Jacobi spectral Galerkin methods for Volterra integral equations with weakly singular kernel (Q2787661)

For the weakly singular Volterra integral equation \(y(t)=\int_0^t(t-\tau)^{-\gamma}K(t,\tau)y(\tau)d\tau+f(t)\) with \(0<\gamma<1\), the author discusses a polynomial Galerkin method based on a Jacobi weight function. In addition, a fully discrete variant of this method is proposed where the integrals that arise in the construction of the method are approximately computed via a Gauss quadrature scheme with the same weight function. A few error estimates are given. These estimates require a certain degree of smoothness of the solution. This assumption may be difficult to satisfy in practical applications due to the singularity in the kernel of the integral operator.