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<p>In this study, the Fokker-Planck model, which describes the time-evolution of probability density functions, is reformulated within a fuzzy-fractional structure to incorporate uncertainties. Triangular fuzzy numbers are employed to represent uncertainty by incorporating fuzzy parameter in the initial conditions. After fuzzification, He-Laplace framework is applied to derive approximate analytical solutions. To evaluate the efficiency and reliability of the proposed methodology, residual errors are computed for both upper and lower bound solutions. The numerical results are further analyzed numerically through comparative tables, and visually as 2D, 3D and contour plots. The results demonstrate that the observed errors remain negligible, supporting the efficacy of proposed methodology and validity of obtained solutions. The proposed framework provides an efficient tool for analyzing fuzzy–fractional Fokker–Planck models and can be extended to other nonlinear fuzzy fractional systems.</p>

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Fuzzy fractional modeling and analysis of stochastic Fokker Planck equations under uncertainty and memory effects

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月5日

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  • Mubashir Qayyum,
  • Warda Rashid,
  • Faisal Z. Duraihem,
  • Efaza Ahmad,
  • Omar Khan,
  • Syed Tauseef Saeed,
  • Feyisa Edosa Merga

摘要

In this study, the Fokker-Planck model, which describes the time-evolution of probability density functions, is reformulated within a fuzzy-fractional structure to incorporate uncertainties. Triangular fuzzy numbers are employed to represent uncertainty by incorporating fuzzy parameter in the initial conditions. After fuzzification, He-Laplace framework is applied to derive approximate analytical solutions. To evaluate the efficiency and reliability of the proposed methodology, residual errors are computed for both upper and lower bound solutions. The numerical results are further analyzed numerically through comparative tables, and visually as 2D, 3D and contour plots. The results demonstrate that the observed errors remain negligible, supporting the efficacy of proposed methodology and validity of obtained solutions. The proposed framework provides an efficient tool for analyzing fuzzy–fractional Fokker–Planck models and can be extended to other nonlinear fuzzy fractional systems.