A two-index generalization of conformable operators with potential applications in engineering and physics

Abstract

We developed a somewhat novel fractional-order calculus workbench as a certain generalization of the Khalil’s conformable derivative. Although every integer-order derivate can naturally be consistent with fully physical-sense problem’s quotation, this is not the standard scenario of the non-integer-order derivatives, even aiming physics systems’s modelling, solely.We revisited a particular case of the generalized conformable fractional derivative and derived a differential operator, whose properties overcome those of the integer-order derivatives, though preserving its clue advantages.Worthwhile noting, that two-fractional indexes differential operator we are dealing, departs from the single-fractional index framework, which typifies the generalized conformable fractional derivative. This distinction leads to proper mathematical tools, useful in generalizing widely accepted results, with potential applications to fundamental Physics within fractional order calculus. The later seems to be especially appropriate for exercising the Sturm-Liouville eigenvalue problem, as well as the Euler-Lagrange equation and to clarify several operator algebra matters.

The non-integer local order calculus

“…In a work from the same year (cf.) 58 another conformable derivative is defined in a very similar way. Let f be a function of ( )…”

Mathematical model to estimate volumetric oxygen transfer coefficient in bioreactors using conformable calculus

“…More recently in (Khalil et al, 2014), the first conformable calculus was introduced that is an alternative to conformal calculus, but with local operators, without memory. These conformable operators can improve the performance of conformal operators for example (see Meléndez-Vázquez et al, 2021;Reyes-Luis et al, 2021;Meléndez-Vázquez et al, 2020), since these operators can introduce a function that conformal operators cannot, regardless of the non-integer order of these derivatives.…”