Boundary integral equation method for conformal mapping of multiply connected regions

This work presents two methods for numerical conformal mappings of unbounded multiply connected regions onto several classes of canonical slit regions. The first method is only limited to conformal mapping of unbounded multiply connected regions onto the first category of Koebe's canonical regi...

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Main Author:	Mohd. Yunus, Arif Asraf
Format:	Thesis
Language:	English
Published:	2013
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/36643/1/ArifAsrafMohdYunusPFS2013.pdf
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spelling	my-utm-ep.366432017-09-19T03:24:08Z Boundary integral equation method for conformal mapping of multiply connected regions 2013 Mohd. Yunus, Arif Asraf QA Mathematics This work presents two methods for numerical conformal mappings of unbounded multiply connected regions onto several classes of canonical slit regions. The first method is only limited to conformal mapping of unbounded multiply connected regions onto the first category of Koebe's canonical regions. It is based on three boundary integral equations formed with the classical adjoint Neumann kernel, adjoint generalized Neumann kernel and modified Neumann kernel. These integral equations are constructed from a boundary relationship satisfied by an analytic function on the unbounded multiply connected regions. By adding some normalizing conditions, the integral equations are uniquely solvable. The second method is for numerical conformal mapping and its inverse of unbounded multiply connected regions onto the first, second, third and fourth category of Koebe’s canonical regions. It is based on reformulating the conformal mapping problem as a Riemann-Hilbert problem and an adjoint Riemann-Hilbert problem. Two integral equations formed with the adjoint generalized Neumann kernel are constructed. With some normalizing conditions, the integral equations are uniquely solvable. For both methods, discretizing the integral equations with their normalizing conditions lead to systems of linear algebraic equations which are solved by Gaussion elimination method to obtain the boundary values of the mapping functions. The interior values of the mapping functions are then determined by using Cauchy’s integral formula. Cauchy’s integral formula is also used to approximate the interior values of the inverse mapping functions for the second method. Some numerical examples are presented to illustrate the effectiveness of both methods for computing the conformal mappings of unbounded multiply connected regions. 2013 Thesis http://eprints.utm.my/id/eprint/36643/ http://eprints.utm.my/id/eprint/36643/1/ArifAsrafMohdYunusPFS2013.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:82889?queryType=vitalDismax&query=Boundary+integral+equation+method+for+conformal+mapping+of+multiply+connected+regions&public=true phd doctoral Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Mohd. Yunus, Arif Asraf Boundary integral equation method for conformal mapping of multiply connected regions
description	This work presents two methods for numerical conformal mappings of unbounded multiply connected regions onto several classes of canonical slit regions. The first method is only limited to conformal mapping of unbounded multiply connected regions onto the first category of Koebe's canonical regions. It is based on three boundary integral equations formed with the classical adjoint Neumann kernel, adjoint generalized Neumann kernel and modified Neumann kernel. These integral equations are constructed from a boundary relationship satisfied by an analytic function on the unbounded multiply connected regions. By adding some normalizing conditions, the integral equations are uniquely solvable. The second method is for numerical conformal mapping and its inverse of unbounded multiply connected regions onto the first, second, third and fourth category of Koebe’s canonical regions. It is based on reformulating the conformal mapping problem as a Riemann-Hilbert problem and an adjoint Riemann-Hilbert problem. Two integral equations formed with the adjoint generalized Neumann kernel are constructed. With some normalizing conditions, the integral equations are uniquely solvable. For both methods, discretizing the integral equations with their normalizing conditions lead to systems of linear algebraic equations which are solved by Gaussion elimination method to obtain the boundary values of the mapping functions. The interior values of the mapping functions are then determined by using Cauchy’s integral formula. Cauchy’s integral formula is also used to approximate the interior values of the inverse mapping functions for the second method. Some numerical examples are presented to illustrate the effectiveness of both methods for computing the conformal mappings of unbounded multiply connected regions.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Mohd. Yunus, Arif Asraf
author_facet	Mohd. Yunus, Arif Asraf
author_sort	Mohd. Yunus, Arif Asraf
title	Boundary integral equation method for conformal mapping of multiply connected regions
title_short	Boundary integral equation method for conformal mapping of multiply connected regions
title_full	Boundary integral equation method for conformal mapping of multiply connected regions
title_fullStr	Boundary integral equation method for conformal mapping of multiply connected regions
title_full_unstemmed	Boundary integral equation method for conformal mapping of multiply connected regions
title_sort	boundary integral equation method for conformal mapping of multiply connected regions
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2013
url	http://eprints.utm.my/id/eprint/36643/1/ArifAsrafMohdYunusPFS2013.pdf
_version_	1747816437486977024

Boundary integral equation method for conformal mapping of multiply connected regions

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