Development of the Nystr\"{o}m Method for Weakly Singular Functional Integral Equations

In this research, we apply the standard product integration method (Nystr"{o}m method) for solving the delay nonlinear weakly singular Volterra integral equations. Typically, in weakly singular integral equations, the singularity of the kernel leads to the derivatives of the solution becoming singular at the boundary of the domain. The Chelyshkov polynomials serving as orthogonal polynomials, find application in numerical integration. Here, we use roots of these polynomials to make Lagrange interpolating polynomial for approximating the kernel functions in weakly singular functional integral equation. The proposed method's convergence analysis is developed, and numerical examples demonstrate the method's reliability and efficiency.