RBF-Based Numerical Solution for Nonlinear Volterra Integral Equation Systems of the Second Kind

This paper presents a numerical method for solving systems of nonlinear Volterra integral equations of the second kind using compactly supported radial basis functions (RBFs). Such equations often appear in population models, epidemic dynamics, viscoelastic processes, and nonlinear transport, where the unknown function enters the kernel in a nonlinear form. Compared with the linear case, these problems are more complex and require iterative strategies for reliable approximation. The proposed method approximates the solution in a discrete space spanned by Wendland functions. Their compact support ensures sparse matrices and efficient computation, whereas their smoothness provides good approximation accuracy. The nonlinear residual system is formulated at collocation nodes and solved iteratively. Two strategies are considered: a fixed-point iteration, which is simple and stable, and a Gauss–Newton scheme, which accelerates convergence for strongly nonlinear kernels. Convergence is analyzed under Lipschitz conditions on the kernel, ensuring stability and consistency. Numerical experiments, for both scalar and system cases, demonstrate that the RBF-based approach yields accurate results with modest computational effort. This work extends RBF techniques from linear to nonlinear Volterra systems, offering a robust and efficient tool for such integral problems.