Representation of Uncertainty for Limit Processes

TL;DR

The paper introduces a mathematical framework using fuzzy limits and derivatives to model and handle uncertainty arising from measurement and computational approximations in limit-based mathematical models.

Contribution

It proposes the concept of fuzzy limits and derivatives to incorporate uncertainty into differential models, extending classical calculus to uncertain environments.

Findings

Developed a mathematical technique for fuzzy derivatives.

Applied fuzzy limits to differential models with uncertainty.

Provided a new approach to quantify uncertainty in measurements.

Abstract

Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve only approximate results. In some cases, this discrepancy between theoretical schemes and practical actions changes drastically outcomes of a research and decision-making resulting in uncertainty of knowledge. In the paper, a mathematical approach to such kind of uncertainty, which emerges in computation and measurement, is suggested on the base of the concept of a fuzzy limit. A mathematical technique is developed for differential models with uncertainty. To take into account the intrinsic uncertainty of a model, it is suggested to use fuzzy derivatives instead of conventional derivatives of functions in this model.