A Fuzzy Fractional Initial Value Problem with Applications Under New Conformable Fractional Granular Differentiability
Bibliographic record
Abstract
This paper introduces a novel definition of the fractional order derivative for fuzzy setvalued functions (FSVF), utilizing the concept of granular difference, which we refer to as the new conformable fractional granular derivative (NCFGD).It also presents the corresponding integral, termed the new conformable fractional granular integral (NCFGI).In terms of results, the basic properties of both the NCFGD and NCFGI are rigorously established and proven, providing a strong mathematical foundation for these new operators.Several illustrative examples demonstrate the effectiveness and applicability of the proposed definitions.Additionally, the paper explores methods for solving the new conformable fractional granular initial value problem (NCFG IVP), which is integral to understanding the behavior of fuzzy set-valued functions under fractional calculus.We apply these methods to solve new conformable fractional granular differential equations (NCFG DEqs) related to growth and decay models, offering a more flexible approach to modeling dynamic systems with fuzzy uncertainty.This approach offers a more flexible and adaptive framework for modeling dynamic systems characterized by fuzzy uncertainty, as it allows for fractional order differentiation.The incorporation of these concepts into the solution of fuzzy fractional differential equations ensures more precise and adaptable models.Ultimately, these findings contribute significantly to advancing both fuzzy calculus and fractional calculus, enhancing the understanding and modeling of complex systems with uncertainty.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".