Manifestation of interval uncertainties for fractional differential equations under conformable derivative

Abstract

We propose a generalization of conformable calculus for Type-2 interval-valued functions. We investigated the differentiability and integrability properties of such functions. The conformable generalized Hukuhara (gH) differentiability of fractional order is introduced in this study. We prove a number of essential theorems on the conformable differentiability of the sum, gH difference, and product in a Type 2 interval setting. Furthermore, we define conformable Laplace transformation of Type-2 interval-valued functions. We interpret uncertain linear differential equations by using proposed theories. Several examples are given in detail to illustrate and clarify these rules and theorems. Applications to solving Type-2 interval differential equations with conformable derivatives are shown. Type-2 interval generalizes the interval uncertainty. On the other hand, conformable calculus extends the notion of integer calculus. This paper contributes a generalized theory that includes several existing results of classical integral and differential calculus and their conformable extensions in crisp and interval environments.