Nonlinear Volterra Integral Equation with Solution-Dependent Singular Kernels: Well-Posedness, Regularity, and Numerical Approximation

Abstract

In this paper, we consider a collocation method for solving a nonlinear Volterra integral equation involving the solution-dependent singular kernel (t−s)−α(u(s))F(s,u(s)), where F(s,u(s))=sσϕ(s,u(s)), 0<σ<1, and ϕ is a suitably defined nonlinear function. We first establish the existence and uniqueness of a continuous solution to the equation using Schaefer’s fixed-point theorem. We then develop a collocation method on a graded mesh for its numerical approximation and derive corresponding error estimates. Finally, numerical experiments are presented to verify the theoretical convergence rates.