The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f (t), by a fractional integral operator applied to the derivative f (t). We define a new fractional operator by substituting for this f (t) a more general proportional derivative. This new operator can also be written as a Riemann-Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann-Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

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BackgroundStudies of ethnobotanical usages in south-eastern Turkey are rare. To widen this field of knowledge, we conducted an ethnobotanical study in Midyat (Mardin Province), Turkey.MethodsThe field study was completed during three years (2007–2010). Our aim was to document the ethnobotanical uses of local plants and to make an ethnobotanical inventory of uncommon plants using qualitative interviews.ResultsDuring field studies, 368 voucher specimens were collected in the investigated area. Ninety-two traditionally used plant species were reported from Midyat and surrounding vicinities in Turkey. Among the 92 taxa (129 usages), 35% were used for medical purposes, 22% for food, 13% for animal fodder, 7% as ornamental plants and dyes, 6% as brooms, 4% for latex and as fragrance, 4% for herbal tea, molasses and wine preparation, 3% for agricultural purposes, and 6% for other purposes. Comparative assessment showed that Teucrium polium (0.51), Matricaria aurea (0.26), Alcea setosa (0.21), and Malva neglecta (0.21) have the highest recorded UVs, and the following taxa had UVs between 0.10–0.20: Anthemis cotula (0.12), Allium cepa (0.13), Alcea striata subsp. striata (0.14), Crupina crupinastrum (0.12), Papaver rhoeas (0.13), Salvia multicaulis (0.14), Thymbra spicata (0.11), and Vicia pannonica subsp. striata (0.15). We reported the ethnobotanical usages of 21 taxa for the first time, in addition to indicating usages previously recorded in the literature. We also recorded four endemic plant usages in the area: Alkanna trichophila var. mardinensis, Centaurea kurdica, Centaurea stapfiana, and Sideritis libanotica subsp. linearis. They have variable leaf and flower morphology that are used traditionally. They are present as well-developed populations and thus their conservation status is not compromised. Additionally, Thymbra sintenisii is a recorded species that is classified as a rare and extensively used species in the region.ConclusionsThese results contribute to the fundamental knowledge of ethnobotanical usages in Midyat. To date, ethnobotanical studies have not been carried out in this region. This investigation uncovered usages of endemic medicinal plant species and traditional knowledge of Midyat communities living in a mixed culture. The people of Midyat, Batman, and Şırnak are Turkish citizens from various ethnic backgrounds, such as Kurdish, Arabic, and Syriac. We compared our data with results from other studies conducted in Turkey, particularly in south-eastern and eastern regions, as well as with studies from bordering countries, Iraq, Jordan, Syria, and Iran. Nonetheless, more work needs to be conducted to extend the present knowledge for locals to contribute to and evaluate economic potential in the region.

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In this work, we study the fractional order Lane-Emden differential equations by using the reproducing kernel method. The exact solution is shown in the form of a series in the reproducing kernel Hilbert space. Some numerical examples are given in order to demonstrate the accuracy of the present method. The results obtained from the method are compared with the exact solutions and another method. The obtained numerical results are better than the ones provided by the collocation method. Results of numerical examples show that the presented method is simple, effective, and easy to use.

The surge of resistant food pathogens is a major threat worldwide. Previous research conducted on phytochemicals has shown their antibacterial activity against pathogenic bacteria. The design of antimicrobial agents to curb pathogenic disease remains a challenge demanding critical attention. Flavonoids such as apigenin and quercetin were evaluated against Gram-positive and Gram-negative bacteria. The results indicated that the antibacterial activity of each flavonoid occurred at a different minimum inhibitory concentration. However, the antimicrobial activity results of the modified flavonoids were also reported, and it was observed that the Gram-positive bacteria were more susceptible in comparison to the Gram-negative bacteria. The cell wall structure of the Gram-positive and Gram-negative bacteria could be the main reason for the bacteria susceptibility. Modified flavonoids could be used as a suitable alternative antimicrobial agent for the treatment of infectious diseases. Our results indicated 100% inhibition of Listeria monocytogenes , Pseudomonas aeruginosa , and Aeromonas hydrophila with modified flavonoids.

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