New modifications of the collocation method for a class of linear equations with a Hadamard integral (Q5951201)

The author considers the linear integral equation \(Ax\equiv x(t)+(Kx)(t)=y(t)\), \(t \in \langle 0,1\rangle\), \(Kx \equiv \int_0^1 K(t,s)[u(s)]^{-1}x(s) ds\), \(u(t)=t^{p_1}(1-t)^{p_2}\), \(p_1,p_2>0.\) He studies solvability of a given equation by means of constructing a Fredholm theory for this equation suggesting a new modification of the collocation method suitable for the approximate solution. Existence and uniqueness theorems for the corresponding approximate equations are proved. The stability and the condition numbers of the approximating equations, are also studied.