Conformable fractional noise modeling with exponential-spline architecture

This work presents a unified computational scheme for solving stochastic partial differential equations (SPDEs) governed by fractional dynamics. The proposed numerical technique combines the modified non-polynomial cubic spline technique and the conformable fractional derivative (CFD) for both noise and time within a single effective framework. By redefining fractional white noise through conformable derivatives, the proposed method overcomes the limitations of classical kernel-based formulations that often lead to singularities and high computational cost. The developed Conformable Fractional Noise Stochastic Ten Non-Polynomial Cubic Spline method integrates ten spline variants within a finite difference structure, achieving fourth-order spatial precision while efficiently capturing temporal fractional effects. A series of comparative simulations demonstrates that the proposed approach achieves accuracy comparable to that of more advanced wavelet-based techniques while requiring fewer random realizations approximately 80% fewer samples in practice. Its stability and consistency were further tested over various weights of the Hurst parameter. In addition, the flexibility of the proposed framework allows it to be adapted to a broad class of fractional stochastic models with different noise structures. This highlights its potential as a reliable and efficient computational tool for practical applications involving complex random phenomena. All computations were performed in Mathematica 12 to ensure consistency and reproducibility throughout the numerical results.