A general fractional scale derivative is introduced and studied. Its relation with the Hadamard derivatives established reformulated. new similar to Grünwald–Letnikov’s deduced. Tempered versions are also introduced. Scale-invariant systems described exemplified. For solving corresponding differential equations, a logarithmic Mittag-Leffler series proposed.

This paper presents a series of new results in modeling of the Grünwald-Letnikov discretetime fractional difference by means of discrete-time Laguerre filers. The introduced Laguerrebased difference LD and combined fractional/Laguerre-based difference CFLD are shown to perfectly approximate its fractional difference original, for fractional order α ∈ 0, 2 . This paper is culminated with the pre...

In this paper, we further discuss the properties of three kinds of fractional derivatives: the Grünwald–Letnikov derivative, the Riemann–Liouville derivative and the Caputo derivative. Especially, we compare the Riemann–Liouville derivative with the Caputo derivative. And sequential property of the Caputo derivative is also derived, which is helpful in translating the higher fractional-order di...

Phytophthora ramorum is an aggressive Oomycete pathogen introduced into western North America and western Europe in the late twentieth century by the ornamental plant trade (Goss et al. 2011, Grünwald et al. 2012, Mascheretti et al. 2008, Prospero et al. 2007). The pathogen attacks a wide range of trees and shrubs, causing foliage blights and bleeding stem lesions both in nurseries and in the f...

This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several subdomains, and a fractional derivative (FDs) at a time node point is approximated using a modified Grünwald-Letnikov ap...

Bone marrow biopsy showed infiltration by numerous histiocytes. Some of the histiocytes contained vacuolated cytoplasm whilst others contained granules that stained deep blue with the May-Grünwald-Giemsa stain (Figure 2). This histological pattern was indicative of Niemann-Pick disease. Confirmation of this diagnosis was obtained by lysosomal enzyme assay that showed undetectable sphingomyelina...