New Polynomials for the Solution of Certain Classes of Singular Integral Equations
In this thesis,we propose a new classes of orthogonal polynomials Zᴷ(I,n)(x),i= { 1,2,3,4} n=0,1,2,…, namely Extended Chebyshev polynomials (ECPs) of the first, second, third and fourth kinds respectively, of order k, where k is positive odd integer and the polynomials are the extension of Chebyshev...
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Universiti Sains Islam Malaysia |
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Extended Chebyshev polynomials (ECPs) Polynomials Integral equations -- Numerical solutions |
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Extended Chebyshev polynomials (ECPs) Polynomials Integral equations -- Numerical solutions Hasham M.H. Al-Hawamda New Polynomials for the Solution of Certain Classes of Singular Integral Equations |
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In this thesis,we propose a new classes of orthogonal polynomials Zᴷ(I,n)(x),i= { 1,2,3,4} n=0,1,2,…, namely Extended Chebyshev polynomials (ECPs) of the first, second, third and fourth kinds respectively, of order k, where k is positive odd integer and the polynomials are the extension of Chebyshev polynomials. It is found that ECPs of first- fourth kinds are orthogonal with respect to a certain weight functions on the interval [-1,1]. The main characteristics of ECPs sequences { Z²m+1(x)} km=0;i= {1,2,3,4};n= 0,1,2,3,… are described and explained when k is sufficiently large odd integer.It is shown that ECPs sequences of the first and second kinds converge to one of the three values {-1,0,1} depending on n= {2l+1,2 (2l+1),4l} , l=0,1,2,….respectively. Morever ECPs sequences of the third kind approaches into two values {-1,1} depending on n= {1,2,5,6,…;0,3,4,7,8,…} respectively, and ECPs sequences of the fourth kind approaches into two values {-1,1} depending on n= {2,3,6,7,…;0,1,4,5,8,9,…} respectively. Recurrence relations and the relationships between each other kinds are proved. Spectral properties of the proposed new orthogonal polynomials are obtained and used to solve the following equations: • Special class of the Logarithmic Singular Integral Equations (first and second kind) of order k (LogSIEk). • Special class of Singular Integral Equations of the first kind of order k (SSIEk). Eigenvalues and corresponding eigenfunctions are found for the LogSIEk of the first kind.For the second kind inhomogeneous LogSIEk highly accurate approximate solution are obtained. For SSIEk of the first kind, we have considered the following cases: • Case (I): The solution is unbounded at both end points x = +-1, • Case (II): The solution is bounded at both end points x = +-1, • Case (III): The solution is bounded at the end points x = -1 and unbounded at x=1 • Case (IV): The solution is bounded at the end points x = 1 and unbounded at x=-1 It is found that the approximate solution is highly accurate for the four cases. Numerical examples and comparisons with other methods are also provided to illustrate the effective-ness and accuracy of the proposed methods. |
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Thesis |
| author |
Hasham M.H. Al-Hawamda |
| author_facet |
Hasham M.H. Al-Hawamda |
| author_sort |
Hasham M.H. Al-Hawamda |
| title |
New Polynomials for the Solution of Certain Classes of Singular Integral Equations |
| title_short |
New Polynomials for the Solution of Certain Classes of Singular Integral Equations |
| title_full |
New Polynomials for the Solution of Certain Classes of Singular Integral Equations |
| title_fullStr |
New Polynomials for the Solution of Certain Classes of Singular Integral Equations |
| title_full_unstemmed |
New Polynomials for the Solution of Certain Classes of Singular Integral Equations |
| title_sort |
new polynomials for the solution of certain classes of singular integral equations |
| granting_institution |
Universiti Sains Islam Malaysia |
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https://oarep.usim.edu.my/bitstreams/7d959ca7-2b60-4f93-bd0e-b863ea3194c0/download https://oarep.usim.edu.my/bitstreams/9959dff4-6fcf-492b-b367-6e0e20430d6f/download https://oarep.usim.edu.my/bitstreams/4c2abf35-6c20-49b0-8ccb-3f812ce69342/download https://oarep.usim.edu.my/bitstreams/2fb532b7-c46e-4c5e-9896-60a504cb7d82/download https://oarep.usim.edu.my/bitstreams/425ac1a8-7a1f-4411-9f2a-d0fdae874514/download https://oarep.usim.edu.my/bitstreams/7d47ea4a-0084-4664-a0fe-cef6baa41bfd/download https://oarep.usim.edu.my/bitstreams/c8297b95-bbe2-46ea-9423-4d9f8fd2bad1/download https://oarep.usim.edu.my/bitstreams/6948a8b5-c606-4aa9-8953-35d4109c340a/download https://oarep.usim.edu.my/bitstreams/dbc22a8b-0703-4081-94d5-87a82d9794a3/download |
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1812444892256796672 |
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my-usim-ddms-130012024-05-29T20:17:29Z New Polynomials for the Solution of Certain Classes of Singular Integral Equations Hasham M.H. Al-Hawamda In this thesis,we propose a new classes of orthogonal polynomials Zᴷ(I,n)(x),i= { 1,2,3,4} n=0,1,2,…, namely Extended Chebyshev polynomials (ECPs) of the first, second, third and fourth kinds respectively, of order k, where k is positive odd integer and the polynomials are the extension of Chebyshev polynomials. It is found that ECPs of first- fourth kinds are orthogonal with respect to a certain weight functions on the interval [-1,1]. The main characteristics of ECPs sequences { Z²m+1(x)} km=0;i= {1,2,3,4};n= 0,1,2,3,… are described and explained when k is sufficiently large odd integer.It is shown that ECPs sequences of the first and second kinds converge to one of the three values {-1,0,1} depending on n= {2l+1,2 (2l+1),4l} , l=0,1,2,….respectively. Morever ECPs sequences of the third kind approaches into two values {-1,1} depending on n= {1,2,5,6,…;0,3,4,7,8,…} respectively, and ECPs sequences of the fourth kind approaches into two values {-1,1} depending on n= {2,3,6,7,…;0,1,4,5,8,9,…} respectively. Recurrence relations and the relationships between each other kinds are proved. Spectral properties of the proposed new orthogonal polynomials are obtained and used to solve the following equations: • Special class of the Logarithmic Singular Integral Equations (first and second kind) of order k (LogSIEk). • Special class of Singular Integral Equations of the first kind of order k (SSIEk). Eigenvalues and corresponding eigenfunctions are found for the LogSIEk of the first kind.For the second kind inhomogeneous LogSIEk highly accurate approximate solution are obtained. For SSIEk of the first kind, we have considered the following cases: • Case (I): The solution is unbounded at both end points x = +-1, • Case (II): The solution is bounded at both end points x = +-1, • Case (III): The solution is bounded at the end points x = -1 and unbounded at x=1 • Case (IV): The solution is bounded at the end points x = 1 and unbounded at x=-1 It is found that the approximate solution is highly accurate for the four cases. Numerical examples and comparisons with other methods are also provided to illustrate the effective-ness and accuracy of the proposed methods. Universiti Sains Islam Malaysia 2018-07 Thesis en_US https://oarep.usim.edu.my/handle/123456789/13001 https://oarep.usim.edu.my/bitstreams/f85e7ee6-3812-4ad4-9c8f-d31dfa8ad4c8/download 8a4605be74aa9ea9d79846c1fba20a33 https://oarep.usim.edu.my/bitstreams/7d959ca7-2b60-4f93-bd0e-b863ea3194c0/download 120c0af7bd8923672f641dd7a39f3d70 https://oarep.usim.edu.my/bitstreams/9959dff4-6fcf-492b-b367-6e0e20430d6f/download dd84eaa5a3b6225c1e54de1d8d029093 https://oarep.usim.edu.my/bitstreams/4c2abf35-6c20-49b0-8ccb-3f812ce69342/download 581dab085688e888243bddfb274f5eb2 https://oarep.usim.edu.my/bitstreams/2fb532b7-c46e-4c5e-9896-60a504cb7d82/download 708e90c4b7528376eb42f88e9a784999 https://oarep.usim.edu.my/bitstreams/425ac1a8-7a1f-4411-9f2a-d0fdae874514/download d13c49260d8dc15b1a0e71736b0f8b3c https://oarep.usim.edu.my/bitstreams/7d47ea4a-0084-4664-a0fe-cef6baa41bfd/download efadfa9e15be3862da8c1d504ed1d4ab https://oarep.usim.edu.my/bitstreams/c8297b95-bbe2-46ea-9423-4d9f8fd2bad1/download 3ff494cdd343bc83bdc9de0df4fd6c79 https://oarep.usim.edu.my/bitstreams/6948a8b5-c606-4aa9-8953-35d4109c340a/download 5411bf4010292a151c96053e80fbcfe2 https://oarep.usim.edu.my/bitstreams/dbc22a8b-0703-4081-94d5-87a82d9794a3/download 9966647112c2220e8b2b95dd240f6236 https://oarep.usim.edu.my/bitstreams/4feccf11-cb26-4879-a039-363e81fbdcdd/download 68b329da9893e34099c7d8ad5cb9c940 https://oarep.usim.edu.my/bitstreams/56861740-5ca0-447e-93b7-86551f343cf6/download 1bb607118047afc5c385b82385dd931f https://oarep.usim.edu.my/bitstreams/d198f28f-a809-4101-b6c9-48ff86685d15/download 6d93d3216dc4a7f5df47d4876fbec4d3 https://oarep.usim.edu.my/bitstreams/01dd3dff-0232-494a-b7ce-508c9ca4edbe/download 429079e52f342c49301a02e207bfde06 https://oarep.usim.edu.my/bitstreams/b6306b3d-e86a-4cb6-af01-93278d13aebb/download 884e307a96265ab0e1bf7bcc1f89c892 https://oarep.usim.edu.my/bitstreams/c03d7037-011b-4c90-b07a-afd75025fb76/download 9d176d1f5f1b1954e72f6a2eb8776b1a https://oarep.usim.edu.my/bitstreams/0d496c8c-ddc8-4aad-81ff-8acc12487f57/download 1b70f87af6fdb6f73d267b35669fa6d4 https://oarep.usim.edu.my/bitstreams/cdfd0d6a-195f-43b7-b14d-6940d8b7768a/download 6d93d3216dc4a7f5df47d4876fbec4d3 https://oarep.usim.edu.my/bitstreams/9de6cbf8-8479-48cb-a8be-89c79388008b/download 36c6f0b2061da514c400c0bc2749b5cf Extended Chebyshev polynomials (ECPs) Polynomials Integral equations -- Numerical solutions |