Product rules over the square of weakly singular double integrals

Keywords. Double integrals; numerical integration; product rules. Double integrals of the form ∫ 1 0 ∫ 1 0 log |x − y|f(x, y)dxdy, or ∫ 1 0 ∫ 1 0 log |x − y| √ xy(1 − x)(1 − y) f(x, y)dxdy, are of interest in the linear theory of the aerodynamics of slender bodies of revolution [1]. In this talk we consider the double integrals of the form ∫ 1 −1 ∫ 1 −1 k(|x − y|)f(x, y)dxdy, |x|, |y| < 1, with k(|x−y|) = |x−y|,or k(|x−y|) = log |x−y|and the function f(x, y) is a smooth function on [−1, 1]2. In this paper we consider product rules of interpolatory type, based on suitable Jacobi zeros. A different approach was recently proposed in [2], but the numerical method presented in the paper requires more computational efforts. For the proposed method convergence results are proved and numerical tests are given. References [1] Ashley H. , Landahl M. (1965) Aerodynamics of Wings and Bodies. Addison- Wesley, Reading, MA. [2] Gautschi W. (2012) Numerical integration over the square in the presence of algebraic/ logarithmic singularities with an application to aerodynamics. Num. Algor., Vol. 61, pp. 275–290.