Spline-Collocation Formulation for Nonlinear Fractional Volterra–Hammerstein Systems with Variable Coefficients

This paper focuses on developing a method for solving a nonlinear fractional integro-differential system of Volterra–Hammerstein (NFIDS–VH) type with variable coefficients. The proposed technique employs a cubic B-spline curve as the approximate solution, while a quadratic B-spline curve is used as a predictor at a single mesh point to initiate the scheme, together with a collocation approach. Furthermore, all definite integral terms are approximated using the Gauss–Legendre quadrature formula. Following these computations, the NFIDS–VH type is transformed into a nonlinear algebraic system involving unknown control points. This nonlinear system is then solved iteratively using Newton's method to obtain accurate solutions. The proposed technique is implemented and evaluated through two illustrative examples to demonstrate its effectiveness, robustness and computational efficiency, with errors decreasing systematically as the degree of Legendre polynomial increases. Four distinct algorithms are developed to implement the proposed method, and custom MATLAB (version 9.2) code is written for each algorithm to solve the illustrative examples. In addition, comparisons between the exact and approximate solutions are presented through graphs and tables. Finally, a comparative analysis with existing methods is conducted to demonstrate the accuracy of the proposed approach.