The paper studied the integration of logarithmic singularity problem J ((y) over bar) = integral integral(del) zeta((y) over bar) log vertical bar(y) over bar = (y) over bar (0)*vertical bar dA, where (y) over bar = (alpha, beta), (y) over bar (0) = (alpha(0), beta(0)), the domain del is rectangle (y) over bar [r(1), r(2)] x [r(3), r(4)]; the arbitrary point (y) over bar is an element of del and the fixed point (y) over bar (0) is an element of del. The given density function zeta((y) over bar), is smooth on the rectangular domain del and is in the functions class C-2,C-tau (del). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle del is constructed by applying type (0, 2) modified spline function D-Gamma(P). The results obtained by testing the density functions zeta((y) over bar) as linear and absolute value functions shows that the constructed C-F is highly accurate.

In this note, singular integration problems of the form H alpha(h) = integral integral(Omega) h(x,y)/vertical bar(x) over bar - (x) over bar (0)vertical bar(2-alpha)dA, 0 <= alpha <= 1, where Omega = [a(1), a(2)] x [b(1), b(2)], (x) over bar = (x, y) is an element of Omega and fixed point (x) over bar (0) = (x(0), y(0)) is an element of Omega; is considered. The density function h (x, y) is assumed given, continuous and smooth on the rectangle Omega and belong to the class of functions C-2,C-alpha(Omega). Cubature formula for double integrals with algebraic singularity on a rectangle is constructed using the modified spline function S-Omega( 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.