New Derivatives on the Fractal Subset of Real-Line

Abstract

Abstract:In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on the fractals subset of real-line lies in the fact that they are better at modeling processes with memory effect.

Phase Engineering of High‐Entropy Alloys

“…BCC phase, which is highly demanded for the aerospace industry, is basically composed of refractory elements and turns out high thermal stability and hardness 36,37. It was demonstrated by Senkov et al that the single‐phase BCC HEAs Ta 25 Nb 25 Mo 25 W 25 (TaNbMoW) and Ta 25 Nb 25 V 25 Mo 25 W 25 (TaNbVMoW) were synthesized by vacuum arc melting 36.…”

The concept of fractal experiments: New possibilities in quantitative description of quasi-reproducible measurements

“…Before these two trends (including many papers and books that cannot be listed here) that are developed almost independently from each other, many researches tried to establish new relationships between fractals and fractional calculus. Here we want to mark especially the book [12] and papers [13][14][15] , where new links between these two trends have been established. This paper put forward another concept: the imposed fractal experiments that can create the desired self-similarity during the process of the realized measurements.…”

Accurate relationships between fractals and fractional integrals: New approaches and evaluations

“…(8) Here Ω n = 2πn/ ln ξ is a set of frequencies providing a periodicity with ln of product (4). Taking into account this decomposition one can present the limiting solution of (1) in the form…”

“…The interest to relationship between fractals and fractional calculus is renewed again. Some original approach (but without proper physical interpretation) was outlined in papers [16,4,5]. One of the basic problems that did not accurately solved yet in the fractional calculus community is the finding of the justified and accurate relationships between the smoothed functions averaged over fractal objects and fractional operators.…”