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.