scholarly journals Conformal Invariants in Simply Connected Domains

2020 ◽  
Vol 20 (3-4) ◽  
pp. 747-775
Author(s):  
Mohamed M. S. Nasser ◽  
Matti Vuorinen
Keyword(s):  
Integral Equation ◽  
Analytic Solutions ◽  
Boundary Integral ◽  
Hyperbolic Distance ◽  
Conformal Invariants ◽  
Numerical Examples ◽  
Reduced Modulus ◽  
Simply Connected ◽  
Simply Connected Domains ◽  
Neumann Kernel

AbstractThis paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numerical examples are presented. The performance and accuracy of the presented method is validated by considering several model problems with known analytic solutions.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Mohamed M. S. Nasser ◽  
Ali H. M. Murid ◽  
Samer A. A. Al-Hatemi
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Boundary Integral ◽  
Numerical Examples ◽  
Multiply Connected ◽  
Generalized Neumann Kernel ◽  
Multiply Connected Regions ◽  
Neumann Kernel

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.

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

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Ali H. M. Murid ◽  
Mohmed M. A. Alagele ◽  
Mohamed M. S. Nasser
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Gaussian Elimination ◽  
Linear Equations ◽  
Boundary Integral ◽  
System Of Linear Equations ◽  
Generalized Neumann Kernel ◽  
Simply Connected ◽  
Neumann Kernel

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.

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

2014 ◽  
Vol 71 (1) ◽  
Author(s):  
Siti Zulaiha Aspon ◽  
Ali Hassan Mohamed Murid ◽  
Mohamed M. S. Nasser ◽  
Hamisan Rahmat
Keyword(s):  
Integral Equation ◽  
Dirichlet Problem ◽  
Linear System ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Integral ◽  
Generalized Neumann Kernel ◽  
Integral Equation Approach ◽  
Gauss Elimination ◽  
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.

A New Boundary Integral Equation Formulation for Linear Elastic Solids

1992 ◽  
Vol 59 (2) ◽  
pp. 344-348 ◽  
Author(s):  
Kuang-Chong Wu ◽  
Yu-Tsung Chiu ◽  
Zhong-Her Hwu
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Analytic Solutions ◽  
Boundary Integral ◽  
Tangential Displacement ◽  
Infinite Body ◽  
Equation Formulation ◽  
Integral Equation Formulation ◽  
Elasticity Problems

A new boundary integral equation formulation is presented for two-dimensional linear elasticity problems for isotropic as well as anisotropic solids. The formulation is based on distributions of line forces and dislocations over a simply connected or multiply connected closed contour in an infinite body. Two types of boundary integral equations are derived. Both types of equations contain boundary tangential displacement gradients and tractions as unknowns. A general expression for the tangential stresses along the boundary in terms of the boundary tangential displacement gradients and tractions is given. The formulation is applied to obtain analytic solutions for half-plane problems. The formulation is also applied numerically to a test problem to demonstrate the accuracy of the formulation.

scholarly journals The boundary integral equation method for the solution of a class of problems in anisotropic elasticity

1981 ◽  
Vol 22 (4) ◽  
pp. 394-407 ◽  
Author(s):  
David L. Clements ◽  
Oscar A. C. Jones
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Integral Method ◽  
Integral Equation Method ◽  
Anisotropic Elasticity ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Important Class ◽  
Numerical Examples ◽  
Integral Procedure

AbstractA boundary integral procedure for the solution of an important class of problems in anisotropic elasticity is outlined. Specific numerical examples are considered in order to provide a comparison with the standard boundary integral method.

Improved Integral Formulation for Acoustic Radiation Using Integral Equation of Frequency Averaged Quadratic Pressure

2017 ◽  
Vol 25 (01) ◽  
pp. 1750004 ◽  
Author(s):  
Honglin Gao ◽  
Sheng Li
Keyword(s):  
Integral Equation ◽  
Boundary Conditions ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Acoustic Radiation ◽  
Boundary Integral ◽  
Uniqueness Of Solutions ◽  
High Frequencies ◽  
Numerical Examples ◽  
Integral Equation Formulation

This paper is concerned with the problem of obtaining a unique solution for radiation at irregular frequencies when an integral equation of frequency averaged quadratic pressure (FAQP) is used to get robust predictions at medium and high frequencies. It is proved that there is no unique solution of the integral equation of FAQP at irregular frequencies, and existence and uniqueness of solutions under four types of boundary conditions are discussed. A combined energy boundary integral equation formulation (CEBIEF) is presented and proves to be efficient to overcome the nonuniqueness of the integral equation of FAQP. The numerical examples are given to demonstrate the versatility of the CEBIEF method with a proposed function correctly indicating a solution.

scholarly journals An Integral Equation Related to the Exterior Riemann- Hilbert Problem on Region with Corners

2014 ◽  
Vol 4 (2) ◽  
Author(s):  
Zamzana Zamzamir ◽  
Munira Ismail ◽  
Ali H. M. Murid
Keyword(s):  
Integral Equation ◽  
Hilbert Problem ◽  
Boundary Integral ◽  
Connected Region ◽  
Fredholm Alternative ◽  
Alternative Theorem ◽  
Smooth Region ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Riemann Hilbert Problem

Nasser in 2005 gives the first full method for solving the Riemann-Hilbert problem (briefly the RH problem) for smooth arbitrary simply connected region for general indices via boundary integral equation. However, his treatment of RH problem does not include regions with corners. Later, Ismail in 2007 provides a numerical solution of the interior RH problem on region with corners via Nasser’s method together with Swarztrauber’s approach, but Ismail does not develop any integral equation related to exterior RH problem on region with corners. In this paper, we introduce a new integral equation related to the exterior RH problem in a simply connected region bounded by curves having a finite number of corners in the complex plane. We obtain a new integral equation that adopts Ismail’s method which does not involve conformal mapping. This result is a generalization of the integral equation developed by Nasser for the exterior RH problem on smooth region. The solvability of the integral equation in accordance with the Fredholm alternative theorem is presented. The proof of the equivalence of our integral equation to the RH problem is also provided.

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

2014 ◽  
Vol 7 (1) ◽  
Author(s):  
Ali H.M. Murid ◽  
Ali W. Kareem Sangawi ◽  
M.M.S. Nasser
Keyword(s):  
Integral Equation ◽  
Conformal Mapping ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Science And Engineering ◽  
Multiply Connected ◽  
Circular Slit ◽  
Multiply Connected Regions ◽  
Multiply Connected Region ◽  
Neumann Kernel

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 .

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