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.

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.

scholarly journals Concentration of dynamic stresses in an elastic space with twoperiodic array of elliptical cracks

2019 ◽  
pp. 18-25
Author(s):  
Igor Zhbadynskyi
Keyword(s):  
Integral Equation ◽  
Green’S Function ◽  
Boundary Integral Equation ◽  
Green's Function ◽  
Scattering Problem ◽  
Mapping Method ◽  
Boundary Integral ◽  
Elastic Space ◽  
Wide Range ◽  
Elliptical Cracks

Normal incidence of the plane time-harmonic longitudinal wave on double-periodic array of coplanar elliptical cracks, which are located in 3D infinite elastic space is considered. Corresponding symmetric wave scattering problem is reduced to a boundary integral equation for the displacement jump across the crack surfaces in a unit cell by means of periodic Green’s function, which is presented in the form of Fourier integrals. A regularization technique for this Green’s function involving special lattice sums in closed forms is adopted, which allows its effective calculation in a wide range of wave numbers. The boundary integral equation is correctly solved by using the mapping method. The frequency dependencies of mode-I stress intensity factor in the vicinity of the crack front points for periodic distances in the system of elliptical cracks are revealed.

Integral equation approach to the problem of scattering by a number of fixed scatterers

1977 ◽  
Vol 55 (16) ◽  
pp. 1442-1452
Author(s):  
M. Hron ◽  
M. Razavy
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inhomogeneous Term ◽  
Integral Equation Approach ◽  
An Array Of Cylinders

In the derivation of the Lippmann–Schwinger integral equation for scattering of a wave ψ(r) by the potential ν(r), one constructs the Green's function for the operator [Formula: see text], and treats νψ as the inhomogeneous term. However, in certain cases, it is desirable to formulate the scattering problem in terms of an integral equation by obtaining the Green's function for the operator [Formula: see text], and by considering (−k2ψ) as the inhomogeneous term. An important aspect of this formulation is that the resulting integral equation can be used to generate a low energy expansion of the wave function for some separable and nonseparable systems. For two-dimensional scattering, if the geometry of the scatterers is simple enough, the Laplace equation with the prescribed boundary conditions on the surface of the scatterers is separable in a certain coordinate system, then one can write the solution of the wave equation as an inhomogeneous integral equation. In this way the problems of scattering by two cylinders, an array of cylinders, and a grating can be formulated in terms of integral equations. For three-dimensional scattering, one can consider either the spherically symmetric cases or nonseparable problems. In the former case, for certain types of force laws, a Volterra integral equation in one variable can be found for the wave function. In the latter case, integral equations in two or three variables can be obtained for scattering by two spheres or by a torus.

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.

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