A theory of calculus for A-correlated fuzzy processes

Abstract

This article presents a new concept of calculus for fuzzy number-valued functions based on Zadeh's extension principle. More precisely, we propose a calculus theory for A-correlated processes which correspond to functions of the form fˆ(t,A), where fˆ denotes the Zadeh extension of a function of f(t,⋅) at a fixed fuzzy number A. The notions of derivatives and integrals for these functions are defined in natural way by means of the Zadeh extension of the derivatives and integral operators of the function f. We show that these definitions coincide with the notions of derivative and integral based on arithmetic operations with interactive operands. In addition, we apply the proposed calculus to study certain fuzzy differential equations. We argued that the proposed fuzzy calculus theory is adequate to deal with certain fuzzy processes that represent epistemic states of single-valued variables.