Voltammetric behavior of rutin at a boron-doped diamond electrode. Its electroanalytical determination in a pharmaceutical formulation

This paper examined for the first time, the possibilities of the usage of a boron-doped diamond electrode for the redox behavior of rutin using cyclic and adsorptive stripping voltammetry. The cyclic voltammograms showed a pair of redox peaks at lower potentials followed by an irreversible oxidation peak at higher positive potential. Using square-wave adsorptive stripping voltammetry, the compound yielded a well-defined voltammetric response in Britton-Robinson buffer, pH 4.0 at +0.48 V (vs. Ag/AgCl) (after 60 s accumulations at a fixed potential of 0.2 V). The calibration curve was linear in the concentration range from 0.01 µg mL^?1 to 0.1 µg mL^?1 (1.64×10^?8 M ? 1.64×10^?7 M). A detection limit of 0.0017 µg mL^?1 (2.78×10^?9 M) was observed without any chemical modifications and electrochemical surface pretreatments. As an example, the practical applicability of boron-doped diamond electrode was tested with the measurement of rutin in dietary supplement products.      

Analytical and numerical results for the Swift-Hohenberg equation

The problem of the Swift-Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter α and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter α and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Diffusion of chemically reactive species in a porous medium over a stretching sheet

Solutions for a class of nonlinear second-order differential equations, arising in diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet, are obtained. Furthermore, using the Brouwer fixed point theorem, existence results are established. Moreover, the exact analytical solutions (for some special cases) are obtained. The results obtained for the diffusion characteristics reveal many interesting behaviors that warrant further study of the effects of reaction rate on the transfer of chemically reactive species.     

Discretization and simulation of stochastic differential equations

We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being either the mean square error or the error induced by the approximation on the value of the expectation of a functional of the solution. We finally give and comment on a few comparative simulation results.     

A new method for homoclinic solutions of ordinary differential equations

Consideration is given to the homoclinic solutions of ordinary differential equations. We first review the Melnikov analysis to obtain Melnikov function, when the perturbation parameter is zero and when the differential equation has a hyperbolic equilibrium. Since Melnikov analysis fails, using Homotopy Analysis Method (HAM, see [Liao SJ. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman & Hall/CRC Press; 2003; Liao SJ. An explicit, totally analytic approximation of Blasius’ viscous flow problems. Int J Non-Linear Mech 1999;34(4):759–78; Liao SJ. On the homotopy analysis method for nonlinear problems. Appl Math Comput 2004;147(2):499–513] and others [Abbasbandy S. The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 2006;360:109–13; Hayat T, Sajid M. On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys Lett A, in press; Sajid M, Hayat T, Asghar S. Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn, in press]), we obtain homoclinic solution for a differential equation with zero perturbation parameter and with hyperbolic equilibrium. Then we show that the Melnikov type function can be obtained as a special case of this homotopy analysis method. Finally, homoclinic solutions are obtained (for nontrivial examples) analytically by HAM, and are presented through graphs.     

Spectral approximation for drainage of an Elastico‐viscous liquid and error analysis

A Fourier‐Galerkin spectral method is proposed and used to analyze a system of quasilinear partial differential equations governing the drainage of liquids of the Oldroyd four‐constant type. It is shown that, Fourier‐Galerkin approximations are convergent with spectral accuracy. An efficient and accurate algorithm based on the Fourier‐Galerkin approximations to the system of quasilinear partial differential equations are developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 492–505, 2012     

Viscoelastic stresses in the stagnation flow of a dilute polymer solution

In this paper, we consider viscoelastic stresses T₁₁, T₁₂ and T₂₂ arising in the stagnation flow of a dilute polymer solution; in particular, we consider an upper convected Maxwell (UCM) fluid. We present exact solutions to the coupled partial differential equations describing the viscoelastic stresses and deduce the results for the stress T₂₂ of Becherer et al. [P. Becherer, A.N. Morozov, W. van Saarloos, Scaling of singular structures in extensional flow of dilute polymer solutions, J. Non-Newtonian Fluid Mech. 153 (2008) 183–190]. As we considered the viscoelastic stresses over two spatial variables, we are able to study the effect of variable boundary data at the inflow. As such, our results are applicable to a wider range of fluid flow problems.     

Laguerre spectral approximation of Stokes’ first problem for third-grade fluid

A Laguerre–Galerkin method is proposed and analyzed for Quasilinear parabolic differential equation which arises from Stokes’ first problem for a third-grade fluid on a semi-infinite interval. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations to the transformed equations is developed and implemented. Effects of non-Newtonian parameters on the flow phenomena are analyzed and documented.