The paper studied the integration of logarithmic singularity problem J(y)=???(y)log|y-y0?|dA, where y=(?,?), y0=(?0,?0) the domain ? is rectangle ? = [r1, r2] � [r3, r4], the arbitrary point y? and the fixed point y0?. The given density function ?(y), is smooth on the rectangular domain ? and is in the functions class C2,? (?). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle ? is constructed by applying type (0, 2) modified spline function D?(P). The results obtained by testing the density functions ?(y) as linear and absolute value functions shows that the constructed CF is highly accurate. � 2016 Author(s).

In this note, singular integration problems of the form H? (h) = ??? h(x,y)/|-x0|2-? dA, 0 ? ? ? 1, where ? = [x0,y0] � [b1, b2], x= (x,y) ? ? and fixed point x 0 = (x0,y0) ? ? is considered. The density function h(x, y) is assumed given, continuous and smooth on the rectangle ? and belong to the class of functions C2,?(?). Cubature formula for double integrals with algebraic singularity on a rectangle is constructed using the modified spline function S?(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. � 2015 IEEE.

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed. � 2017 Author(s).

Modified homotopy perturbation method (HPM) is used to solve the hypersingular integral equations (HSIEs) of the second kind on the interval [-1, 1] with the assumption that the kernel in the form K(x, t)(x-t)-c0 where K(x, t) is a constant on the diagonal of the domain. This method introduced selective functions as Chebyshev polynomials of second kind and unknown parameters that leads to two step iterations and gives exact solution. Example are presented to prove the efficiency and realiability of the method. � 2016 Author(s).

A hypersingular integral equations (HSIEs) of the first kind on the interval [?1, 1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.

In this note, we consider a supersingular integral equations (SuperSIEs) of the first kind on the interval [-1,1] with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain D = [-1,1]×[-1,1]. Projection method together with Chebyshev polynomials of the first, second, third and fourth kinds are used to find bounded, unbounded and semi-bounded solutions of SuperSIEs respectively. Exact calculations of singular integrals for Chebyshev polynomials allow us to obtain high accurate approximate solution. Gauss- Chebyshev quadrature formulas are used for high accurate computations of regular kernel integrals. Two examples are provided to verify the validity and accuracy of the proposed method. Comparisons with other methods are also given. Numerical examples reveal that approximate solutions are exact if solution of SuperSIEs is of the polynomial forms with corresponding weights. It is worth to note that proposed method works well for large value of node points and errors are drastically decreases. © 2020 Author(s).