A.7 Calculation of Weakly Singular Integrals on Non-Smooth Surfaces

In Sections 5 and 6 asymptotically optimal methods for calculating weakly singular integrals defined on the squares [0,2 ?] ² or [-1, 1] ² were constructed.

It is of interest to study optimal methods for calculating weakly singular integrals on piecewise-Lyapunov surfaces.

Consider the integral

(A.44)

where G is a Lyapunov surface of class L _s( B, ?).

We show that the results derived in Sections 5 and 6 can be partially generalized to the integrals (A.44).

Calculate integrals (A.44) by the formula:

(A.45)

where t = ( t ₁ , t ₂ , t ₃) , v = ( v ₁ , v ₂ , v ₃), v = v ₁ + v ₂ + v ₃ , f ^{( v )}( t ₁ , t ₂ , t ₃) = .

The error of formula (A.45) is:

Assume f ? ? ₁, and G ? ? ₂. Then the error of formula (A.45) on the classes ? ₁ and ? ₂ is:

Let

A cubature formula with nodes and weights is called optimal, asymptotically optimal, optimal with respect to order on the class of functions ? ₁ and surfaces ? ₂, if

respectively.

Let ? ₁ = H _?(1), 0 < ? ? 1, and ? ₂ = L ₁( B, ?)...