Quadrature formula for evaluating left bounded Hadamard type hypersingular integrals

Left semi-bounded Hadamard type Hypersingular integral (HSI) of the form H(h,x)=1/π1+x/1-x λ-1∗∗1 1-t/1+t h(t)(t-x)2dt,x∈(-1.1), Where h(t) is a smooth function is considered. The automatic quadrature scheme (AQS) is constructed by approximating the density function h(t) by the truncated Chebyshev polynomials of the fourth kind. Numerical results revealed that the proposed AQS is highly accurate when h(t) is choosing to be the polynomial and rational functions. The results are in line with the theoretical findings

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

Numerical Solution of System of Linear Volterra Integral Equations via Hybrid of Taylor and Block-pulse Functions

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An automatic quadrature schemes and error estimates for semibounded weighted Hadamard type hypersingular integrals

The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class of h(t) ∈ CN,a [-1, 1]. Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density function h (t)is any polynomial or rational functions. The results are in line with the theoretical findings

An Automatic Quadrature Schemes and Error Estimates for Semibounded Weighted Hadamard Type Hypersingular Integrals

The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class of ℎ( ) ∈ , [−1, 1]. Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density function ℎ ( ) is any polynomial or rational functions. The results are in line with the theoretical findings

Construction of cubature formula for double integration with algebraic singularity by spline polynomial

In this note, singular integration problems of the form Hα (h) = ∫Ω∫ h(x,y)/|-x0|2-α dA, 0 ≤ α ≤ 1, where Ω = [x0,y0] × [b1, b2], x= (x,y) ϵ Ω and fixed point x 0 = (x0,y0) ϵ Ω is considered. The density function h(x, y) is assumed given, continuous and smooth on the rectangle Ω and belong to the class of functions C2,α(Ω). Cubature formula for double integrals with algebraic singularity on a rectangle is constructed using the modified spline function SΩ(P) of type (0, 2). Highly accurate numerical results for the proposed method is given for both tested density function h(x, y) as linear, quadratic and absolute value functions. The results are in line with the theoretical findings

An accurate spline polynomial cubature formula for double integration with logarithmic singularity

The paper studied the integration of logarithmic singularity problem J(ӯ) = ∫∫Δζ(ӯ)log|ӯ - ӯ 0∗|dA, where ӯ=(α,β), y0=(α0,β0) the domain Δ is rectangle Δ = [r1, r2] × [r3, r4], the arbitrary point ӯ ϵ Δ and the fixed point ӯ0 ϵ Δ. The given density function ζ(ӯ), is smooth on the rectangular domain Δ and is in the functions class C2,τ (Δ). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle Δ is constructed by applying type (0, 2) modified spline function DΓ(P). The results obtained by testing the density functions ζ(ӯ) as linear and absolute value functions shows that the constructed CF is highly accurate