Modified spline functions and chebyshev polynomials for the solution of hypersingular integrals problems

The research work studied the singular integration problems of the form. The density function h(x, y) is given, continuous and smooth on the rectangle Ω and belong to the class of functions C 2,γ (Ω). Cubature formulas for double integrals with algebraic and logarithmic singularities on a rectangle Ω are constructed using the modified spline function SΛ(P) of type (0,2). Exactness of the cubature formulas for the two cases k ∈ {1,2} together with tested examples are shown each for linear and quadratic functions. Highly accurate numerical results for the cubature formulas are given for both tested density function h(x, y) as linear and quadratic functions. The results are in line with the theoretical findings. Hend Mohamed Bouseliana Further more, Hadamard type hypersingular integral (HSI) of the form Hi (h, x) = wi (x) π = Z 1 −1 h(t) wi (t)(t − x) 2 d t, x ∈ (−1,1), i ∈ {1,2,3,4}, where w1(t) = p 1− t 2, w2(t) = 1 p 1− t 2 , w3(t) = vt1− t 1+ t and w4(t) = vt1+ t 1− t are the weights and h(t) is a smooth function, are considered. Automatic quadrature schemes (AQSs) in each case for i ∈ {1,2,3,4} are constructed via approximating the density function h(t) by the first, second, third and fourth kind truncated series of Chebyshev polynomials, respectively. Error estimations in the cases i ∈ {1,2,3,4} are obtained via approximating the density function by truncated series of Chebyshev polynomials of the first, second, third and fourth kind, respectively, in the class of function C N,α[−1,1]. Exactness of the methods each for i ∈ {1,2,3,4} are shown for the degree 3 polynomial functions and the results of tested examples are presented and discussed. Numerical results of the obtained quadrature schemes revealed that the proposed methods are highly accurate for the tested density function h(t) as polynomial and rational functions. Comparisons made with other known methods showed that the automatic quadrature schemes (AQSs) constructed in this research has better results than others. The results are in line with the theoretical findings.

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