Construction of Cubature Formula for Double Integration with Algebraic Singularity by Spline Polynomial

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.