An Accurate Spline Polynomial Cubature Formula for Double Integration with Logarithmic Singularity

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.