$ hp $-version collocation method for a class of nonlinear Volterra integral equations of the first kind

Abstract: In this paper, we present a collocation method for nonlinear Volterra integral equation of the first kind. This method benefits from the idea of $hp$-version projection methods. We provide an approximation based on the Legendre polynomial interpolation. The convergence of the proposed method is completely studied and an error estimate under the $L^2$-norm is provided. Finally, several numerical experiments are presented in order to verify the obtained theoretical results.