A fractional derivative with non-singular kernel for interval-valued functions under uncertainty

The purpose of the current investigation is to generalize the concept of fractional derivative in the sense of Caputo-Fabrizio derivative (CF-derivative) for interval-valued function under uncertainty. The reason to choose this new approach is originated from the non-singularity property of the kernel that is critical to interpret the memory aftermath of the system, which was not precisely illustrated in the previous definitions. We study the properties of CF-derivative for interval-valued functions under generalized Hukuhara-differentiability. Then, the fractional differential equations under this notion are presented in details. We also study three real-world systems such as the falling body problem, Basset and Decay problem under interval-valued CF-differentiability. Our cases involve a demonstration that this new notion is accurately applicable for the mechanical and viscoelastic models based on the interval CF-derivative equations.