An accurate spline polynomial cubature formula for double integration with logarithmic singularity

The paper studied the integration of logarithmic singularity problem J ( y ¯ ) = ∬ ∇ ζ ( y ¯ ) l o g | y ¯ − y ¯ 0 * | d A , where y ¯ = ( α , β ) , y ¯ 0 = ( α 0 , β 0 ) the domain ∇ is rectangle ∇ = [r 1, r 2] × [r 3, r 4], the arbitrary point y ¯ ∈ ∇ and the fixed point y ¯ 0 ∈ ∇ . The given density function ζ( y ¯ ) , is smooth on the rectangular domain ∇ and is in the functions class C 2,τ (∇). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle ∇ is constructed by applying type (0, 2) modified spline function D Γ(P). The results obtained by testing the density functions ζ( y ¯ ) as linear and absolute value functions shows that the constructed CF is highly accurate.