Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering

Wavelet method is a recently developed tool in applied mathematics. Investigation of various wavelet methods, for its capability of analyzing various dynamic phenomena through waves gained more and more attention in engineering research. Starting from ‘offering good solution to differential equations’ to capturing the nonlinearity in the data distribution, wavelets are used as appropriate tools at various places to provide good mathematical model for scientific phenomena, which are usually modeled through linear or nonlinear differential equations. Review shows that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations. This review intends to provide the great utility of wavelets to science and engineering problems which owes its origin to 1919. Also, future scope and directions involved in developing wavelet algorithm for solving reaction–diffusion equations are addressed.     

Regression and ANN models for estimating minimum value of machining performance

Surface roughness is one of the most common performance measurements in machining process and an effective parameter in representing the quality of machined surface. The minimization of the machining performance measurement such as surface roughness (R_a) must be formulated in the standard mathematical model. To predict the minimum R_a value, the process of modeling is taken in this study. The developed model deals with real experimental data of the R_a in the end milling machining process. Two modeling approaches, regression and Artificial Neural Network (ANN), are applied to predict the minimum R_a value. The results show that regression and ANN models have reduced the minimum R_a value of real experimental data by about 1.57% and 1.05%, respectively.     

A mathematical model for the 3d location estimation of 2D echocardiography data

The recovery of 3D left ventricle(LV) shape using 2D echocardiography is very attractable topic in the field of ultrasound imaging. In this paper, we propose a mathematical model to determine the 3D position of LV contours extracted from multiple 2D echocardiography images. We formulate the proposed model as a non-convex constrained minimization problem. To solve it, we propose a proximal alternating minimization algorithm with a solver OPTI for quadratically constrained quadratic program. For validating the proposed model, numerical experiments are performed with real ultrasound data. The experimental results show that the proposed model is promising and available for real echocardiography data.     

Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.     

关于极限理论的学习经验

分析极限理论在数学分析学习中的重要性，介绍学习极限理论的三点经验与体会，即充分理解基本内容；加强基本功训练；试着多问自己一些问题，学会举出各种各样的例子。     

The external demagnetizing factor and the static characteristic loop

In this paper a proposal is put forward for the calculation of the shearing factor N_S linked to the well known, demagnetization coefficient N_D, from the data of a sheared saturation loop, obtained by a non-toroidal, open circuit hysteresis measurement. Following that, the paper illustrates the way to recover the real static loop data from an open circuit, measured data. The proposal is based on the hyperbolic model. The proposed method was illustrated on the tests carried out on two closed and open toroidal samples, mimicking the demagnetization effect of the open circuit VSM measurement, made of NO Fe–Si electrical steel sheets. These test results, presented here, agree very well with the calculated results, based on the hyperbolic modeling.     

Application of supercritical fluid extraction coupled with counter–current chromatography for extraction and online isolation of unstable chemical components from Rosa damascena

Supercritical fluid extraction (SFE) coupled with high‐speed counter‐current chromatography (HSCCC) was successfully used for the extraction and online isolation of the unstable compounds from Rosa damascene in a single extraction and separation operation in two stages. The solvent systems of SFE/HSCCC were optimized with the help of multiexponential function model. At the first stage, the upper phase of the solvent system of n‐butanol–tert‐butyl methyl ether–acetonitrile–0.1% aqueous TFA (1.7:1.0:0.8:4.0, v/v/v/v) was used as both the SFE entrainer and the HSCCC stationary phase, and the target compounds were eluted with the corresponding lower phase to separate the hydrophobic compounds. At the second stage, the upper phase of the solvent system of n‐hexane–ethyl acetate–methanol–water (3.2:1.0:2.8:2.6, v/v/v/v) was used as both the SFE entrainer and the HSCCC stationary phase, followed by elution with the corresponding lower phase to separate the moderate hydrophobic compounds. Six compounds including formononetin, delphinidin, cyaniding, 5,6,4′‐trihydroxy‐7,8‐dimethoxy flavone, 5,3′‐dihydroxy‐7,8‐dimethoxy flavone, and 5‐hydroxy‐6,7,8,3′,4′‐pentamethoxy flavone were successfully separated in one extraction–separation operation within 300 min. The targeted compounds were identified by MS and NMR spectroscopy. This research has opened up great prospects for industrial application of SFE/HSCCC to the extraction and separation of unstable compounds.