Luciana Takata Gomes

Fuzzy differential equations: An approach via fuzzification of the derivative operator

Abstract In this paper we study fuzzy differential equations (FDEs) in terms of derivative for fuzzy functions, in a different way from the traditional Hukuhara derivative defined for set valued functions. The derivative we use is obtained by means of fuzzification of the classical derivative operator for standard functions. We discuss the relation of this approach to fuzzy differential inclusions (FDIs) and Hukuhara and strongly generalized derivatives. A theorem of existence of a solution is studied, with hypothesis similar to those assumed for FDIs. Some examples are explored in order to illustrate the theory.

Fuzzy Differential Equations in Various Approaches

1. Introduction. -2. Basic Concepts. -3. Fuzzy Calculus. -4. Fuzzy Differential Equations. -Mathematical Background. -Index.

A note on the generalized difference and the generalized differentiability

This note reveals that the assertion in Stefanini (2008) 2 that the generalized difference between two fuzzy numbers is always a fuzzy number is incorrect. As a consequence, the theorem of existence of the generalized differentiability in Bede and Stefanini (2013) 7 is also incorrect. We propose a modification in the definition of the generalized difference in order to assure the mentioned results.

Fuzzy calculus via extension of the derivative and integral operators and fuzzy differential equations

We define the concepts of derivative and integral of fuzzy functions using the extension principle of Zadeh on the corresponding classical operators. Here are some of its properties and we articulate a version of the Fundamental Theorem of Calculus for fuzzy functions and ensure the existence of a solution of fuzzy initial value problem under certain conditions. A method of solving fuzzy initial value problem is presented and an application of a decay model is solved and interpreted within a fuzzy context.

Solution to Convex Variational Problems with Fuzzy Initial Condition Using Zadeh's Extension.

This paper investigates conditions to solve a fuzzy variational problem using Zadeh’s extension. The fuzzy problem is obtained by extending a classical one in the initial condition. The solution to the problem is a fuzzy bunch of functions (fuzzy set of functions) obtained by extending the classical solution in the initial condition. For convex functionals the resulting functional value is a fuzzy number that is proved to be the smallest element in a partial order.

Zadeh’s Extension of the Solution of the Euler-Lagrange Equation for a Quadratic Functional Type

This article aims to provide some necessary conditions so that the solution of the Euler-Lagrange equation, arising from the necessary conditions of a variational calculation problem is increasing with respect to the initial condition. This result is important for the study of the variational calculus problems with fuzzy initial condition and arbitrary final condition, since it establishes conditions to apply Zadeh’s extension with respect to the initial condition for a quadratic functional type.

Uma Abordagem direta para o Problema de Controle Ótimo de Pragas

Uma das preocupacoes dos produtores rurais e e a incidencia de pragas nas lavouras. Elas costumam impedir o desenvolvimento das plantas e se nao controladas, podem trazer muitos prejuizos. Propomos um modelo de controle otimo de pragas atraves da insercao de inimigos naturais no ecossistema a fim de manter a populacao de pragas em um determinado limiar que nao cause danos economicos ao produtor. Resolvemos o problema de controle otimo atraves de uma abordagem direta das condicoes de otimalidade do problema.

A new interpretation on fuzzy differential equations: Linguistic concepts evolving over time

Fuzzy differential equations have been explored in many different ways, including theoretical aspects and in applications. The interpretation that is given to the fuzziness is a uncertainty due to fluctuations accumulated in the measurement process or due to partial information. This has much to do with a probabilistic interpretation. We propose to interpret the fuzziness as Zadeh first stated: membership to a non-sharp concept. As example we solve a fuzzy initial value problem whose independent variable is the time and the unknown variable is regarded as a single concept (the word “tall”) whose compatibility (membership) function varies over time. In summary, the main novelty is that the fuzzy differential equation models a concept evolving over time.

Diferenças entre números fuzzy

Apresentamos diversas definicoes de diferencas entre numeros fuzzy presentes na literatura: tradicional, Hukuhara, generalizada de Hukuhara, generalizada, CIA e via distribuicoes. Exemplificamos atraves de um problema de transmissao direta e comparamos as diferentes abordagens.