A numerical method for a class of Volterra integral equations with logarithmic perturbation kernel

We consider a class of integral equations of Volterra type with constant coefficients containing a logarithmic difference kernel. This class coincides for a=0 with the Symm's equation. We can transform the general integral equation into an equivalent singular equation of Cauchy type which allows us to give an explicit formula for the solution g. The numerical method proposed in this paper consists in substituting the Lagrange polynomial interpolating the known function f in the expression of the solution g. Then, with the aid of the invariance properties of the orthogonal polynomials for the Cauchy integral equations, we obtain an easy expression for the approximate solution. Moreover, we show that the previous numerical method is a collocation method where the coefficient of the polynomial approximating the solution can be easily computed. We give weighted norm estimates for the error of this method. The paper concludes with some numerical examples.