A Geometric Approach to Solve Fuzzy Linear Systems of Differential Equations

TL;DR

This paper introduces a geometric method for solving fuzzy linear differential systems, representing solutions as fuzzy regions with nested parallelepipeds, providing a novel perspective on fuzzy differential equations.

Contribution

It presents a new geometric approach that considers solutions as fuzzy sets of vector-functions, differing from previous methods by using fuzzy regions in coordinate space.

Findings

Solution forms fuzzy regions in coordinate space

Alfa-cuts are nested parallelepipeds

Method effectively solves examples

Abstract

In this paper, systems of linear differential equations with crisp real coefficients and with initial condition described by a vector of fuzzy numbers are studied. A new method based on the geometric representations of linear transformations is proposed to find a solution. The most important difference between this method and methods offered in previous papers is that the solution is considered to be a fuzzy set of real vector-functions rather than a fuzzy vector-function. Each member of the set satisfies the given system with a certain possibility. It is shown that at any time the solution constitutes a fuzzy region in the coordinate space, alfa-cuts of which are nested parallelepipeds. Proposed method is illustrated on examples.