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...
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2007
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| الوصول للمادة أونلاين: | http://psasir.upm.edu.my/id/eprint/5021/1/FS_2007_30.pdf |
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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 |