Variational iteration technique for solving a system of nonlinear equations

In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving a system of nonlinear equations. We prove that the new method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new iterative methods. Our results can be viewed as a refinement and improvement of the previously known results.     

Auxiliary Principle Technique for Solving Bifunction Variational Inequalities

In this paper, we use the auxiliary principle technique to suggest and analyze an implicit iterative method for solving bifunction variational inequalities. We also study the convergence criteria of this new method under pseudomonotonicity condition.     

Synthesis of a C-2 Stapled Bis-Lexitropsin

Summary. A convenient synthesis method was developed for the preparation of C-stapled homodimeric bis-lexitropsins connected through the nitrogen atoms of the central pyrrole ring with a bis-methylene linker. This lexitropsin is designed as a standard for other bis-lexitropsins with longer chains in biological evaluation and NMR studies. The key step in this method is the treatment of ethyl 4-nitropyrrole-2-carboxylate with flame-dried potassium carbonate in DMF followed by the addition of 1,2-dibromoethane to form the 1,2-dipyrroloethane derivative.     

Unifying fourth-order family of iterative methods

In this work, we develop a new two-parameter family of iterative methods for solving nonlinear scalar equations. One of the parameters is defined through an infinite power series consisting of real coefficients while the other parameter is a real number. The methods of the family are fourth-order convergent and require only three evaluations during each iteration. It is shown that various fourth-order iterative methods in the published literature are special cases of the developed family. Convergence analysis shows that the methods of the family are fourth-order convergent which is also verified through the numerical work. Computations are performed to explore the efficiency of various methods of the family.     

Algorithms for nonlinear mixed variational inequalities

In this paper, we establish the equivalence between the generalized nonlinear mixed variational inequalities and the generalized resolvent equations. This equivalence is used to suggest and analyze a number of iterative algorithms for solving generalized variational inequalities. We also discuss the convergence analysis of the proposed algorithms. As special cases, we obtain various known results from our results.     

On the Parametric Cubic Spline Approach for Solving Second Order Boundary Value Problems

In this article some comments on the paper “parametric cubic spline approach to the solution of a system of second order boundary value problems” in (Khan and Aziz, J. Optim. Theory Appl. 118:45–54, 2003) are given. This paper concerns with a numerical method for solving a second order boundary value problem associated with obstacle, unilateral and contact problems. Corrections are given for the convergence analysis of the numerical method and the computational experiments.     

Templating Ag on DNA/polymer hybrid nanowires: Control of the metal growth morphology using functional monomers

Ag nanowires and nanoparticles have been formed on hybrid λ-DNA/conducting polymer templates. The strong, but non-covalent, interaction of the conducting polymer with the double helix allowed us to incorporate chemical functionalities (alkynyl) into the DNA/conducting polymer strands by synthesis of functional monomers. Oxidative polymerisation of alkynyl-thienylpyrrole in the presence of λ-DNA produced conductive nanowires bearing alkyne groups; we show, using a combination of AFM, cAFM and EFM phase measurements that the alkyne functionality strong influences the subsequent templating reaction of Tollens’ reagent to produce uniform conductive nanowires comprised of many connected Ag clusters.     

On nonconvex bifunction variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which is called the nonconvex bifunction variational inequality. We suggest and analyze some iterative methods for solving nonconvex bifunction variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only pseudomonotonicity, which is weaker condition than monotonicity. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.     

Cubic spline method for solving two-point boundary-value problems

In this paper, we use uniform cubic spline polynomials to derive some new consistency relations. These relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem. The present method outperforms other collocations, finite-difference and splines methods of the same order. Numerical illustrations are provided to demonstrate the practical use of our method.