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

Abstract

The paper studied the integration of logarithmic singularity problem J(y¯)=∬∇ζ(y¯)log|y¯−y¯0*|dA, where y¯=(α,β), y¯0=(α0,β0) the domain ∇ is rectangle ∇ = [r1, r2] × [r3, r4], 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 C2,τ (∇). Cubature formula (CF) for double integration with logarithmic singularities (LS) on a rectangle ∇ is constructed by applying type (0, 2) modified spline function DΓ(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.