrical duct for an infinite and finite duct is also covered in the review.

scholarly journals Green’s function for discrete problems with nonlocal boundary conditions

2011 ◽  
Vol 52 ◽  
pp. 291-296
Author(s):  
Svetlana Roman ◽  
Artūras Štikonas
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Green's Function ◽  
Homogeneous Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence Condition ◽  
Discrete Problem ◽  
Discrete Problems ◽  
Inhomogeneous Problems

In this article, we investigate an m-order discrete problem with additional conditions which are described by m of linearly independent linear functionals. We have presented a formula and the existence condition of Green’s function, if the general solution of a homogeneous equation is known. We have obtained the relation between two Green’s functions of two inhomogeneous problems. It allows us to find Green’s function for the same equation but with different additional conditions. The obtained results are applied to problems withnonlocal boundary conditions. This research was funded by a grant (No. MIP-051/2011) from the Research Council of Lithuania

scholarly journals Generalized Green’s functions for second-order discrete boundary-value problems with nonlocal boundary conditions

2012 ◽  
Vol 53 ◽  
pp. 96-101 ◽  
Author(s):  
Gailė Paukštaitė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Conditions ◽  
Green's Functions ◽  
Green’S Functions ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence Condition ◽  
Second Order ◽  
Linear Functionals ◽  
Discrete Green’S Function ◽  
Sufficient Existence Condition

In this paper, generalized Green’s functions for second-order discrete boundaryvalueproblems with nonlocal boundary conditions are investigated, where the necessaryand sufficient existence condition of discrete Green’s function is not satisfied and nonlocalboundary conditions are described by linear functionals.

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