Fuzzy differential equations and fuzzy algebra with applications

Abstract

The fuzzy set theory was developed to analyze and characterize sets withvague frontiers. There are several ramifications within the area of fuzzy sets, in which this project dedicates to study two of them. The first one is the theory offuzzy differential equations (FDEs), in which the imprecision may be associated to the parameters involved, initial/boundary conditions or direction field. In modelling, some concepts are considered, such as fuzzy numbers, which extends the concept of real numbers, and also fuzzy derivatives and fuzzy integrals. Recently, it has been proved that the class of fuzzy numbers RF, under certain conditions, has the structureof a Banach space. This fact has brought several applications in dynamical systems. The condition mentioned above is associated with a fuzzy relationship called interactivity. This project focuses on exploring new results and applications from FDEs, consideringthe fuzzy relation of interactivity. In particular, this project will continue to study two distinct approaches thatare being explored by the candidate. The first considers the sup-J extension principle, which is a method that extends classical operators so that it is possible to have interactive (non-interactive) fuzzy numbers as inputs. The second is to explore dynamic systems, without effectivelyuse technical skills to solve classical differential equations, but instead consider the qualitative knowledge of the phenomenon and study it through a technique called p-fuzzy systems. The second area is the fuzzy algebra theory. In this fields it is possible to work in two perspectives. The first one is based on the algebraic structure of fuzzy numbers and arithmetic operations. The second perspective is based on the study of algebraic structures of fuzzy subsets of classical groups. For each approach, it is intended to apply the theoretic results in problems of exact and biological sciences, more specifically, in the areas of physics, chemistry and biology. (AU)