Special versions of the collocation method for a class of integral equations of the third kind (Q1759308)

The authors consider the integral equation of the third kind \[ (Ax)(t)=(Ux)(t)+(Kx)(t)=y(t) \] where \[ (Ux)(t)=x(t)t^{p_1}(1-t)^{p_2}\prod_{j=1}^q (t-t_j)^{m_j} \] and \[ (Kx)(t)=\int_0^1K(t,s)x(s)ds,~ t\in [0,1] \] where \(p_1, p_2\in \mathbb{R}^+\), \(t_j\in(0,1)\), \(m_j\in N(j=1,\dots,q)\), \(K\) and \(y\) are known continuous functions, \(x\) is the unknown solution. They develop special direct methods for an approximate solution in the space of distributions. Some important properties of the studied methods are given.