Comparison Between Numerical Solutions of Fuzzy Partial Differential Equations via Interactive and Non-interactive Arithmetics: Application to the Heat Equation

Abstract

This paper provides a new numerical solution to partial differential equations, where the initial and boundary conditions are given by interactive fuzzy numbers. The interactivity considered here is the one obtained from the joint possibility distribution I, which is associated with a parameterized family of joint possibility distributions. The proposed method is given by the finite difference method, adapted for arithmetic operations of I-interactive fuzzy numbers. In addition to the proposed solution, a comparison with a numerical solution given by the fuzzy standard arithmetic, which is also known as non-interactive arithmetic, is presented. This paper shows that the numerical solution via interactive arithmetic is more specific than the one via non-interactive arithmetic, which means that the numerical solution via I-interactive arithmetic propagates less uncertainty than the numerical solution via standard arithmetic. The proposed method can be applied in any fuzzy partial and ordinary differential equation. In order to illustrate the results, an application to the heat equation is presented.