Product quasi-interpolation method for weakly singular integral equation eigenvalue problem

A discrete method of accuracy O(hm) is constructed to solve the integral equation eigenvalue problem λϕ(x)=∫−11K(x,y)ϕ(y)dy, where K(x, y) = log|x − y| or K(x, y) = |x − y|−α, 0 < α < 1, ϕ(x) being the eigenfunction associated with the eigenvalue λ. The method is based first on improving the boundary behavior of the exact solution ϕ(x) with the help of a change of variables, and second on the product integration method based on a discrete spline quasi-interpolant (abbr. dQI) of order m.