arXiv subjects

Estevão Esmi

Publications and source records attributed to Estevão Esmi.

3 recordsLinked to original sources

arXiv2026

On Fuzzy Partial Differential Equations for A-Linearly Interactive Fuzzy Complex Processes: Sobolev Spaces and Fourier Analysis

This article develops a mathematical framework to handle fuzzy partial differential equations (PDEs) using Sobolev spaces defined over A-linearly interactive complex fuzzy processes. We introduce the notion of weak derivative in the fuzzy sense to define fuzzy Sobolev spaces that preserve key analytical properties. Furthermore, we introduce a fuzzy Fourier transform adapted to this context and explore its main properties. Applications to fuzzy versions of the heat and Schrödinger equations are presented to demonstrate the effectiveness and generality of the proposed framework. This approach not only extends classical tools to the fuzzy setting, but also provides new insights into the treatment of uncertainty in dynamic systems.

math.GM↗

arXiv2025

Calculus for Functions with Fuzzy Inputs and Outputs: Applications to Fuzzy Differential Equations

This article presents a theory of differential and integral calculus for mapping between Banach spaces formed by subsets of fuzzy numbers called A-linearly correlated fuzzy numbers, where both the domain and codomain are spaces composed of fuzzy numbers. This is one of the main contributions of this study from a theoretical point of view, as well-known approaches to fuzzy calculus in the literature typically deal with fuzzy number-valued functions defined on intervals of real numbers. Notions of differentiability and integrability based on complex functions are proposed. Moreover, we introduce the study of ordinary differential equations for which the solutions are functions from A-linearly correlated fuzzy numbers to A-linearly correlated fuzzy numbers or from real functions to A-linearly correlated fuzzy numbers. For the latter case, we present an initial study of the solution and its phase portrait for two-dimensional differential equation systems. In particular, for the former case, we examine the Lotka Volterra model and analyze its phase portrait.

math.GM↗

arXiv2025

Fuzzy Variational Calculus in Linearly Correlated Space: Part I

This article is the first part of series of articles that aim to present the foundations for fuzzy variational calculus for functions taking values in the space of linearly correlated fuzzy numbers $\mathbb{R}_{\mathcal{F}(A)}$. Recall that the space $\mathbb{R}_{\mathcal{F}(A)}$ is composed by all sums of a real number $r$ and a fuzzy number $qA$, where $A$ is a given asymmetric fuzzy number. Two advantages of this space are that it can be equipped with a Banach space structure and, similar to additive models in Statistics, its elements can be interpreted as the sum of a deterministic expected/predictable value with an uncertain/noise component. The foundation of variational calculus theory requires the definition and establishement of many concepts and results. This aritcle presents a total order relation on $\mathbb{R}_{\mathcal{F}(A)}$ for wihch the notions of local minimal and maximal of a $\mathbb{R}_{\mathcal{F}(A)}$-valued function $f$ can be derived. We present a fuzzy version of the first and second optimality conditions in terms of derivatives of $f$. Finally, we present a generalized fuzzy version of du Bois-Reymond lemma which is essential in variational calculus theory.

math.GM↗