Harmonic Green Function Research Articles

Hyperbolic tessellation: harmonic Green function for a Schweikart triangle in hyperbolic geometry*

Parqueting reflection is used to construct the harmonic Green function for a Schweikart triangle in the hyperbolic plane.

Polyanalytic Neumann-n problem in the unit disk

An iterated Neumann boundary value problem for the polyanalytic operator is investigated in the unit disc of the complex plane. Although this problem can be treated for domains in the complex plane having a harmonic Green function, the unit disk is particularly interesting as it is the origin of complex boundary value problems and the prototype of simply connected domains. Moreover, the resulting formulas for solutions and for solvability conditions are expressed by integral formulas with explicit kernels. By the way, the result verifies the findings for domains with harmonic Green function.

A Neumann problem for the polyanalytic operator in planar domains with harmonic Green function

The higher order Pompeiu integral operator providing a particular solution to the inhomogeneous polyanalytic equation provides via the higher order Cauchy–Pompeiu representation an attempt for solving the respective higher order Dirichlet problem under certain solvability conditions. An alteration of this Pompeiu operator, the Schwarz–Pompeiu integral serves to solve the well-posed Schwarz problem for the polyanalytic operator via the Cauchy–Schwarz–Pompeiu representation. This alteration, explicitly known for the unit disc, is available for domains with the harmonic Green function. Another alteration of the Pompeiu operator with regard to the Neumann problem is introduced also for domains with the harmonic Green function. In fact, the novelty of the new integral operator is not the operator itself. For the unit disc, it is just the known higher order polyanalytic Pompeiu operator. For other domains however, an explicit evaluation of the integrals involved is not possible. The novelty is the combination with the effect of the harmonic Green function, which enables to handle the Neumann problem. As the higher order Neumann problem, however, is also over-determined for the polyanalytic operator, again solvability conditions on the data are required. They are worked out here in combination with the representation formula for the solution.

Polyanalytic boundary value problems for planar domains with harmonic Green function

There are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

An iterative real sequence based on and providing a plane parqueting and harmonic Green functions

The real monotone decreasing sequence converging to 1 is used to construct a parqueting of the complex plane leading to the harmonic Green function and the Poisson kernel for the countably many domains of the parqueting clustering at the point 1.

Harmonic Green and Neumann functions for domains bounded by two intersecting circular arcs

ABSTRACT The parqueting-reflection principle is shown to work for constructing the harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two intersecting circular arcs in with a particular angle . With the explicit expressions of the harmonic Green and Neumann functions, we solve the Dirichlet and Neumann boundary problems to the Poisson equation in these domains by applying the Green and Neumann representation formulas.

Reduced modelling computation of layered soil's harmonic green functions

Ground-borne vibrations are disturbing to human beings. In order to model and reduce these vibrations, the calculation of the harmonic Green's-functions of the soil is highly required since the most effective numerical solution to predict ground-borne vibration is to couple the finite-element and the boundary-element methods. In this work, we elaborate a direct space-frequency formulation that is especially suitable of material exhibiting visco-elastic behavior. Moreover, an original formulation of the Proper Generalized Decomposition (PGD) reduction technique was undertaken in the frequency-space domain. This new approach highly shortens time computing costs. It is mainly valuable when a parametric study is concerned. Moreover, the PGD resolution allows providing an off-line solution, thus, the solution is calculated only once.

A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem revisited

The extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general. They do hold only for certain domains.

On a weighted harmonic Green function and a theorem of Littlewood

We continue our study of weighted harmonic functions. The emphasis is put on properties of conformal invariance and associated Green functions. A classical result of Littlewood about pointwise limits almost everywhere of Green potentials of measures in the disc is, as far as the unit disc is concerned, generalized to the full scale of Green potentials considered.

Harmonic Green functions for the Almaty apple

The parqueting reflection principle is suitable for constructing harmonic Green functions for certain plane domains with boundaries consisting of sub-arcs from circles and straight lines. The (continued) reflection at the boundary parts has to provide a parqueting of the complex plane. Occasionally, however, a parqueting of is not sufficient and multiple parqueting is required, e.g. for the disc sector with opening angle , where a double parqueting is needed, then compensated by applying the square root mapping. The parqueting-reflection principle is more convenient to apply than basing on the conformal invariance of the harmonic Green functions, see e.g., but sometimes the latter is applicable where the first is not. Here the situation is studied for the Almaty apple, a modified half disc. While the bigger of this cut disc is also requiring a double parqueting, for the smaller part a simple one suffices. However, for the bigger part a proper square root reduction to a simple parqueting is still to be determined in order to find the harmonic Green (and Neumann) function. The idea to use this disc cap for a parqueting of the complex plane has occured while two of the authors were visiting a monument of an apple in Almaty, the centre of an area full of wild apple trees considered as the locus of origin of the apple fruit.

A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem

ABSTRACTAccording to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.

Dirichlet Problem for Complex Poisson Equation in a Half Hexagon Domain

The parqueting-reflection method is applied to a nonregular domain and the harmonic Green function for the half hexagon is constructed. The related Dirichlet problem for the Poisson equation is solved explicitly.

Green function for a hyperbolic strip and a class of related plane domains

The parqueting reflection principle serves to construct the harmonic Green function for a hyperbolic strip in the unit disc. The formula obtained represents the Green functions also for the set of strips covering the unit disc and related to the original strip by continued reflection at the boundaries. A complementary unbounded domain is also included. The complex plane is covered by the closures of countably many domains having all the same expression for their Green functions.

Time dependent flexural gravity waves in the presence of current

In the present study, a combined effect of current and compressive force on time dependent flexural gravity wave motion in both the cases of single and two-layer fluids is analyzed in finite and infinite water depths in two dimensions. The roots of the dispersion relation associated with the plane flexural gravity waves are analyzed via contour plots and by plotting various terms of the dispersion relation separately. The characteristic of plane flexural gravity waves is studied by analyzing the phase and group velocities along with the law of conservation of energy flux to understand the combined effect of current and compressive force on the wave motion. The integral form of the time dependent Green's function in the presence of current is obtained using the Laplace transform method and used in Green's identity to derive the time dependent velocity potential for the flexural gravity wavemaker problem. The time harmonic Green's function and velocity potentials are obtained as a special case from the time dependent problems. Numerical results are computed and analyzed in particular cases using the method of stationary phase to obtain the asymptotic results for Green's function and the deflection of ice sheet. The integral form of Green's function derived here will be suitable to deal with physical problem when the roots of the dispersion relation for the flexural gravity wave problem coalesces which were otherwise not possible in the eigenfunction expansion method used for time harmonic problems.

Dirichlet problem for a generalized inhomogeneous polyharmonic equation in an annular domain

In this article, we investigate the solvability of the Dirichlet problems in ring domains for elliptic linear complex partial differential equations having polyharmonic operators as main parts. First, we give higher order Green functions as fundamental solutions of the homogeneous problems using the iteration of harmonic Green functions for ring domains. Second, we introduce some classes of operators related to Dirichlet problems together with their basic properties. Next, we transform the original problems into equivalent singular integral equations. Finally, solvability of the problems is discussed by defining the adjoint problems and using Fredholm alternative.

How to find harmonic Green functions in the plane

There are three different methods to determine harmonic Green functions for planar domains. One is based on the Schwarz boundary value problem for analytic functions. Another one is utilizing the conformal invariance of the harmonic Green function. Finally, the Schwarz reflection serves for domains the boundary of which is composed by segments of circles and lines providing a parqueting of the complex plane. Each of these methods is used exemplarily to find the harmonic Green function for particular domains.

Harmonic Green's functions for flexural waves in semi-infinite plates with arbitrary boundary conditions and high-frequency approximation for convex polygonal plates

This paper provides a method for obtaining the harmonic Green's function for flexural waves in semi-infinite plates with arbitrary boundary conditions and a high frequency approximation of the Green's function in the case of convex polygonal plates, by using a generalised image source method. The classical image source method consists in describing the response of a point-driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate, which represent successive reflections on the boundaries. The proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence. This is achieved by writing the original source as a Fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries. Thus, the image source contributions arise as continuous sums of reflected plane waves. For semi-infinite plates, the exact Green's function is obtained for an arbitrary set of boundary conditions. For polygonal plates, a high-frequency approximation of the Green's function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges. The method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability.

Critical points of the regular part of the harmonic Green function with Robin boundary condition

In this paper we consider the Green function for the Laplacian in a smooth bounded domain Ω ⊂ R N with Robin boundary condition ∂ G λ ∂ ν + λ b ( x ) G λ = 0 , on ∂ Ω , and its regular part S λ ( x , y ) , where b > 0 is smooth. We show that in general, as λ → ∞ , the Robin function R λ ( x ) = S λ ( x , x ) has at least 3 critical points. Moreover, in the case b ≡ const we prove that R λ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when b ≢ const there are critical points of R λ near non-degenerate critical points of b.

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part I: a 2D model

AbstractThe Green's functions for a triclinic half‐space for embedded harmonic line load are considered. Corresponding displacement and stress fields are expressed in terms of double Fourier integrals. The first integral was evaluated using contour integration while the second one was computed through the Gauss–Legendre quadrature. The resulting Green's functions algorithm avoids repeated calculations of the same quantities and utilizes the vector computational features within MATLAB environment. Extensive testing of the results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed procedure for evaluating the Green's functions without any restrictions upon material properties, frequency, and location of the source and observation points. Copyright © 2006 John Wiley & Sons, Ltd.

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part II: a 3D model

AbstractThe displacement and stress Green's functions for a 3D triclinic half‐space with embedded harmonic point load is considered. The resulting displacement and stress fields are expressed in terms of triple Fourier integrals. The first integral was evaluated using contour integration and the 3D Green's functions were obtained as a superposition of 2D results over the azimuthal angle. The resulting algorithm developed for evaluation of the Green's functions avoids repeated calculations of the same quantities and it utilizes the vectorized manipulation within MATLAB environment. The algorithm places no restriction on material properties, frequency and location of source and observation points. Extensive testing of the numerical results was performed for both displacement and stress. The tests confirm the accuracy of the numerical results. Copyright © 2006 John Wiley & Sons, Ltd.