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.     

Investigation of the equality constraint effect on the reduction of the rotational ambiguity in three-component system using a novel grid search method

The obtained results by soft modeling multivariate curve resolution methods often are not unique and are questionable because of rotational ambiguity. It means a range of feasible solutions equally fit experimental data and fulfill the constraints. Regarding to chemometric literature, a survey of useful constraints for the reduction of the rotational ambiguity is a big challenge for chemometrician. It is worth to study the effects of applying constraints on the reduction of rotational ambiguity, since it can help us to choose the useful constraints in order to impose in multivariate curve resolution methods for analyzing data sets. In this work, we have investigated the effect of equality constraint on decreasing of the rotational ambiguity. For calculation of all feasible solutions corresponding with known spectrum, a novel systematic grid search method based on Species-based Particle Swarm Optimization is proposed in a three-component system.     

Three-Component Reaction of Triphenylphosphine,Dimethyl Acetylenedicarboxylate,and Aldehyde Benzoylhydrazones: An Efficient One-Pot Synthesis of Stable Phosphorus Ylides

Protonation of the reactive 1:1 intermediate produced in the reaction between dimethyl acetylenedicarboxylate and triphenylphosphine by benzoylhydrazones leads to vinylphosphonium salts, which undergo Michael addition with the conjugate base of the NH acid to produce highly fanctionalized, salt-free phosphorus ylides in excellent yields.     

An Efficient,One-Pot Synthesis of Dialkyl 5-Hydroxy-4-aryl-2,5-dihydrofuran-2,3-dicarboxylate Derivatives

The reaction between arylglyoxalmonohydrates, dialkyl acetylenedicarboxylates, and triphenylphosphine is described as a simple and efficient method for the synthesis of 5-hydroxy-4-aryl-2,5-dihydrofuran-2,3-dicarboxylate derivatives.