Approximate Solutions of Nonlinear Functional Volterra Integral Equations Using A Collocation Method With Neural Networks Activated by Sigmoidal Functions

ABSTRACT In this paper, we study a broad class of nonlinear integral equations, referred to as nonlinear Volterra‐Urysohn functional integral equations. Our analysis is conducted in the Banach space of all continuous real‐valued functions defined on a non‐empty, closed, and bounded interval . We establish the existence and uniqueness of solutions under suitable smoothness conditions. We then develop a numerical collocation method to approximate the solution. The approximations are represented as neural networks with sigmoidal activation functions, specifically employing Heaviside, logistic, and Gompertz functions. Unlike classical polynomial, spline, or spectral‐based collocation methods, the proposed sigmoidal collocation scheme yields an explicit formula for the coefficients, avoiding iterative solvers such as Newton or fixed‐point methods and ensuring stable computation. This explicit solvability, combined with representational flexibility and practical computational efficiency, leads to competitive accuracy‐versus‐time performance, filling a gap left by existing numerical approaches. The convergence of the approximate solutions to the exact solution is analyzed, and an a priori error bound is derived. Finally, the effectiveness of the proposed method is demonstrated through several numerical examples, and the results are compared with those obtained using other methods.