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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| Format: | Thesis |
| Language: | English English |
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2010
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| Online Access: | http://psasir.upm.edu.my/id/eprint/11990/ http://psasir.upm.edu.my/id/eprint/11990/1/FS_2010_7_A.pdf |
| _version_ | 1848841724783230976 |
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| author | Mahiub, Mohammad Abdulkawi |
| author_facet | Mahiub, Mohammad Abdulkawi |
| author_sort | Mahiub, Mohammad Abdulkawi |
| building | UPM Institutional Repository |
| collection | Online Access |
| 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. |
| first_indexed | 2025-11-15T07:47:47Z |
| format | Thesis |
| id | upm-11990 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-15T07:47:47Z |
| publishDate | 2010 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-119902013-05-27T07:50:38Z http://psasir.upm.edu.my/id/eprint/11990/ Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations 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. 2010-01 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/11990/1/FS_2010_7_A.pdf Mahiub, Mohammad Abdulkawi (2010) Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations. PhD thesis, Universiti Putra Malaysia. Singular integrals Differential equations - Numerical solutions English |
| 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 |
| title | Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
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| title_full | Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
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| title_fullStr | Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
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| title_full_unstemmed | Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
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| title_short | Numerical Solutions Of Cauchy Type Singular Integral Equations Of The First Kind Using Polynomial Approximations
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| title_sort | numerical solutions of cauchy type singular integral equations of the first kind using polynomial approximations |
| topic | Singular integrals Differential equations - Numerical solutions |
| url | http://psasir.upm.edu.my/id/eprint/11990/ http://psasir.upm.edu.my/id/eprint/11990/1/FS_2010_7_A.pdf |