An Effective Numerical Approach to Stochastic Systems with Conformable Fractional Noise: A Unified Analysis of Convergence and Stability

This paper proposes a dual-fractional framework for stochastic differential equations (SDEs) that integrates conformable fractional calculus into both the system dynamics and the stochastic driving noise. For the first time, a conformable formulation of fractional noise is introduced, replacing the traditional Caputo-based representation. This modification eliminates singular kernel functions while preserving the fundamental properties of classical calculus, thereby simplifying both the analysis and numerical implementation. A complete analytical study is presented, rigorously addressing the convergence properties, deriving explicit error estimates, and establishing the numerical stability of the proposed scheme. The framework is realized through an enhanced conformable fractional discrete Temimi–Ansari method (CFDTAM), which accommodates distinct fractional orders for the system dynamics and the stochastic component. The stability and accuracy of the proposed scheme are validated through comparisons with the stochastic Runge–Kutta method (SRK) as implemented in Mathematica 12. Applications to benchmark models—including the fractional Langevin, Ginzburg–Landau, and Davis–Skodje systems—further demonstrate the robustness of the framework, especially in regimes where the Hurst exponent  (Formula presented.)  greater than 0.5. Overall, the results establish the method as a rigorous and efficient tool for modelling and analyzing stochastic fractional systems in finance, biophysics, and engineering.