Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method

In this thesis, characteristic singular integral equations of Cauchy type ()(),LtdtfxxLxtϕ=∈−∫ (1) where L is open or closed contour, are examined. The analytical solutions for equation (1) are described. Some examples of solution for certain functions f (x) are given. A quadrature formula for e...

Full description

Saved in:

Bibliographic Details
Main Author:	Mahiub, Mohammad Abdulkawi
Format:	Thesis
Language:	English English
Published:	2007
Subjects:	Cauchy integrals Numerical analysis
Online Access:	http://psasir.upm.edu.my/id/eprint/5021/1/FS_2007_30.pdf
Tags:	Add Tag No Tags, Be the first to tag this record!

id	my-upm-ir.5021
record_format	uketd_dc
spelling	my-upm-ir.50212013-05-27T07:19:49Z Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method 2007 Mahiub, Mohammad Abdulkawi In this thesis, characteristic singular integral equations of Cauchy type ()(),LtdtfxxLxtϕ=∈−∫ (1) where L is open or closed contour, are examined. The analytical solutions for equation (1) are described. Some examples of solution for certain functions f (x) are given. A quadrature formula for evaluation of Cauchy type singular integral (SI) of the form 11(),11tdtxxtϕ−−<<−∫ (2) is constructed with equal partitions of the interval [−1,1] using modification discrete vortex method (MMDV), where the singular point x is considered in the middle of one of the intervals [tj, tj+1], j=1,…, n. It is known that the bounded solution of equation (1) when L=[−1,1] is 12211,111ftxxdtttxϕπ−−=−−∫ (3) A quadrature formula is constructed to approximate the SI in (3) using MMDV and linear spline interpolation functions, where the singular point x is assumed to be at any point in the one of the intervals [tj,tj+1], j=1,…, n. The estimation of errors of constructed quadrature formula are obtained in the classes of functions C1[−1,1] and Hα(A,[−1,1]) for SI (2) and Hα(A,[−1,1]) for (3). For SI (2), the rate of convergence is improved in the class C1[−1,1], whereas in the class Hα(A,[−1,1]), the rate of convergence of quadrature formula is the same of that of discrete vortex method (MDV). FORTRAN code is developed to obtain numerical results and they are presented and compared with MDV for different functions f(t). Numerical experiments assert the theoretical results. Cauchy integrals Numerical analysis 2007 Thesis http://psasir.upm.edu.my/id/eprint/5021/ http://psasir.upm.edu.my/id/eprint/5021/1/FS_2007_30.pdf application/pdf en public masters Universiti Putra Malaysia Cauchy integrals Numerical analysis Science English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Cauchy integrals Numerical analysis
spellingShingle	Cauchy integrals Numerical analysis Mahiub, Mohammad Abdulkawi Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method
description	In this thesis, characteristic singular integral equations of Cauchy type ()(),LtdtfxxLxtϕ=∈−∫ (1) where L is open or closed contour, are examined. The analytical solutions for equation (1) are described. Some examples of solution for certain functions f (x) are given. A quadrature formula for evaluation of Cauchy type singular integral (SI) of the form 11(),11tdtxxtϕ−−<<−∫ (2) is constructed with equal partitions of the interval [−1,1] using modification discrete vortex method (MMDV), where the singular point x is considered in the middle of one of the intervals [tj, tj+1], j=1,…, n. It is known that the bounded solution of equation (1) when L=[−1,1] is 12211,111ftxxdtttxϕπ−−=−−∫ (3) A quadrature formula is constructed to approximate the SI in (3) using MMDV and linear spline interpolation functions, where the singular point x is assumed to be at any point in the one of the intervals [tj,tj+1], j=1,…, n. The estimation of errors of constructed quadrature formula are obtained in the classes of functions C1[−1,1] and Hα(A,[−1,1]) for SI (2) and Hα(A,[−1,1]) for (3). For SI (2), the rate of convergence is improved in the class C1[−1,1], whereas in the class Hα(A,[−1,1]), the rate of convergence of quadrature formula is the same of that of discrete vortex method (MDV). FORTRAN code is developed to obtain numerical results and they are presented and compared with MDV for different functions f(t). Numerical experiments assert the theoretical results.
format	Thesis
qualification_level	Master's degree
author	Mahiub, Mohammad Abdulkawi
author_facet	Mahiub, Mohammad Abdulkawi
author_sort	Mahiub, Mohammad Abdulkawi
title	Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method
title_short	Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method
title_full	Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method
title_fullStr	Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method
title_full_unstemmed	Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method
title_sort	numerical evaluation of cauchy type singular integrals using modification of discrete vortex method
granting_institution	Universiti Putra Malaysia
granting_department	Science
publishDate	2007
url	http://psasir.upm.edu.my/id/eprint/5021/1/FS_2007_30.pdf
_version_	1747810332972154880

Numerical Evaluation of Cauchy Type Singular Integrals Using Modification of Discrete Vortex Method

Similar Items