Mathematical Methods in Interdisciplinary Sciences

7Arbitrary Order Differential Equations with Fuzzy Parameters

Tofigh Allahviranloo and Soheil Salahshour

Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey

7.1 Introduction

In the last decades, some generalization of theory of ordinary differential equations has been considered to the arbitrary order differential equations by many researchers, the so‐called theory of arbitrary order differential equations (often called as fractional order differential equations [FDEs]) [1–5].

Because of the ability for modeling real phenomena, arbitrary order differential equations have been applied in various fields such as control systems, biosciences, bioengineering [6–8], and references therein.

The extension of fractional calculus is done in several directions, in which two of them are common; the first is proposing several types of fractional integral and fractional derivatives, which is reviewed completely by Teodoro et al. [9], and the second is considering the uncertainty concepts in the structure of FDEs (we emphasize on fuzzy uncertainty) [10–19]. A combination of these two points of view led to propose some new model and obtain new types of solutions and approximations [20,21].

Based on this approach, we provided some new types of arbitrary order differential equations with respect to another function involving fuzzy uncertainty.

In fact, using the definitions for arbitrary order integral and derivative of Riemann–Liouville type with respect to ...