Solution of Lane–Emden type equations using rational Bernoulli functions

In this paper, a numerical method for solving Lane‐Emden type equations, which are nonlinear ordinary differential equations on the semi‐infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane‐Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.     

REPELLERS FOR MULTIFUNCTIONS OF SEMI-BORNOLOGICAL SPACES

In this article the notion of repeller for multifunctions from the viewpoints of semi-bornologicul spaces is considered. The concept of lower semi-continuous multifunctions is extended by the use of semi-bornological spaces. Semi-bornological vector spaces are studied. The notion of conjugacy for semi-bornological multifunctions is considered. The persistence of repeller under conjugate relation is proved.     

Normal and mixed partial second-order subdifferentials in variational analysis and optimization

A partial second-order subdifferential is defined here for extended real valued functions of two variables corresponding to its variables through coderivatives of first-order partial subdifferential mappings. In addition, some rules are presented to calculate these second-order structures along with defining some conditions to insure the equality \(\partial ^2_{yx}\) and \(\partial ^2_{xy}\). Moreover, as an application, some conditions are stated which show the relation between local minimum of a function and positiveness of principal minors of its hessian matrix.     

Numerical Solution for a Variable-Order Fractional Nonlinear Cable Equation via Chebyshev Cardinal Functions

In this paper, a variable-order fractional derivative nonlinear cable equation is considered. It is commonly accepted that fractional differential equations play an important role in the explanation of many physical phenomena. For this reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of class of fractional partial differential equation with variable coefficient of fractional differential equation in various continues functions of spatial and time orders. Our main aim is to generalize the Chebyshev cardinal operational matrix to the fractional calculus. Finally, illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Aluminumdodecatungstophosphate (AlPW12O40) as a reusable Lewis acid catalyst: Facile regioselective ring opening of epoxides with alcohols, acetic acid and thiols

AlPW₁₂O₄₀ as a recyclable Lewis acid and stable solid compound was introduced for efficient ring opening of different epoxides with thiols, acetic acid, and primary, secondary, and tertiary alcohols. These processes were carried out at room temperature with high regio- and stereoselectivity.     

Copper-doped silica cuprous sulfate: A highly efficient heterogeneous nano-catalyst for one-pot three-component synthesis of 1-H-2-substituted benzimidazoles from 2-bromoanilines,aldehydes, and [bmim]N3

A facile and highly efficient one-pot three-component synthesis of 1-H-2-substituted benzimidazole derivatives from readily available substrates catalyzed by copper-doped silica cuprous sulfate (CDSCS) is described. In this method, treatment of diverse 2-bromoanilines, aldehydes, and [bmim]N₃ in DMF at 110 °C in the presence of CDSCS as a highly efficient heterogeneous nano-catalyst affords the corresponding 1-H-2-substituted benzimidazoles in good to excellent yields. The CDSCS is an inexpensive and stable nano-catalyst that could be simply prepared, recovered and reused for many consecutive reaction runs without significant loss of its activity.     

Magnetic Solid Phase Extraction Based on Modified Magnetite Nanoparticles Coupled with Dispersive Liquid–Liquid Microextraction as an Efficient Method for Simultaneous Extraction of Hydrophobic and Hydrophilic Drugs

In this work, surfactant-coated Fe₃O₄@decanoic acid nanoparticles was synthesized as a viable nanosorbent for coextraction of drugs with different polarities (hydrophobic, hydrophilic). To reach desirable enrichment factors, efficient clean-up and low limits of detection (LODs), the method was combined with dispersive liquid–liquid microextraction (DLLME). The coupling of these extraction methods with GC-FID detection was applied to simultaneous extraction and quantification of venlafaxine (VLF) as a hydrophilic model drug and desipramine (DESI) and clomipramine (CLO) as hydrophobic model drugs in urine samples. The effect of sample pH, nanosorbent amount, sorption time, surfactant concentration, eluent type, eluent volume, salt content, elution time in magnetic solid phase extraction step and extraction solvent and its volume along with sample pH in DLLME step were optimized. Under the selected conditions, linearity was achieved within the range of 5–5000 µg L^?1. The LOD values were obtained in the range of 1.5–3.0 µg L^?1 for DESI, 1.2–2.5 µg L^?1 for VLF and 2.0–4.0 µg L^?1 for CLO, respectively. The percent of extraction recoveries and relative standard deviations (n?=?5) were in the range of 82.4–95.9 and 6.1 for DESI, 60.5–92.8 and 6.9 for VLF and 57.2–58.0 and 5.5 for CLO, respectively. Ultimately, the applicability of the new method was successfully confirmed by the extraction and quantification of DESI, VLF and CLO from human urine samples.     

Electrocatalytic Reduction of Dioxygen on the Surface of Glassy Carbon Electrodes Modified with Cobalt Porphyrin Complexes

The preparation and electrocatalytic behavior of glassy carbon electrodes modified with three different cobalt porphyrin complexes were investigated. The electrocatalytic ability of the modified electrodes for the reduction of dioxygen to hydrogen peroxide and water in air‐saturated aqueous solutions was examined by cyclic voltammetry and chronoamperometry techniques. The porphyrin‐adsorbed glassy carbon electrodes possess excellent electrocatalytic abilities for dioxygen reduction with overpotential about 0.5 V lower than that at a plain glassy carbon electrode. The experimental parameters were optimized and the mechanism of the catalytic process was discussed. The possible effects of the electron‐donating properties of groups in the meso‐position of the porphyrin ring were investigated.     

Green synthesis and biological activity investigation of new derivatives of spiroisatins

In this research, the synthesis of novel derivatives of spiroisatins in high yields was investigated. These new compounds were synthesized using a multicomponent reaction (MCR) of isatin, malononitrile, acetophenone derivatives, diethyl oxalate, primary amines, and Et₃N in aqueous media. Under similar conditions, spiropyrroloisatins were prepared using MCRs of synthesized spiroistins. The antioxidant activity of newly synthesized spiroisatins is due to having an NH group which was evaluated by two procedures. Also, the antimicrobial activity of newly generated spiroisatins was evaluated by a disk distribution process utilizing two kinds of Gram-negative bacteria and Gram-positive bacteria and bacterial growth was stopped using these compounds. The advantages of this method are short reaction times, high yields of products, and the easy separation of catalyst and product using simple procedures.     

Immobilization of cerium (IV) and erbium (III) in mesoporous MCM‐41: Two novel and highly active heterogeneous catalysts for the synthesis of 5‐substituted tetrazoles,and chemo‐ and homoselective oxidation of sulfides

Two well‐ordered 2D ‐ hexagonal cerium (IV) and erbium (III) embedded functionalized mesoporous MCM ‐ 41(MCM‐41@Serine/Ce and MCM ‐ 41@Serine/Er) have been developed via functionalization of mesoporous MCM ‐ 41. The surface modification method has been used in the preparation of serine‐grafted MCM ‐ 41 and led to the development of MCM‐41@Serine. The reaction of MCM‐41@Serine with Ce (NH₄)₂(NO₃)₆·2H₂O or ErCl₃·6H₂O in ethanol under reflux led to the organization of MCM‐41@Serine/Ce and MCM‐41@Serine/Er catalysts. The structures of these catalysts were determined using scanning electron microscopy, mapping, energy‐dispersive X‐ray spectroscopy, Fourier transform‐infrared, thermogravimetric analysis, X‐ray diffraction, inductively coupled plasma, and Brunauer–Emmett–Teller analysis. These MCM‐41@Serine/Ce and MCM‐41@Serine/Er catalysts show outstanding catalytic performance in sulfides oxidation and synthesis of 5‐substituted tetrazoles. These catalysts can be recycled for seven repeated reaction runs without showing a considerable decrease in catalytic performance.