Numerical analysis of a high order method for state-dependent delay integral equations

In this paper we study the piecewise collocation method for a class of functional integral equations with state-dependent delays that is, where the delays depend on the solution. It is well known that these equations typically have discontinuity in the solution or its derivatives at the initial point of integration domain. This discontinuity propagated along the integration interval giving rise to subsequent points, called ”singular points”, which can not be determined priori and the solution derivatives in these points are smoothed out along the interval. Most of the known numerical methods for this type of equations are generally very sensitive to the singular points and therefore must have a process that detects these points and insert them into the mesh to guarantee the required accuracy. Here, we present a numerical algorithm based on the piecewise collocation method and an approach for tracking the singular points relying on the recent strategy for implicit delay differential equations which has been proposed by Guglielmi and Hairer in 2008. The convergence analysis of the method is investigated and some numerical experiments are presented for clarifying the robustness of the method.     

Spectroscopic,Thermal, and Structural Studies of Two New Co‐Crystals of [4,4′‐Bithiazole]‐2,2′‐diamine

Two new 2 : 1 co‐crystals based on [4,4′‐bithiazole]‐2,2′‐diamine (=2,2′‐diamino‐4,4′‐bithiazole (DABTZ)) with 2,2′‐bipyridine (bipy) and benzo‐18‐crown‐6 (bk) were synthesized by slow‐evaporation method in MeOH. These co‐crystals were characterized by means of elemental analysis, and IR, and ¹H‐ and ¹³C‐NMR spectroscopy. Also, thermal analyses under air atmosphere and X‐ray crystallography have been performed on these structures. X‐Ray single‐crystal analyses revealed that these networks contain large vacant voids. These structures, [(DABTZ)₂(bipy)] and [(DABTZ)₂(bk)(MeOH)], crystallized in monoclinic and triclinic forms with space groups of P2₁/c and P , respectively. The self‐assembly of these compounds in the solid state is likely caused by both H‐bonding and π? π stacking.     

Electrochemical methods for simultaneous determination of trace amounts of dopamine and uric acid using a carbon paste electrode incorporated with multi-wall carbon nanotubes and modified with α-cyclodextrine

In this work, we investigate the electrochemical activity of dopamine (DA) and uric acid (UA) using both a bare and a modified carbon paste electrode as the working electrode, with a platinum wire as the counter electrode and a silver/silver chloride (Ag/AgCl) as the reference electrode. The modified carbon paste electrode consists of multi-walled carbon nanotubes (>95%) treated with α-cyclodextrine, resulting in an electrode that exhibits a significant catalytic effect toward the electro-chemical oxidation of DA in a 0.2-M Britton–Robinson buffer solution (pH 5.0). The peak current increases linearly with the DA concentration within the molar concentration ranges of 2.0 × 10⁻⁶ to 5.0 × 10⁻⁵ M and 5.0 × 10⁻⁵ to 1.9 × 10⁻⁴ M. The detection limit (signal to noise >3) for DA was found to be 1.34 × 10⁻⁷ M, respectively. In this work, voltammetric methods such as cyclic voltammetry, chronoamperometry, chronocuolometry, differential pulse and square wave voltammetry, and linear sweep and hydrodynamic voltammetry were used. Cyclic voltammetry was used to investigate the redox properties of the modified electrode at various scan rates. The diffusion coefficient (D, cm² s⁻¹ = 3.05 × 10⁻⁵) and the kinetic parameters such as the electron transfer coefficient (α = 0.51) and the rate constant (k, cm³ mol⁻¹ s⁻¹ = 1.8 × 10³) for DA were determined using electrochemical approaches. By using differential pulse voltammetry for simultaneous measurements, we obtained two peaks for DA and UA in the same solution, with the peak separation approximately 136 mV. The average recovery was measured at 102.45% for DA injection.