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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| Summary: | 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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