Numerical computation of weakly singular integrals (Q2718825)

The author considers the numerical evaluation of integrals NEWLINE\[NEWLINE \int\limits_0^b f(x)dx, \quad (b>0), NEWLINE\]NEWLINE where \(f\in C^{k,v}(0,b],\) i.e. \(f(x)\) is \(k\) times, \(k \geq 1\) continuously differentiable for \(x \in (0,b]\) and NEWLINE\[NEWLINE|f^{(i)}(x)|\leq c \begin{cases} 1,& \text{if }i<1-v,\\ 1+|\log x |,& \text{if }i=1-v,\\ x^{1-v-i},& \text{if } i>1-v,\end{cases} NEWLINE\]NEWLINE where \(i=0,1,\ldots,k,\) and the index singularity \(v<2.\) Quadrature rules which use \(N\) values of the integrable function \(f(x)\) with exactness \(O(N^{-k})\) are considered.