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

Abstract

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.