On the Numerical Solution of Ordinary, Interval and Fuzzy Differential Equations by Use of F-Transform

Davide Radi,Laerte Sorini,Luciano Stefanini

doi:10.3390/axioms9010015

Abstract

An interesting property of the inverse F-transform f ^ of a continuous function f on a given interval [ a , b ] says that the integrals of f ^ and f on [ a , b ] coincide. Furthermore, the same property can be established for the restrictions of the functions to all subintervals [ a , p k ] of the fuzzy partition of [ a , b ] used to define the F-transform. Based on this fact, we propose a new method for the numerical solution of ordinary differential equations (initial-value ordinary differential equation (ODE)) obtained by approximating the derivative x · ( t ) via F-transform, then computing (an approximation of) the solution x ( t ) by exact integration. For an ODE, a global second-order approximation is obtained. A similar construction is then applied to interval-valued and (level-wise) fuzzy differential equations in the setting of generalized differentiability (gH-derivative). Properties of the new method are analyzed and a computational section illustrates the performance of the obtained procedures, in comparison with well-known efficient algorithms.

Highlights

The fuzzy transform (F-transform) of a continuous function f : [ a, b] −→ R was introduced by Perfilieva in [1,2]
We propose a numerical method where the inverse F-transform is used to approximate the derivative x (t); the solution x (t) is obtained by exact integration of the approximated derivative: this is allowed by an interesting property which says that the integral of the inverse F-transform of f on [ a, b] coincides with the integral of the function f itself; this idea was presented in a preliminary form in [17]
The algorithm ordinary differential equations (ODEs)-FT described in Section 2 has been implemented in MATLAB and used to solve a series of d-dimensional ODEs with d = 1, 2, 3 and 4

Summary

Introduction

The fuzzy transform (F-transform) of a continuous function f : [ a, b] −→ R was introduced by Perfilieva in [1,2]. This special fuzzy method is appealing and useful to handle many real-world problems and an extensive research activity has both analyzed its properties and its fields of applications; for the literature related to this paper we refer to, e.g., [3,4,5,6,7,8,9,10,11] and the references therein. In the final section of this paper, we will present some comments and a preliminary comparative valuation of the proposed methods

Methods

Results

Conclusion