A novel difference and derivative of linearly correlated fuzzy number-valued functions

In the present paper, the notion of the linearly correlated difference for linearly correlated fuzzy numbers is introduced. Especially, the linearly correlated difference and the generalized Hukuhara difference are coincident for interval numbers or even symmetric fuzzy numbers. Accordingly, an appropriate metric is induced by using the norm and the linearly correlated difference in the set of linearly correlated fuzzy numbers. Based on the symmetry of the basic fuzzy number, the linearly correlated derivative is proposed by the linearly correlated difference of linearly correlated fuzzy number-valued functions. In both non-symmetric and symmetric cases, the equivalent characterizations of the linearly correlated differentiability of a linearly correlated fuzzy number-valued function are established, respectively. Moreover, it is shown that the linearly correlated derivative is consistent with the generalized Hukuhara derivative for interval-valued functions.

Newton’s divided gH-difference interpolation formula with unequal spacing for interval-valued function using Hukuhara difference

Interval interpolation formulae play a significant role to find the value of an unknown function at some points under interval uncertainty. The objective of this paper is to establish Newton’s divided interpolation formula for interval-valued functions using generalized Hukuhara difference of intervals. For this purpose, arithmetic of intervals, Hukuhara difference and its some properties and concept of interval-valued function have been discussed briefly. Using Hukuhara difference of intervals, the definition of Newton’s divided gH-difference for interval-valued function has been introduced. Then Newton’s divided gH-differences interpolation formula has been derived. Finally, with the help of some numerical examples, the proposed interpolation formula has been illustrated.

An optimization technique for national income determination model with stability analysis of differential equation in discrete and continuous process under the uncertain environment

The paper represents a variation of the national income determination model with discrete and continuous process in fuzzy environment, a significant implication in economics planning, by means of fuzzy assumptions. This model is re-recognized and deliberated with fuzzy numbers to estimate its uncertain parameters whose values are not precisely known. Exhibition of imprecise solutions of the concerned model is carried out by using the proposed two methods: generalized Hukuhara difference and generalized Hukuhara derivative (gH-derivative) approaches. Moreover, the stability analysis of the model in two different systems in fuzzy environment is illustrated. Additionally, different illustrative examples for optimization of national income determination model are undertaken with the constructive graph and table for convenience for clarity of the projected approaches.

Some results on the fundamental concepts of fuzzy set theory in intuitionistic fuzzy environment by using α and β cuts

In this paper we have firstly examined the properties of ? and ? cuts of intuitionistic fuzzy numbers in Rn with the help of well-known Stacking and Characterization theorems in fuzzy set theory. Then, we have studied the generalized Hukuhara difference in intuitionistic fuzzy environment by using the properties of ? and ? cuts and support function. Finally, we have extended the strongly generalized differentiability concept from fuzzy set theory to intuitionistic fuzzy environment and proved the related theorems with this concept.

Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.