A Unified Newton-Collocation Framework for Nonlinear Fredholm Integral Equations

We develop a unified and operator-agnostic Newton-collocation framework for the numerical solution of nonlinear Fredholm integral equations of the second kind. The method combines a fixed low-degree polynomial trial space, Gauss-Legendre quadrature, and analytically constructed Jacobians, yielding stable and rapidly convergent Newton iterations without problem-specific tuning. A cross-operator verification strategy is employed in which the computed solution is independently validated using a finer quadrature rule than that used for discretization. The approach is demonstrated on two structurally distinct nonlinear models: a nonlocal epidemic transmission equation with rational saturation and a radiative heat-transfer equation with quartic temperature feedback. In both cases, the same algorithmic structure produces accurate solutions, illustrating the transferability and reliability of the proposed framework for nonlinear Fredholm equations.