Some Fundamental Results on Fuzzy Conformable Differential Calculus

In this paper, we combine fuzzy calculus, and conformable calculus to introduce the fuzzy conformable calculus. We define the fuzzy conformable derivative of order $2\Psi $ and generalize it to derivatives of order $p\Psi $. Several properties on difference, product, sum, and addition of two fuzzy-valued functions are provided which are used in the solution of the fuzzy conformable differential equations. Also, examples in each case are given to illustrate the utility of our results.

Series based on the new order in intuitionistic fuzzy environment

Intuitionistic fuzzy number (IFN) is an effective tool for dealing with the uncertain information, and it has been applied to various fields. According to IFNs, the intuitionistic fuzzy calculus has been developed, which can effectively integrate the continuous uncertain information. Series in intuitionistic fuzzy environment is a part of the intuitionistic fuzzy calculus theory, of which core idea is limit. However, the order used in the existing limit theory is not the one used in intuitionistic fuzzy calculus, causing the separation of the limit theory and intuitionistic fuzzy calculus. Thus, series in intuitionistic fuzzy environment is not closely related to the intuitionistic fuzzy calculus. In order to solve the above problem, we construct the related theories. There are mainly the following three aspects: (1) the limit theory including the sequence limit and the function limit is studied based on the new order. (2) we re-examine the numerical series according to the new tool of researching IFNs: the basis and the coordinates. (3) we discuss the function series and put forward the uniform convergence in intuitionistic fuzzy environment.

“Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators

In this paper, based on the “fuzzy” calculus covering the continuous range of operations between two couples of arithmetic operations (+, –) and (×, :), a new form of the fractional integral is proposed occupying an intermediate position between the integral and derivative of the first order. This new form of the fractional integral satisfies the C1 criterion according to the Ross classification. The new calculus is tightly related to the continuous values of the continuous spin S = 1 and can generalize the expression for the fractional values of the shifting discrete index. This calculus can be interpreted as the appearance of the hidden states corresponding to unobservable values of S = 1. Many well-known formulas can be generalized and receive a new extended interpretation. In particular, one can factorize any rectangle matrix and receive the “perfect” filtering formula that allows transforming any (deterministic or random) function to another arbitrary function and vice versa. This transformation can find unexpected applications in data transmission, cryptography and calibration of different gadgets and devices. One can also receive the hybrid (”centaur”) formula for the Fourier (F-) transformation unifying both expressions for the direct and inverse F-transformations in one mathematical unit. The generalized Dirichlet formula, which is obtained in the frame of the new calculus to allow selecting the desired resonance frequencies, will be useful in discrete signals processing, too. The basic formulas are tested numerically on mimic data.

Foundation of Interval-Valued Intuitionistic Fuzzy Limit and Differential Theory and an Application to Continuous Data

The intuitionistic fuzzy calculus (IFC), based on the basic operational laws of intuitionistic fuzzy numbers (IFNs), has been put forward. However, the interval-valued IFC (IVIFC), based on the basic operational laws of interval-valued IFNs (IVIFNs), is only in the original stage. To further develop the theory of the IVIFC and make it be rigorous, the primary task is to systematically investigate the characteristics of the limits and differentials, which is a foundation of the IVIFC. Moreover, there is quite a lot of literature on IVIFNs; however, the scholars did not reveal the relationships between IFNs and the IVIFNs. To do that, we first investigate the limit of interval-valued intuitionistic fuzzy sequences, and then, we focus on investigating the limit, the continuity, and the differential of IVIFFs in detail and reveal their relationships. After that, due to the fact that the IFC and the IVIFC are based on the basic operational laws of IFNs and IVIFNs, respectively, we reveal the relationships between the IFNs and the IVIFNs via some homomorphic mappings. Finally, a case study about continuous data of IVIFNs is provided to illustrate the advantages of continuous data.