Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations

In this thesis, the exact solutions of the characteristic singular integral equation of Cauchy type 1−1'(t)t − x dt = f(x), −1 < x < 1, (0.1) are described, where f(x) is a given real valued function belonging to the H¨older class and '(t) is to be determined. We also described...

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Main Author:	Mahiub, Mohammad Abdulkawi
Format:	Thesis
Language:	English English
Published:	2010
Subjects:	Singular integrals Differential equations - Numerical solutions
Online Access:	http://psasir.upm.edu.my/id/eprint/11990/1/FS_2010_7_A.pdf
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spelling	my-upm-ir.119902013-05-27T07:50:38Z Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations 2010-01 Mahiub, Mohammad Abdulkawi In this thesis, the exact solutions of the characteristic singular integral equation of Cauchy type 1−1'(t)t − x dt = f(x), −1 < x < 1, (0.1) are described, where f(x) is a given real valued function belonging to the H¨older class and '(t) is to be determined. We also described the exact solutions of Cauchy type singular integral equations of the form /1−1'(t)t − xdt +/ 1−1 K(x, t) '(t) dt = f(x), −1 < x < 1, (0.2) where K(x, t) and f(x) are given real valued functions, belonging to the H¨older class, by applying the exact solutions of characteristic integral equation (0.1) and the theory of Fredholm integral equations. This thesis considers the characteristic singular integral equation (0.1) and Cauchy type singular integral equation (0.2) for the following four cases:Case I. '(x) is unbounded at both end-points x = ±1, Case II. y(x) is bounded at both end-points x = ±1, Case III. y(x) is bounded at x = −1 and unbounded at x = 1, Case IV. y(x) is bounded at x = 1 and unbounded at x = −1. The complete numerical solutions of (0.1) and (0.2) are obtained using polynomial approximations with Chebyshev polynomials of the first kind Tn(x), second kind Un(x), third kind Vn(x) and fourth kind Wn(x) corresponding to the weight functions w1(x) = (1 − x2)−1/2 , w2(x) = (1 − x2)1/2 , w3(x) = (1 + x)1/2 (1 − x)−1/2 andw4(x) = (1 + x)−1/2 (1 − x)1/2 , respectively. Singular integrals Differential equations - Numerical solutions 2010-01 Thesis http://psasir.upm.edu.my/id/eprint/11990/ http://psasir.upm.edu.my/id/eprint/11990/1/FS_2010_7_A.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Singular integrals Differential equations - Numerical solutions Faculty Of Science English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Singular integrals Differential equations - Numerical solutions
spellingShingle	Singular integrals Differential equations - Numerical solutions Mahiub, Mohammad Abdulkawi Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
description	In this thesis, the exact solutions of the characteristic singular integral equation of Cauchy type 1−1'(t)t − x dt = f(x), −1 < x < 1, (0.1) are described, where f(x) is a given real valued function belonging to the H¨older class and '(t) is to be determined. We also described the exact solutions of Cauchy type singular integral equations of the form /1−1'(t)t − xdt +/ 1−1 K(x, t) '(t) dt = f(x), −1 < x < 1, (0.2) where K(x, t) and f(x) are given real valued functions, belonging to the H¨older class, by applying the exact solutions of characteristic integral equation (0.1) and the theory of Fredholm integral equations. This thesis considers the characteristic singular integral equation (0.1) and Cauchy type singular integral equation (0.2) for the following four cases:Case I. '(x) is unbounded at both end-points x = ±1, Case II. y(x) is bounded at both end-points x = ±1, Case III. y(x) is bounded at x = −1 and unbounded at x = 1, Case IV. y(x) is bounded at x = 1 and unbounded at x = −1. The complete numerical solutions of (0.1) and (0.2) are obtained using polynomial approximations with Chebyshev polynomials of the first kind Tn(x), second kind Un(x), third kind Vn(x) and fourth kind Wn(x) corresponding to the weight functions w1(x) = (1 − x2)−1/2 , w2(x) = (1 − x2)1/2 , w3(x) = (1 + x)1/2 (1 − x)−1/2 andw4(x) = (1 + x)−1/2 (1 − x)1/2 , respectively.
format	Thesis
qualification_name	Doctor of Philosophy (PhD.)
qualification_level	Doctorate
author	Mahiub, Mohammad Abdulkawi
author_facet	Mahiub, Mohammad Abdulkawi
author_sort	Mahiub, Mohammad Abdulkawi
title	Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
title_short	Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
title_full	Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
title_fullStr	Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
title_full_unstemmed	Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
title_sort	numerical solutions of cauchy type singular integral equations of the first kind using polynomial approximations
granting_institution	Universiti Putra Malaysia
granting_department	Faculty Of Science
publishDate	2010
url	http://psasir.upm.edu.my/id/eprint/11990/1/FS_2010_7_A.pdf
_version_	1747811290529660928

Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations

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