A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion

Abstract

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.

Well-posedness and stability for fuzzy fractional differential equations

“…Recently, Ngo presented results on the existence and uniqueness of solutions for two kinds of fractional fuzzy functional integral equations and fuzzy functional differential equations using the contraction mapping principle and the successive approximation method [11,12]. For research on solutions of initial boundary value problems for fractional fuzzy differential equations, more information can be found in [1,4,6,13,22,27,32,35] and the references therein.…”

Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition

“…Various papers have been published on fractional differential equations (FDEs) (see, e.g., in [1][2][3][4][5][6]). Over the years, hybrid fractional differential equations have attracted much attention.…”

New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

“…Fractional calculus refers to the integration and differentiation of a non-integer order and is as old as the classical (integer order) calculus [1]. It is a subject that has gained much popularity and importance in the last few decades and has been applied in several fields of knowledge, such as mechanics [2,3], bioengineering [4], signal and image processing [5], physics [6,7], viscoelasticity [8], electrical engineering [9], economics [10], epidemiology [11,12], control theory [13,14], energy supply-demand systems [15], and fuzzy problems [16].…”