Some integral equations related to the Ahlfors map for multiply connected regions

2015 ◽  
Author(s):  
Kashif Nazar ◽  
Ali H. M. Murid ◽  
Ali W. K. Sangawi
Keyword(s):  
Integral Equations ◽  
Multiply Connected ◽  
Ahlfors Map ◽  
Multiply Connected Regions

scholarly journals Circular Slits Map of Bounded Multiply Connected Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Ali W. K. Sangawi ◽  
Ali H. M. Murid ◽  
M. M. S. Nasser
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Connected Region ◽  
Numerical Conformal Mapping ◽  
Multiply Connected ◽  
Circular Slit ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

We present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a circular slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized, and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.

scholarly journals Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing

2020 ◽  
Vol 44 (1) ◽  
pp. 171-202
Author(s):  
Ali W. K. Sangawi ◽  
Ali H. M. Murid ◽  
Khiy Wei Lee
Keyword(s):  
Image Processing ◽  
Integral Equations ◽  
Complex Geometry ◽  
Boundary Integral Equations ◽  
Fast Multipole Method ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Brain Image ◽  
Multiply Connected ◽  
Multiply Connected Regions

AbstractIn this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is $$O((M + 1)n)$$ O ( ( M + 1 ) n ) , where $$M+1$$ M + 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require $$O((M+1)^3 n^3)$$ O ( ( M + 1 ) 3 n 3 ) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.

scholarly journals Conformal Mapping of Unbounded Multiply Connected Regions onto Canonical Slit Regions

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Arif A. M. Yunus ◽  
Ali H. M. Murid ◽  
Mohamed M. S. Nasser
Keyword(s):  
Integral Equation ◽  
Conformal Mapping ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Linear Boundary ◽  
Multiply Connected ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels.

scholarly journals Numerical Conformal Mapping of Unbounded Multiply Connected Regions onto Circular Slit Regions

2014 ◽  
Vol 8 (1) ◽  
Author(s):  
A.A.M. Yunus ◽  
A.H.M. Murid ◽  
M.M. S. Nasser
Keyword(s):  
Conformal Mapping ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Numerical Conformal Mapping ◽  
Multiply Connected ◽  
Circular Slit ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

This paper presents a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto circular slit regions. Three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the classical and the adjoint generalized Neumann kernels. Several numerical examples are presented.

Integral Equation for the Ahlfors Map on Multiply Connected Regions

2015 ◽  
Vol 73 (1) ◽  
Author(s):  
Kashif Nazar ◽  
Ali H. M. Murid ◽  
Ali W. K. Sangawi
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Connected Region ◽  
Multiply Connected ◽  
Multiple Method ◽  
Boundary Correspondence ◽  
Ahlfors Map ◽  
Multiply Connected Regions ◽  
Multiply Connected Region

This paper presents a new boundary integral equation with the adjoint Neumann kernel associated with  where  is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for  using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of    we solve for the boundary correspondence function  which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed method.

Conformal and Quasiconformal Mappings of Close Multiply-Connected Domains

2002 ◽  
Vol 9 (2) ◽  
pp. 367-382
Author(s):  
Z. Samsonia ◽  
L. Zivzivadze
Keyword(s):  
Integral Equations ◽  
Quasiconformal Mappings ◽  
Multiply Connected Domains ◽  
Multiply Connected

Abstract Doubly-connected and triply-connected domains close to each other in a certain sense are considered. Some questions connected with conformal and quasiconformal mappings of such domains are studied using integral equations.

scholarly journals A characterization of internal functions on multiply connected regions

1984 ◽  
Vol 96 ◽  
pp. 23-28
Author(s):  
Lee A. Rubel
Keyword(s):  
Hardy Spaces ◽  
Multiply Connected ◽  
Multiply Connected Regions ◽  
Internal Function

The notion of internal function enters naturally in the study of factorization of function in Lumer’s Hardy spaces—see [RUB], where this aspect is developed in some detail.

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