Abstract Weakly singular Volterra integral equations (VIEs) of the second kind, where the kernel of the integral term has a factor of the form (t-s) ^- (t) with 0 (t) 1, are studied on the interval 0 t 1. It is known that typical solutions of these problems lie in the Vainikko space C^m, (0) (0, 1], so they have a weak singularity at the initial time t=0. Estimates for best approximations of functions in C^m, (0, 1] by polynomials defined on t₁, 1 are derived in the L^ (t₁, 1) norm, where (-, 1) and t₁ 0, 1) are arbitrary, and these bounds are shown to be sharp in the most singular case 1- m/2; such estimates were not previously known and are of independent interest. To solve these VIEs numerically, an analytical technique of Huang & Stynes (2017, Spectral Galerkin methods for a weakly singular Volterra integral equation of the second kind. IMA J. Numer. Anal. , 37, 1411–1436) is used on an initial small interval [0, t₁, and then collocation by globally defined polynomials at suitably chosen Jacobi–Gauss collocation points is used on t₁, 1. Error bounds for the solution of this hybrid method are derived in the L^ (0, 1) norm. Numerical experiments show that these error bounds are often sharp.
Ma et al. (Tue,) studied this question.