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Fractional Calculus and the Study of Fuzzy Fractional Differential Equations in the Framework of Interactivity
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DOI：10.1016/j.fss.2026.110024.png)
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Abstract
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In this paper, we introduce new concepts of fuzzy fractional derivatives and fuzzy fractional integrals for S -linearly correlated fuzzy processes, which provide a suitable framework for modeling uncertainty with interactive effects. We establish their fundamental properties, including product rules and integration by parts formulas. We propose a novel fuzzy product, called the fuzzy exponent rule product, which facilitates the analysis and solution of fuzzy fractional differential equations. Using both the Φ-fuzzy Caputo and Φ-fuzzy Caputo-Fabrizio derivatives, we study fuzzy fractional differential equations with fuzzy coefficients. As an application, we introduce a modified fuzzy fractional SIR model and analyze it through numerical solutions and interpretation. Moreover, we show that the fuzzy space S(1,A,…,An) provides a natural and mathematically consistent framework for incorporating hierarchical levels of uncertainty. Examples are provided to illustrate our results.
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