Stochastic improved Simpson for solving nonlinear fractional-order systems using product integration rules

This research introduces a novel and robust numerical approach, the stochastic improved Simpson Method, specifically developed to solve Itô and Stratonovich stochastic nonlinear system of differential equations with fractional order. By extending the classical Simpson's one-third rule with the explicit product integration rectangle rule, the proposed method efficiently handles fractional derivatives of orders between 0 and 1, based on the Caputo derivative. The novelty of this approach lies in its enhanced accuracy and stability in addressing the unique challenges posed by both Itô and Stratonovich systems, outperforming traditional numerical techniques. Rigorous order analysis, conducted with Mathematica 12 software, demonstrates the robustness and precision of the method. Its effectiveness is further validated through four distinct numerical case studies, highlighting its superior performance in solving stochastic nonlinear system of differential equations with fractional order.