Pterocarya fraxinifolia (Lam.) Spach., a relict tree species of the Juglandaceae family, is native to the Great Caucasus, Anatolia, and to the Hyrcanian forests of the southern Azerbaijan and Northern Iran. In this study, the phylogenetic relationship of the species, sampled in selected Iranian populations, and the global biogeography of the genus Pterocarya were addressed. Leaves were collected from 8 to 10 trees from three geographically isolated habitats. The samples were analyzed with nuclear (internal transcribed spacer [ITS] regions) and chloroplast (trnH-psbA) DNA markers. The obtained results were compared and analyzed with the data registered in NCBI GenBank. It is reported that the ITS regions varied from 644 to 652 for Pterocarya genus, but we did not observe polymorphisms for Iranian Pterocarya. The phylogenetic tree divided the Pterocarya genus in three clades: clade 1 grouping exclusively the samples P. fraxinifolia, clearly separated from the East Asiatic taxa; clade 2 that includes the species P. hupehensis and P. macroptera; clade 3 clustering P. stenoptera and P. tonkinensis. Although the Iranian Pterocarya samples and P. fraxinifolia from the Caucasus were in the same clade, they presented two different secondary structures. The Iranian populations showed the maximum genetic distance with P. stenoptera and P. tonkinensis. Our analysis demonstrates that the traditional division of all the six species sampled throughout their distribution area as well as the phylogeny of the genus Pterocarya needs to be reviewed.

We present here a high-order numerical formula for approximating the Caputo fractional derivative of order for 0 < < 1. This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is 4 − , while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The effectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial differential equations is presented by constructing a finite difference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.

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