Neumann Kernel Research Articles

SOLVING THE DIRICHLET PROBLEM WITH DISCONTINUOUS COEFFICIENTS IN BOUNDED MULTIPLY CONNECTED REGIONS USING A BOUNDARY INTEGRAL EQUATION WITH THE GENERALIZED NEUMANN KERNEL

This paper presents a numerical method for solving the Dirichlet problem with discontinuous coefficients in bounded multiply connected regions. The method is based on reducing the problem of solving the Dirichlet problem with discontinuous coefficients to a problem of solving Dirichlet problem with continuous coefficients. The Dirichlet problem with continuous coefficients is then solved using a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the Fast Multipole Method. Numerical results and comparison are given to illustrate the efficiency of the suggested method. Received: March 5, 2014 c © 2014 Academic Publications, Ltd. url: www.acadpubl.eu Correspondence author 448 M. Aghaeiboorkheili, A.H.M. Murid AMS Subject Classification: 30E25, 31B10

Integral Equation Approach for Computing Green’s Function on Doubly Connected Regions via the Generalized Neumann Kernel

This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presented.

Solving a Mixed Boundary Value Problem via an Integral Equation with Generalized Neumann Kernel on Unbounded Multiply Connected Region

In this paper, we solve the mixed boundary value problem on unbounded multiply connected region by using the method of boundary integral equation. Our approach in this paper is to reformulate the mixed boundary value problem into the form of Riemann-Hilbert problem. The Riemann-Hilbert problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. As an examination of the proposed method, some numerical examples for some different test regions are presented.

Integral and differential equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits

Conformal mapping is a useful tool in science and engineering. On the other hand exact mapping functions are unknown except for some special regions.In this paper we present a new boundary integral equation with classical Neumann kernel associated to f f , where f is a conformal mapping ofbounded multiply connected regions onto a disk with circular slit domain. This boundary integral equation is constructed from a boundary relationshipsatisfied by a function analytic on a multiply connected region. With f f known, one can then treat it as a differential equation for computing f .

Integral equation with the generalized neumann kernel for computing green’s function on simply connected regions

This research is about computing the Green’s functions on simply connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The numerical method for solving this integral equation is the Nystrӧm method with trapezoidal rule which leads to a system of linear equations. The linear system is then solved by the Gaussian elimination method. Mathematica plot of Green’s function for atest region is also presented.

Numerical conformal mapping and its inverse of unbounded multiply connected regions onto logarithmic spiral slit regions and straight slit regions.

This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions. For each canonical region, two integral equations are solved before one can approximate the boundary values of the mapping function. Cauchy's-type integrals are used for computing the mapping function and its inverse for interior points. This method also works for regions with piecewise smooth boundaries. Three examples are given to illustrate the effectiveness of the proposed method.

A boundary integral equation with the generalized Neumann kernel for a mixed boundary value problem in unbounded multiply connected regions

Abstract In this paper we propose a new method for solving the mixed boundary value problem for the Laplace equation in unbounded multiply connected regions. All simple closed curves making up the boundary are divided into two sets. The Dirichlet condition is given for one set and the Neumann condition is given for the other set. The mixed problem is reformulated in the form of a Riemann-Hilbert (RH) problem which leads to a uniquely solvable Fredholm integral equation of the second kind. Three numerical examples are presented to show the effectiveness of the proposed method.

A Fast Boundary Integral Equation Method for Conformal Mapping of Multiply Connected Regions

This paper presents a fast boundary integral equation method for approximating the conformal mapping from bounded and unbounded multiply connected regions of connectivity $m+1$ onto the canonical region obtained by removing $m$ rectilinear slits from a strip. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the fast multipole method. The presented method requires $O((m+1)n\ln n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component. The numerical results of some test calculations illustrate that our method has the ability to handle regions with complex geometry and very high connectivity. Besides the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's 39 canonical slit regions.

Numerical conformal mapping of multiply connected regions onto the fifth category of Koebe’s canonical slit regions

This paper presents a boundary integral method for approximating the conformal mapping from bounded multiply connected regions onto the fifth category of Koebe’s classical canonical slit regions. The method is based on a uniquely solvable boundary integral equation with generalized Neumann kernel. The results of some test calculations illustrate the performance of the presented method.

Boundary Integral Equations for Potential Flow Past Multiple Aerofoils

Two uniquely solvable boundary integral equations for calculating the incompressible potential flow past multiple aerofoils with smooth boundaries are presented. The kernels of the integral equations are the Neumann kernel and the adjoint Neumann kernel. Numerical examples reveal that the present method offers an effective solution technique for the potential flow problem.

A Boundary Integral Equation with the Generalized Neumann Kernel for a Certain Class of Mixed Boundary Value Problem

We present a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving two‐dimensional Laplace’s equation on multiply connected regions with mixed boundary condition. Two numerical examples are presented to verify the accuracy of the proposed method.

The structure of homomorphisms of connected locally compact groups into compact groups

We obtain consequences of the theorem concerning the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the commutator subgroup of the group and also of an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of soluble Lie groups. In particular, we prove that an almost connected locally compact group admitting a (not necessarily continuous) injective homomorphism into a compact group also admits a continuous injective homomorphism into a compact group, and thus the given group is a finite extension of the direct product of a compact group and a vector group. We solve the related problem of describing the images of (not necessarily continuous) homomorphisms of connected locally compact groups into compact groups. Moreover, we refine the description of the von Neumann kernel of a connected locally compact group and describe the intersection of the kernels of all (not necessarily continuous) finite-dimensional unitary representations of a given connected locally compact group. Some applications are mentioned. We also show that every almost connected locally compact group admitting a (not necessarily continuous) locally bounded injective homomorphism into an amenable almost connected locally compact group is amenable.

Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebeʼs canonical slit domains

This paper presents a boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region G onto the second, third and fourth categories of Koebeʼs canonical slit domains. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The method is an extension of the authorʼs method for the first category of Koebeʼs canonical slit domains (see [M.M.S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput. 31 (2009) 1695–1715]). Three numerical examples are presented to illustrate the performance of the proposed method.

Bootstrapping Lexical Knowledge from Unsegmented Text using Graph Kernels

Extraction of named entitiy classes and their relationships from large corpora often involves morphological analysis of target sentences and tends to suffer from out-of-vocabulary words. In this paper we propose a semantic category extraction algorithm called Monaka and its graph-based extention g-Monaka, both of which use character n-gram based patterns as context to directly extract semantically related instances from unsegmented Japanese text. These algorithms also use ``bidirectional adjacent constraints,'' which states that reliable instances should be placed in between reliable left and right context patterns, in order to improve proper segmentation. Monaka algorithms uses iterative induction of instaces and pattens similarly to the bootstrapping algorithm Espresso. The g-Monaka algorithm further formalizes the adjacency relation of character n-grams as a directed graph and applies von Neumann kernel and Laplacian kernel so that the negative effect of semantic draft, i.e., a phenomenon of semantically unrelated general instances being extracted, is reduced. The experiments show that g-Monaka substantially increases the performance of semantic category acquisition compared to conventional methods, including distributional similarity, bootstrapping-based Espresso, and its graph-based extension g-Espresso, in terms of F-value of the NE category task from unsegmented Japanese newspaper articles.

Neumann problem on a half-space

In this paper, a solution of the Neumann problem on a half-space for a slowly growing continuous boundary function is constructed by the generalized Neumann integral with this boundary function. The relation between this particular solution and certain general solutions is discussed. A solution of the Neumann problem for any continuous boundary function is also given explicitly by the Neumann integral with the generalized Neumann kernel depending on this boundary function.

Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions

This paper presents a new boundary integral method for the solution of Laplace’s equation on both bounded and unbounded multiply connected regions, with either the Dirichlet boundary condition or the Neumann boundary condition. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. Numerical results are presented to illustrate the efficiency of the proposed method.

Von Neumann kernels of connected lie groups, revisited and refined

In the note, a 30-year old result concerning the von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional continuous complex unitary representations of the group) is corrected and the smallest von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional (not necessarily continuous) complex unitary representations of the group) is described.

Numerical Conformal Mapping via a Boundary Integral Equation with the Generalized Neumann Kernel

We present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region $G$ onto the five classical canonical slit domains. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. Using the proposed method, the approximate mapping functions onto the five canonical slit domains can be computed in a unified way by solving linear systems with a common coefficient matrix. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries.

An integral equation method for conformal mapping of doubly connected regions involving the neumann kernel

We present an integral equation method for conformal mapping of doubly connected regions onto a unit disc with a circular slit of radius μ $lt$ 1. Our theoretical development is based on the boundary integral equation for conformal mapping of doubly connected region derived by Murid and Razali [15]. In this paper, using the boundary relationship satisfied by the mapping function, a related system of integral equations via Neumann kernel is constructed. For numerical experiment, the integral equation is discretized which leads to a system of linear equations, where μ is assumed known. Numerical implementation on a circular annulus is also presented. Keywords: Conformal mapping; integral equations; doubly connected regions; Neumann kernel.

A boundary integral method for the Riemann–Hilbert problem in domains with corners

This article presents a boundary integral method for the Riemann–Hilbert problems in simply connected regions with piecewise smooth boundaries. The method is based on two boundary integral equations with the generalized Neumann kernel. To illustrate the results, several numerical examples are presented where the integral equations are solved using a Nyström method based on the trapezoidal rule with a graded mesh.