A spline method for the solution of integral equations of the third kind (Q2479659)

The authors consider the linear integral equation of the third kind \[ \begin{aligned} (Ax)(t)&= (Ux)(t)+(Kx)(t)=y(t),\\ \text{where} (Ux)(t)&= x(t)t^{p_1}(1-t)^{p_2}\prod _{j=1}^q(t-t_j)^{m_j},\\ (Kx)(t)&= \int_0^1K(t,s)x(s)\,ds, \quad t\in [0,1],\;p_1,p_2\in\mathbb R^{+}, \end{aligned} \] \(t_j\in (0,1)\), \(m_j\in \mathbb N\) \((j=1,\dots,q)\), \(K,y\) are known continuous functions. The authors propose and substantiate one special direct method for the approximate solution in a special space of distributions. It is proved that the constructed method is optimal by the order of exactness on a certain class of functions.