Nonclassical Method Research Articles

On the calibration of size parameters related to non-classical continuum theories using molecular dynamics simulations

The topic presented in this research is the calibration of small-scale parameters of non-classical continuum theories such as nonlocal strain gradient theory, strain gradient theory, stress-driven nonlocal elasticity, and strain-driven nonlocal elasticity. Governing equations of vibrational behavior of circular nanoplate and associated boundary conditions for each method derived using Hamilton's principle. Obtained governing differential equations from non-classical methods were solved using the general differential quadrature rule (GDQR). Then, the first natural frequencies for different radiuses and different size parameters were obtained. On the other hand, the first natural frequencies of circular nanoplates are calculated using molecular dynamics simulation based on AIREBO and Tersoff potentials for different radiuses. Fast Fourier transform (FFT) was utilized to calculate natural frequencies based on the molecular dynamics simulation. Using the accurate size parameter is an important point in the application of non-classical continuum theories. To obtain the size parameters related to different non-classical methods, the results of molecular dynamics compared to those of nan-classical methods and simulated annealing (SA) algorithm optimization technique was utilized. Results show that stress-driven nonlocal, strain-driven nonlocal, and strain gradient methods cannot predict the behavior predicted by molecular dynamics for all ranges of radius. In other words, the responses of these three methods for any value of the size parameters (in some interval radius) and results of the molecular dynamics method are not equal for a few numbers of studied radii. In contrast to these three methods, the nonlocal strain gradient method predicts the results obtained by molecular dynamics well for all radii. The results of this paper are very useful for researchers in the field of non-classical continuum mechanics.

Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

Some new exact solutions of (3+1)-dimensional Burgers system via Lie symmetry analysis

In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the (3+1)-Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the (3+1)-Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in the sciences and technology.

A lowest-order virtual element method for the Helmholtz transmission eigenvalue problem

In this paper, we introduce a $$C^{0}$$ virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth-order non-selfadjoint eigenvalue problem. We consider the mixed formulation of the eigenvalue problem discretized by the lowest-order virtual elements. This discrete scheme is based on a conforming $$H^{1}(\varOmega )\times H^{1}(\varOmega )$$ discrete formulation, which makes use of lower regular virtual element spaces. However, the discrete scheme is a non-classical mixed method due to the non-selfadjointness, then we cannot use the framework of classical eigenvalue problem directly. We employ the spectral theory of compact operator to prove the spectral approximation. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

May the Patient’s Will, Expressed by Means of Assistive Communication Technologies, be Admissible as Evidence in Court Proceedings?

The article is dedicated to hallmark the problem of accepting evidence of a patient’s will by a court in diverse proceedings, and it being communicated by non-classical means. Modern technologies allow patients to communicate utilizing various electrified appliances, in case the disabled person suffers from an ailment, or a health disorder affecting speech and mental abilities. Such appliances may truly enhance the patient’s quality of life; however, it is uncertain whether such patient may be found to be a competent witness, or the information reflecting their will obtained in a non-classical method may be found to be as convincing evidence by the court. Currently, there is very little judicial precedent dealing with obtaining evidence of the patient’s will by means of assistive communication technologies, though recent Italian legacy has shown such evidence may be accepted by the court, in case forensic-psychiatric examination approves adequacy of the cognitive abilities of the patient, rendering their will competent. Diverse legal systems render the question of patient’s competence differently, and the issue of accepting information as evidence obtained by means of assistive communication technologies will surely become more frequent in disputes relating to testament validity or determining the patient’s will to undergo or forego medical treatment. Such cases may be of high relevance in civil proceedings on withdrawal of life-supporting treatment, which will ultimately result in the patient’s demise. Keywords: patient’s autonomy, patient’s legal capacity, withdrawal of life-supporting treatment, assistive communication technologies, theory of evidence, critically-ill patients.

On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation

The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries.

Pseudospectral solutions of the Fokker-Planck equation for Pearson diffusion that yields a Kappa distribution; the associated SUSY Schrödinger equation

Numerous satellite measurements of the energy distributions of electrons and ions in the solar system and beyond can be fitted with the Kappa distribution. The origin of these Kappa distributions has attracted considerable attention. In particular, it has been used to support the Tsallis entropy formalism that purports to explain the origin of these distributions. In this paper, we compare the Kappa distribution of space plasmas modelled with a particular Fokker–Planck equation for a two component system with the linear Fokker–Planck equation that has been used to study the Student t-distribution. Pseudospectral solutions of the Fokker–Planck equation for the Student-t distribution based on the steady solution are not convergent owing to its asymptotic power law behaviour. We transform the Fokker–Planck equation to the isospectral Schrödinger equation which yields a Rosen-Morse II potential. The eigenfunctions and eigenvalues of this Schrödinger equation are obtained with a nonclassical pseudospectral method in comparison with a Fourier method and a method based on Hermite polynomials. The nonclassical pseudospectral method converges faster than these other basis sets and agrees with the known eigenvalues.

Robust Estimation methods of Generalized Exponential Distribution with Outliers

This paper discussed robust estimation for point estimation of the shape and scale parameters for generalized exponential (GE) distribution using a complete dataset in the presence of various percentages of outliers. In the case of outliers, it is known that classical methods such as maximum likelihood estimation (MLE), least square (LS) and maximum product spacing (MPS) in case of outliers cannot reach the best estimator. To confirm this fact, these classical methods were applied to the data of this study and compared with non-classical estimation methods. The non-classical (Robust) methods such as least absolute deviations (LAD), and M-estimation (using M. Huber (MH) weight and M. Bisquare (MB) weight) had been introduced to obtain the best estimation method for the parameters of the GE distribution. The comparison was done numerically by using the Monte Carlo simulation study. The two real datasets application confirmed that the M-estimation method is very much suitable for estimating the GE parameters. We concluded that the M-estimation method using Huber object function is a suitable estimation method in estimating the parameters of the GE distribution for a complete dataset in the presence of various percentages of outliers.

Structure and conductivity of polyaniline prepared by non-classical interfacial polymerization

Non-classical interfacial polymerization has been recently focused on due to the alternative process benefiting the control of material structure and properties. By applying two non-classical interfacial polymerization methods, e.g. one with asymmetrical bi-phase volumes and another with varied interface area size, several polyanilines, PANIs, were prepared and characterized. Results showed that the increase of the bottom oil phase volume can allow the dopant dispersion uniformly to benefit the PANI doping, and the reduction of the top aqueous phase volume can reduce the distance to benefit the PANI formed at the interface to fast diffuse into and stabilize in the upper phase. Results further showed that the reduction of the interface area is similarly to the increase of the bottom phase volume positively for PANI formation. The increase of the bottom phase volume or reduce the interface area both can enhance the crystallinity and conductivity for PANI.

Nonclassical symmetries and analytic solutions to Kawahara equation

The focus of this paper is on Kawahara equation (KE) which models diverse variety of waves enforced in many branches of applied sciences. Symmetries to the considered equation are successfully obtained by practicing nonclassical method. According to nonclassical symmetries, KE has been reduced to numerous ODEs. Further, these ODEs are treated such that new analytic solutions are determined by applying power series, ansätz and [Formula: see text]-expansion method.

Analysis of the assortment of ice wine in online stores in Ukraine

Unbalanced structure of the assortment leads to the decrease in the level of proft, loss of the competitive position of enterprises, including wine ones, in promising markets. In the conditions of the modern wine market, there is a tendency to increase consumer influence on the formation of the assortment range of wineries, focusing on wines with original consumer properties, in particular ice. Ice wine is a rare wine of a special type, the technology of which involves the use of non-classical methods of processing grapes. Therefore, ice wine is characterized by original consumer properties leading to increased interestamong consumers who increasingly use distance means of buying goods.In this regard, in this article the monitoring of the ice wine assortment was conducted, which is presented on the sites of Ukrainian online stores, both general retail and specialized wine networks.Coefcient estimation methods were used to evaluate the assortment of ice wines based on comparison with the benchmark system. According to the results of the study, the low coefcients of breadth and depth of ice wine assortment presented in the onlinestoresin Ukraine were obtained. It was determined that most ice wines are wines of foreign producers made from Riesling grapes. The Rozetka online store has the entire list of ice wines on sale, but the largest assortment of ice wines from around the world is not available for orders on the website of the specialized wine online store Wine Style. It should be noted that not a single position of ice wine was found in the online stores of Ukrainian grocery retail chains. Therefore, the subsequent formation of an assortment of ice wines is an urgent task, which will allow us to correctly propose effective methods to increase the competitiveness of ice wines on the Ukrainian retail market. It was determined that most ice wines are wines of foreign producers made from Riesling grapes. The Rozetka online store has the entire list of ice wines on sale, but the largest assortment of ice wines from around the world is not available for orders on the website of the specialized wine online store Wine Style. It should be noted that not a single position of ice wine was found in the online stores of Ukrainian grocery retail chains. Therefore, the subsequent formation of an assortment of ice wines is an urgent task, which will allow us to correctly propose effective methods to increase the competitiveness of ice wines on the Ukrainian retail market. Key words: assortment, market, online store, ice wine, depth ofassortment, breadth of assortment.

MATRIX-STRUCTURAL ANALYSIS OF THE STRESS-DEFORMED STATUS OF THE FLEXIBLE INTERFACES AND MEMS COMPONENTS

The designs of modern electronic technology are complex mechanical systems with many rigid bonds, and their microminiature in many cases has reached the physical limit of the dimensional, charge and energy quantization of components. For such mechanical systems, with non-classical methods for mechanics of fastening of separate structural elements, it is difficult to build a design model, simple enough and at the same time one that well reflects physical and dynamic properties. To ensure mechanical strength of interconnects and ultra-thin electronic components, it is necessary to determine the stress-strain state of structural elements. These reasons necessitate the use of numerical methods to calculate the dynamic parameters of the design of electronic equipment interconnections, which will improve the reliability of the developed tools that meet the requirements of regulatory technical documentation on mechanical characteristics, shorten the time and cost of their creation. Subject matter of this work is the study of beam deformation of constant cross section under the action of axial forces, bending moments and torque relative to the longitudinal axis. Goal this work is to determine the deformation of the elements of the structure of the devices using the stiffness matrix with a known vector of external forces. To achieve this goal, it is necessary to solve the following tasks: to consider the existing methods of description and analysis, which take into account the specificity of the structures and technological processes used to manufacture specific structures MEMS; explore the nature of these methods; calculate displacements and deformations using the finite element method; using the stiffness matrix to determine the deformation of the structural elements of the devices; to calculate the deformation of the beam of constant cross-section under the action of axial forces, bending moments and torque relative to the longitudinal axis. Conclusions: stiffness matrix, constructed using functional analysis, allows for the calculation of deformation of MEMS elements and electronic interconnect flexible.

Evolutionary and genetic algorithms for design of metadevices working on electric dipole resonance

All-dielectric nanophotonics is a rapidly growing field of modern science. Metasurfaces and other planar devices based on all-dielectric nanoparticles lead to manage the light propagation at the nanoscale. Impressive effects such as perfect absorption, invisibility, chirality effects, negative refraction, light focusing in the area with size smaller than wavelength, nano-lasing etc - can be achieved with all-dielectric technologies. While it is needed to use more and more complicated designs for solution of modern nanophotonics’ currents tasks, non-classical methods of optimization become relevant. For example, to design reconfigurable metalenses with an additional degree of freedom such as polarizability or temperature dependence, evolutionary or genetic algorithms show their high applicability. In this work, we show a new approach to design metalenses with evolutionary and genetic algorithms.

Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.

A Lowest-Order Mixed Finite Element Method for the Elastic Transmission Eigenvalue Problem

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, nonselfadjoint, and of fourth order. In this paper, we construct a lowest order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. This scheme is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the nonselfadjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed method which does not fit into the framework of classical theoretical analysis. In stead, we obtain the convergence analysis based on the spectral approximation theory of compact operators. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.

Rapidly Oscillating Systems on the Plane: Weak Resonances and Asymptotic Control

Rapidly Oscillating Systems (ROS) is a class of perturbed dynamical systems with a small parameter that contains oscillating part with the period proportional to the parameter. These include a lot of classical perturbed systems that found wide applications in celestial mechanics, electronics, mechatronics, nanotechnology, etc. However, many important systems of the class cannot be treated classically. In recent years these non-classical ROS found important applications in control of bipedal walk, rapidly varying media, frequency demodulation, Atomic Force Microscopy, etc. A characteristic specific of the non-classical ROS is that oscillations depend not only on time, but also on phase variables. This allows to manipulate the oscillation functions and to use them for asymptotic control. The purpose of this paper is to illustrate non-classical ROS methods and their properties on simple, but representative, 2D examples without technical details of real applications. In some cases these also provide a new technique to the classical systems.

ПРИМЕНЕНИЕ BIG DATA ПРИ АНАЛИЗЕ БОЛЬШИХ ДАННЫХ В КОМПЬЮТЕРНЫХ СЕТЯХ

The concept of Big Data includes the totality of all data sets, the total size of which is several times larger than the capabilities of conventional databases.it is also necessary to note the use of non-classical data processing methods. For example, in the management, analysis of information received, or simply storage. Big Data algorithms have emerged in parallel with the introduction of the first high-performance servers of their kind, such as the mainframe, which have sufficient resources required for operational information processing, as well as corresponding to computer calculations with subsequent analysis. The algorithms are based on performing series-parallel calculations, which significantly increases the speed of performing various tasks. Entrepreneurs and scientists are interested in Big Data, who are concerned with issues related to not only high-quality, but also up-to-date interpretation of data, as well as creating innovative tools for working with them. A huge amount of data is processed in order for the end user to get the results they need for their further effective use. Big Data enables companies to expand the number of their customers, attract new target audiences, and also helps them implement projects that will be in demand not only among current customers, but also attract new ones. Active implementation and subsequent use of Big Data correspond to the solution of these problems. In this paper, we compare the main types of databases and analyze intrusion detection using the example of distributed information system technologies for processing Big Data. Timely detection of intrusions into data processing systems is necessary to take measures to preserve the confidentiality and integrity of data, as well as to correctly correct errors and improve the protection of the data processing system.

Approximate nonclassical symmetries for the time-fractional KdV equations with the small parameter

In this paper, the Lie symmetry analysis is presented for the time-fractional KdV equation with the Riemann-Liouville derivative. We introduce a generalized approximate nonclassical method that is applied to differential equations with fractional order. In the sense of this symmetry, the vector fields of fractional KdV equation are obtained. The similarity reduction corresponding to the symmetries of the equation is constructed.

Probability distributions of one-day noise indicators in the process of the type A uncertainty evaluation of long-term noise indicators

The estimation of long-term noise indicators using a small sample involves type A measurement uncertainty which is dependent upon the size of the sample and its probability distribution. Ifthedistribution function of the data is different from normal, alternative methods of uncertainty assessment must be applied rather than the classical law of uncertainty propagation. This paper presents an analysis of the probability distributions of one-day noise indicators calculated from data provided by noise monitoring of the city of Gdańsk. The measurement data under analysis was comprehensive and diverse – the data included a four-year period and sixty-nine measurement stations; noise of different origins was recorded: traffic noise, railway noise, industrial noise and aircraft noise. The data set was analysed in terms of determining the forms of probability distributions. Only for 3% of the analysed one-day noise indicators, were the distribution functions confirmed to be normal. In this case, using the classical method (the law of uncertainty propagation) for determining type A measurement uncertainty may lead to the erroneous determination of the uncertainty interval. Following analysis of the data set, the possibility of modelling the distributions of one-day noise indicators with a mixture of two normal distributions was verified. Such an approach would significantly simplify uncertainty determination using the non-classical method based on probability distribution propagation. It was indicated that 94% of the analysed samples are characterised by adistribution that is a mixture of two normal distributions.

Investigation on Torsion Stiffness of Thick-Walled Composite Hollow Shaft

A non-classical mechanical analysis method was used to derive the equivalent torsional stiffness <GIp> of thick-walled CFRP drive shaft. The equivalent shear modulus calculated by classical lamination theory, finite element analysis method and non-classical mechanical analysis method were compared. Experimental work was carried out on six composite tubes made of same dimensions but different lay-ups. All the results show that the layers with ±45° fiber orientation angle located on the outside of the wall thickness can increase the torsion stiffness of the composite tube. The non-classical mechanical analysis method predicts the torsion stiffness of thick-walled composite tube more precisely than the theory classical lamination, and agree well with experimental results. This method provides a convenient, accurate and fast method for the torsion stiffness.