Advancing the Numerical Treatment of Caputo Fractional Lotka–Volterra Systems Through Non-Polynomial Spline Approximation and Neural Network Verification

Vollständiger Abstract

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This paper presents a modified non-polynomial fractional cubic spline (MNPFCS) method for solving fractional Lotka–Volterra prey–predator systems governed by the Caputo fractional derivative. The proposed method integrates non-polynomial spline interpolation with fractional calculus to obtain accurate numerical approximations for nonlinear coupled systems. The stability and convergence of the method are established theoretically. To further verify its effectiveness, the numerical solutions are compared with those predicted by a feed-forward artificial neural network (ANN). Two benchmark examples demonstrate excellent agreement between the numerical and ANN solutions, while the MSE, MAE, and R2 metrics confirm the high predictive accuracy of the ANN. The obtained results show that the proposed MNPFCS method is an accurate, stable, and efficient approach for solving nonlinear fractional dynamical systems.