A new approach to the numerical solution of weakly singular Volterra integral equations

We consider linear weakly singular Volterra integral equations of the second kind, with kernels of the form k(x,v)=|x−v|−αK(x,v), 0<α<1, or k(x,v)=log|x−v|K(x,v), K(x,v) being a smooth function. The solutions of such equations may exhibit a singular behaviour in the neighbourhood of the initial point of the interval of integration. By a transformation of the unknown function we obtain an equation which is still weakly singular, but whose solution is as smooth as we like. This resulting equation is then solved by standard product integration methods.