An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.     

Phosphorylated PEG (PPEG) as a new support for generation of nano‐Pd(0): application to the Heck–Mizoroki and Suzuki–Miyaura coupling reactions

Polyethylene glycols (PEGs) with different molecular weights (M_w = 200, 400, 1000) were phosphorylated to their bis‐diphenyl phosphinite derivatives as stable solids which are melted in the range 140–160°C. These phosphorylated PEGs were used as ligands and reducing agents to generate nano‐Pd(0) catalysts in 2.5–8.3 nm. The nano‐Pd(0) particles supported on phosphorylated PEG₂₀₀ were applied for the efficient Heck–Mizoroki carbon–carbon coupling reactions of ArX (X = Cl, Br, I) at 80–100°C under solvent‐free conditions and for the Suzuki–Miyaura coupling reaction in ethanol at 70°C. The catalyst was recycled easily and reused for several runs. Copyright © 2013 John Wiley & Sons, Ltd.     

Pd(0)-guanidine@MCM-41: a very effective catalyst for rapid production of bis (pyrazolyl)methanes

In this research, a rapid, green and efficient protocol for synthesis of bis (pyrazolyl)methane derivatives in the presence of Pd(0)-guanidine@MCM-41 catalysts under solvent-free conditions by the following two methods has been reported: (i) via the one-pot pseudo five-component reaction among phenylhydrazine (2 equivalents), ethyl acetoacetate (2 equivalents) and aromatic aldehydes (1 equivalent); and (ii) the one-pot pseudo three-component reaction between 3-methyl-1-phenyl-1H-pyrazol-5(4H)-one (2 equivalents) and aromatic aldehydes (1 equivalent). Some advantages of this protocol include: green conditions, extremely short times, high efficiency, proper one-pot operation, generality of method, easy work-up and recyclability, and reusability of the catalyst up to five times without significant loss in catalytic activity.     

Quantum chemical study of the nature of interactions between the boraphosphinine and alumaphosphinine with some of the mono- and divalent cations: cation–π or cation–lone pair?

Structural Chemistry - Quantum chemical study of the nature of interactions between the boraphosphinine (BP) and alumaphosphinine (AlP) with some of the alkali metal cations (Li+, Na+, K+) and...