Multistep Methods of the Hybrid Type and Their Application to Solve the Second Kind Volterra Integral Equation

There are some classes of methods for solving integral equations of the variable boundaries. It is known that each method has its own advantages and disadvantages. By taking into account the disadvantages of known methods, here was constructed a new method free from them. For this, we have used multistep methods of advanced and hybrid types for the construction methods, with the best properties of the intersection of them. We also show some connection of the methods constructed here with the methods which are using solving of the initial-value problem for ODEs of the first order. Some of the constructed methods have been applied to solve model problems. A formula is proposed to determine the maximal values of the order of accuracy for the stable and unstable methods, constructed here. Note that to construct the new methods, here we propose to use the system of algebraic equations which allows us to construct methods with the best properties by using the minimal volume of the computational works at each step. For the construction of more exact methods, here we have proposed to use the multistep second derivative method, which has comparisons with the known methods. We have constructed some formulas to determine the maximal order of accuracy, and also determined the necessary and sufficient conditions for the convergence of the methods constructed here. One can proved by multistep methods, which are usually applied to solve the initial-value problem for ODE, demonstrating the applications of these methods to solve Volterra integro-differential equations. For the illustration of the results, we have constructed some concrete methods, and one of them has been applied to solve a model equation.

VALIDATED SPECTROPHOTOMETRIC METHODS FOR THE ESTIMATION OF CINNARIZINE IN BINARY MIXTURE WITH PARACETAMOL IN BULK AND TABLETS

Objectives: The aim of this work was to develop and validate new, simple, accurate, and selective spectrophotometric methods (derivative and derivative ratio spectrophotometric methods) for the determination of these drugs. These methods can be used as analytical tools in routine examination in quality control laboratories. Methods: The first method was derivative method in which the first derivative method (1D) for determination of PCM and the second derivative method (2D) for determination of CIN. The second method was the first derivative ratio spectrophotometric method (1DD) for determination of CIN and PCM. Results: In first method, the first derivative spectrum (1D) of PCM where PCM was determined by measuring the amplitude of the valley at 235 nm while CIN showed zero crossing spectrum, and the second derivative spectrum (2D) of CIN where CIN was determined by measuring the amplitude of the peak at 287.5 nm while PCM showed a zero value. In the second method, the first derivative ratio spectrophotometry (1DD) for CIN and PCM determination, where the amplitude at 290 and 291 nm, was selected for the determination of CIN and PCM, respectively. Conclusions: The developed methods were applied for the determination of the cited drugs in tablets containing binary drug mixture. The methods are simple and precise and can be used for routine analysis of the labeled drugs in combined dosage forms in quality control laboratories.

DERIVATIVE SPECTROPHOTOMETRY FOR NO3- DETERMINATION IN SANITARY SEWAGES

The objective of this study was to observe the feasibility of using the UV spectrophotometric method of the second derivative in the determination of nitrogen in the form of nitrate in sanitary sewers, since this method is indicated for water with low content of organic matter. Therefore, the results obtained in that method were compared with the salicylate methodology. The Mann-Whitney test showed a significant difference (? = 5%) between the results obtained in both methods. However, when compared, the difference between them was less than 0.25 mg L-1 in half of the observations. Despite this significant difference, the use of the second derivative method for sanitary sewage samples showed satisfactory performance.

DETERMINATION OF THE SURFACE FUNCTIONALITY OF NANOCARBON ALLOTROPES BY BOEHM TITRATION

Critical point (CP) in a titration curve to quantify carboxylic, lactonic, and phenolic groups on multi-walled carbon nanotube (CNT) and Vulcan carbon (VC). Titration technique used was Boehm type and tipping point measurement from titration curve was done by second derivative method. The calculation functional groups on nanocarbons (NCs) presents lowest uncertainty previously published. An order of [Formula: see text]meq/g is the precision of the values obtained, one order of magnitude below reported. Both NCs display a high quantity of carboxylic groups created after being treated with regard to the other functional groups generated, mainly for functionalized Vulcan carbon VC-F.

Quantitative Measurement of Functional Groups on Nanocarbon Allotropes Surface by Boehm Titration

Various nanocarbons (NCs) were used to study their surface groups under standardized Bohem titration, including: multiwalled carbon nanotube (CNT), graphene (G), Vulcan carbon (VC), and nanodiamond (ND). Endpoint-measured titration using second derivative method to quantify carboxylic, lactonic, and phenolic groups created on treated carbon surfaces shows a high precision comparable to other recent reports and with errors of 1 order of magnitude lower. The results exhibit major concentration of carboxyl group increased after the NCs were oxidized compared to the amount of other functional groups like phenols and lactonic groups. It is important highlight, the concentration ratio of carboxyl group with VC:VC-O was showed at 1:77, exhibited a major result regarding other NCs which exhibited ratios of 1:4.5, 1:1.4, and 1:2.5 for ND:ND-O, CNT:CNT-O, and G:G-O, respectively. It is concluded that VC is a NC that competes and excels in its capacity of oxidation with respect to the popular NCs as CNT, graphene (G), and ND.

Different Spectrophotometric and Chromatographic Methods for Determination of Mepivacaine and Its Toxic Impurity

Stability-indicating spectrophotometric, TLC-densitometric, and ultra-performance LC (UPLC) methods were developed for the determination of mepivacaine HCl (MEP) in the presence of its toxic impurity, 2,6-dimethylanaline (DMA). Different spectrophotometric methods were developed for the determination of MEP and DMA. In a dual-wavelength method combined with direct spectrophotometric measurement, the absorbancedifference between 221.4 and 240 nm was used for MEPmeasurements, whereas the absorbance at 283 nm was used for measuring DMA in the binary mixture. In the second-derivative method, amplitudes at 272.2 and 232.6 nm were recorded and used for the determination of MEP and DMA, respectively. The developed TLC-densitometric method depended on chromatographic separation using silica gel 60 F254 TLC plates as a stationary phase and methanol–water–acetic acid (9 + 1 + 0.1, v/v/v) as a developing system, with UV scanning at 230 nm. The developed UPLC method depended on separation using a C18 column (250 × 4.6 mm id, 5 μm particle size) as a stationary phase and acetonitrile–water (40 + 60, v/v; pH 4 with phosphoric acid) as a mobilephase at a flow rate of 0.4 mL/min, with UV detection at 215 nm. The chromatographic run time was approximately 1 min. The proposed methods were validated with respect to International Conference on Harmonization guidelines regarding precision, accuracy, ruggedness, robustness, and specificity.