A strategy for third-kind Hammerstein-type nonlinear Volterra integral equations

This paper presents a discontinuous Galerkin (DG) scheme for a category of nonlinear Volterra integral equations (VIEs) of third-kind, specifically those of Hammerstein-type. These equations feature time-dependent singularities through the vanishing coefficient tβ with 0<β<1, and exhibit a structural equivalence to nonlinear cordial VIEs. The proposed DG scheme is constructed on piecewise discontinuous polynomial spaces based on the shifted Legendre polynomials. We establish the solvability of the discrete formulation via a contractive fixed-point mapping and derive rigorous error estimates confirming high-order convergence. The method's compactness and stability are investigated under standard smoothness and Lipschitz conditions. Numerical experiments are utilized to show the spectral accuracy of the approach and to validate the theoretical results.