New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

A New Second-Derivative Spectrophotometric Method for the Determination of Vildagliptin in Pharmaceutical dosage form

An accurate derivative spectrophotometric method was developed and validated for the determination of dipeptidyl peptidase inhibitor vildagliptin in the pharmaceutical dosage form. The second derivative of the UV spectra has enabled the estimation of vildagliptin absorbance at 217 nm without any interferences. Linearity, precision, accuracy, detection (LOD), and quantification (LOQ) limits were established for method validation. Calibration curve was linear in the range of 10-60 µg/mL with a regression coefficient of 0.998. The method was validated as per the International Conference on Harmonization (ICH Q2 (R1)). The limit of detection and the limit of quantification were found to be 2.06 µg/mL and 6.25 µg/mL, respectively. Intra and interday precision data illustrated that the method has acceptable reproducibility as the percentage relative standard deviation (RSD) was less than 2 %, which indicates the precision of the method. The recovery was 98.39 % by the standard addition method. The percentage assay of vildagliptin was 98.06 % which showed good applicability. The following results indicate that the procedure is accurate, precise, and reproducible while being simple and less time-consuming. The method was demonstrated to be adequate for routine analysis in quality control.

Deprived of Second Derivative Iterated Method for Solving Nonlinear Equations

Non-linear equations are one of the most important and useful problems, which arises in a varied collection of practical applications in engineering and applied sciences. For this purpose, in this paper has been developed an iterative method with deprived of second derivative for the solution of non-linear problems. The developed deprived of second derivative iterative method is convergent quadratically, and which is derived from Newton Raphson Method and Taylor series. The numerical results of the developed method are compared with the Newton Raphson Method and Modified Newton Raphson Method. From graphical representation and numerical results, it has been observed that the deprived of second derivative iterative method is more appropriate and suitable as accuracy and iteration perception by the valuation of Newton Raphson Method and Modified Newton Raphson Method for estimating a non-linear problem.

EVALUATION OF THE VESSEL AGE BY APPLYING THE ACCELERATION PLETHYSMOGRAM ANALYSIS

В работе приведены понятия жесткости, эластичности и тонуса сосудов, а также же их взаимосвязь с общим состоянием сосудистой стенки. Описан индекс, объединяющий влияние вышеперечисленных факторов на состояние сосудистой системы и дающий представление о возрасте сосудов пациента, а также показана связь этого индекса с возрастом человека. Представлен обзор способов определения возраста сосудов с помощью контурного анализа пульсовой волны. Среди предложенных способов был выделен подход на основе контурного анализа сигнала пульсовой волны, а также ее второй производной. В данном исследовании проводилась разработка алгоритма расчета показателя возраста сосудов (VA), базирующаяся на анализе сигнала и его второй производной. При этом особое внимание уделялось физической интерпретации параметров, входящих в состав расчетной формулы. С помощью представленного алгоритма в группе из трех испытуемых был определен сосудистый возраст. Из анализа полученных результатов было выявлено влияние физиологических факторов на значение возраста сосудов. Предложены методики, позволяющие исключить влияние этих факторов на значения показателя VA и тем самым получить более точные результаты. Также представлены стратегии дальнейшего развития исследований в этом направлении In The paper presents the concepts of rigidity, elasticity and tone of blood vessels, as well as their relationship with the general state of the vascular wall. An index is described that combines the influence of the above factors on the state of the vascular system and gives an idea of the age of the patient's vessels, and also shows the relationship of this index with the age of a person. An overview of the methods for determining the age of blood vessels using the contour analysis of the pulse wave is presented. Among the proposed methods, an approach based on the contour analysis of the pulse wave signal, as well as its second derivative, was singled out. In this study, an algorithm was developed for calculating the indicator of vascular age (VA), based on the analysis of the signal and its second derivative. In this case, special attention was paid to the physical interpretation of the parameters included in the calculation formula. Using the presented algorithm, vascular age was determined in a group of three subjects. From the analysis of the results obtained, the influence of physiological factors on the value of the age of the vessels was revealed. Methods are proposed that allow to exclude the influence of these factors on the values of the VA indicator and thereby obtain more accurate results. Also presented are strategies for the further development of research in this direction

Rolling Bearing Fault Vibration Signal Denoising Based on Adaptive Morphological Wavelet Perona–Malik Filter Algorithm

This paper proposes an adaptive Perona–Malik filtering algorithm based on the morphological Haar wavelet, which is used for vibration signal denoising in rolling bearing fault diagnosis with strong noise. First, the morphological Haar wavelet operator is utilized to presmooth the noisy signal, and the gradient of the presmooth signal is estimated. Second, considering the uncertainty of gradient at the strong noise point, a strong noise point recognition operator is constructed to adaptively identify the strong noise point. Third, the two-step gradient average value of the strong noise point in the same direction is used to substitute, and the second derivative is introduced into the diffusion coefficient. Finally, diffusion filtering is performed based on the improved Perona–Malik model. The simulation experiment result indicated that not only the algorithm can denoise effectively, but also the average gradient and second derivative in the same direction can effectively suppress the back diffusion of strong noise points to improve the denoising signal-to-noise ratio. The experimental results of rolling bearing show that the algorithm can adaptively filter out strong noise points and keep the information of peak in the signal well, which can improve the accuracy of rolling bearing fault diagnosis.