This paper is devoted to the following nonlinear weakly singular Volterra-Urysohn integral equation \[ y(t)-\int\limits_{0}^{t}(t-v)^{-\gamma}K(t,v,y(v))dv=g(t), \quad t\in[0,1], \quad 0<\gamma<1, \] with respect to the unknown function \(y(t)\) to be determined in a Banach space. The exact solutions \(y(t)\) of these type of weakly singular integral equations may be nonsmooth at the initial point of integration \(t=0\).
The Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods to obtain the approximate solution of this equation are discussed. The convergence results in both the infinity and weighted \(L^2\)-norm are obtained in the paper. Under suitable assumptions on the kernel \(K\) the superconvergence of the approximate solutions are also derived. For finding the improved convergence results, the authors also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted \(L^2\)-norm. It is proved that the iterated Jacobi spectral multi-Galerkin method improves over the iterated Jacobi spectral Galerkin method. Several numerical examples are also presented to confirm the theoretical results.