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

A discrete method of accuracy O( h m ) is constructed to solve the integral equation eigenvalue problem λ ϕ ( x ) = ∫ − 1 1 K ( x , y ) ϕ ( y ) d y , 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.