The numerical solution of Volterra integral equations with nonsmooth solutions based on sinc approximation (Q1189142)

The author discretizes the equation \[ y(x)=g(x)+\int^ x_ 0{k(x,t)\over (x-t)^ \alpha}y(t)dt,\quad 0\leq x\leq a, \] by the collocation method using the trial functions \(S(j,h,X)(x)=\text{sinc}((\phi_ X(x)-jh)/h)\), \(\{\phi_ X(z)=\log(z/(X-z))\), \(h>0\), \(0<X\leq a\). Assuming the analyticity of the data he proves the exponential convergence of the approximate solution.