A New fractional derivative for differential equation of fractional order under interval uncertainty

Abstract

In this article, we develop a new definition of fractional derivative under interval uncertainty. This fractional derivative, which is called conformable fractional derivative, inherits some interesting properties from the integer differentiability which is more convenient to work with the mathematical models of the real-world phenomena. The interest for this new approach was born from the notion that makes a dependency just on the basic limit definition of the derivative. We will introduce and prove the main features of this well-behaved simple fractional derivative under interval arithmetic uncertainty. The actualization and usefulness of this approach are validated by solving two practical models.

Uncertain fractional operator with application arising in the steady heat flow

“…in which 3 . u ∈   The most significant definition of the interval derivative, based on the fuzzy differentiability notion presented in [24,25], is stated as:…”

Asymptotic behaviour of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation

“…Thus conformable fractional derivative is a natural extension of the classical derivative (since it can be expressed as a first derivative multiplied by a fractional factor or power in some cases) and has many advantages over other fractional derivatives as mentioned above. The conformable fractional derivative combines the best characteristics of well-known fractional derivatives, seems more appropriate to describe the behavior of classical viscoelastic models under interval uncertainty, see [24] and gives models that agree and are consistent with experimental data, see [2]. Conformable fractional derivatives are applied in certain classes of conformable differentiable linear systems subject to impulsive effects and establish quantitative behavior of the nontrivial solutions (stability, disconjugacy, etc), see [4,10]; and used to develop the Swartzendruber model for description of non-Darcian flow in porous media [4,28].…”

Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

“…The challenge in studying stochastic Cauchy equation with conformable fractional derivative is that there is no singular kernel of the form (t − s) −α generated for 0 < α < 1 which reflect the nonlocality and the memory in the fractional operator as in the case of R-L and C-D fractional derivatives. Thus, despite the fact that conformable fractional derivative combines the best characteristics of known fractional derivatives, seems more appropriate to describe the behaviour of classical viscoelastic models under interval uncertainty, see [19] and gives models that agree and are consistent with experimental data, see [20], it does not possess or satisfy a semigroup property unlike the R-L and C-D fractional operators that have well-behaved semigroup properties.…”