A comparison of the error function and the tanh transformation as progressive rules for double and triple singular integrals
Numerical integrations are performed over cubic regions in two and three dimensions on singular integrands using appropriate transformations onto the whole of \(\mathbb R^2\) or \(\mathbb R^3\) to enable truncated product trapezoidal, or trapezoidal and Clenshaw-Curtis rules [\textit{C. W. Clenshaw} and \textit{A. R. Curtis}, Numer. Math. 2, 197--205 (1960; Zbl 0093.14006)] to be employed effectively. Transformations of the form \(x=\tanh (g(t))\) with \(g(t)=t^n\) or \(g(t)=c \sinh (t)\) (double exponential DE), as well as the more complicated error function type are used. There are tables of interesting illustrative examples resulting from significant computation. Both end-point and non end-point singularities are considered in two and in three dimensions. The non end-point singularities are handled by splitting the integrals into sums of end-point type integrals which are separately scaled to \([-1,1]\) before transformation. The different transformations and rules are compared. In the three- dimensional case comparisons are also made with a method of \textit{M. Sugihara} [J. Comput. Appl. Math. 17, 197--213 (1977; Zbl 0639.65014)] which in a particular sense approaches optimality.
scientific article; zbMATH DE number 4164014