Analysis and Approximation of a Weakly Singular Volterra Integral Equation with Nonlinear Exponent

A weakly singular Volterra integral equation (VIE) whose kernel has a nonlinear exponent [Formula: see text] is considered. After showing the compactness of the nonlinear Volterra integral operator, we apply Schaefer’s fixed-point theorem to establish existence of a continuous solution [Formula: see text] to the VIE. Uniqueness and regularity of this solution are then proved; it has a weak singularity at the initial time [Formula: see text] that is characterized by [Formula: see text]. A nonlinear collocation method is constructed on uniform and graded meshes. Well-posedness and stability of this method are shown, and an estimate of its error, which is sharp up to a log factor, is derived using completely novel techniques. Numerical experiments support the theoretical findings.