Mathematical Physics Research Articles

MODELING OF MAXIMALLY PARALLEL STRUCTURES OF ALGORITHMS FOR SOLVING THERMAL PROBLEMS

The paper demonstrates the possibility of creating a maximum parallel form of computational algorithms to solve thermal problems and their mapping to the architecture of multiprocessor systems based on solving thermal problems of mathematical physics. It is shown that an effective tool for studying heat and mass transfer problems in metallurgical production could be parallel computing technologies on distributed cluster systems with a relatively low cost and reasonably easily scalable both in the number of processors and in the amount of RAM. Tridiagonal structure systems' parallelization was implemented by a numerical-analytical approach, which predetermined their maximally parallel algorithmic form. That approach is facilitated by the minimum possible implementation time of the developed algorithm on parallel computing systems. Furthermore, during the arithmetic expressions parallel computations, the developed algorithm separates the error in the output data from rounding operations. Thus, the parallelization of tridiagonal systems based on numerical-analytical discretization methods does not impose any restrictions on the topology of the mesh nodes of the computational domain.Furthermore, as applied to the parallel computation of arithmetic expressions, it separates the initial data error from a real PC's rounding operations. That approach eliminates the recurrent structure of computing the sought-for decision vectors, which, as a rule, leads to the round-off errors accumulation. Such a parallel form of the constructed algorithm is maximal and has the shortest possible implementation time of the algorithm on parallel computing systems. The developed approach to parallelizing the mathematical model is stable for various types of input data. It has the most parallel form and is distinguished by the minimum time for solving the problem as applied to multiprocessor computing systems. That is explained as follows. If it is hypothesized that one processor can be assigned to one processor and one processor can be assigned to one node of the computational mesh domain, the computations can be processed in parallel and simultaneously for all nodes of the computational mesh domain. The simulation process was implemented on a PC cluster. It follows from the simulation results analysis that the developed method for solving the heat conduction problem effectively minimizes residuals.

Varying-Coefficient Stochastic Differential Equations with Applications in Ecology

Stochastic differential equations (SDEs) are popular tools to analyse time series data in many areas, such as mathematical finance, physics, and biology. They provide a mechanistic description of the phenomenon of interest, and their parameters often have a clear interpretation. These advantages come at the cost of requiring a relatively simple model specification. We propose a flexible model for SDEs with time-varying dynamics where the parameters of the process are nonparametric functions of covariates, similar to generalized additive models. Combining the SDE and nonparametric approaches allows for the SDE to capture more detailed, non-stationary, features of the data-generating process. We present a computationally efficient method of approximate inference, where the SDE parameters can vary according to fixed covariate effects, random effects, or basis-penalty smoothing splines. We demonstrate the versatility and utility of this approach with three applications in ecology, where there is often a modelling trade-off between interpretability and flexibility.Supplementary materials accompanying this paper appear online.

Generalized lump solutions, classical lump solutions and rogue waves of the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like equation

Under investigation in this paper is the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada-like (CDGKS-like) equation. Based on bilinear neural network method, the generalized lump solution, classical lump solution and the novel analytical solution are constructed by giving some specific activation functions in the single hidden layer neural network model and the “3-2-2” neural network model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained with the help of Maple software. These results fill the blank of the CDGKS-like equation in the existing literature. Via various three-dimensional plots, curve plots, density plots and contour plots, dynamical characteristics of these waves are exhibited. The effective methods used in this paper is helpful to study the nonlinear evolution equations in plasmas, mathematical physics, electromagnetism and fluid dynamics.

The dually flat structure for singular models

The dually flat structure introduced by Amari-Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, and such an excellent geometric structure is rarely defined on the entire space. To fix this trouble, in the present paper, we propose a novel generalization of the dually flat structure for a certain class of singular models from the viewpoint of Lagrange and Legendre singularity theory - we introduce a quasi-Hessian manifold endowed with a possibly degenerate metric and a particular symmetric cubic tensor, which exceeds the concept of statistical manifolds and is adapted to the theory of (weak) contrast functions. In particular, we establish Amari-Nagaoka's extended Pythagorean theorem and projection theorem in this general setup, and consequently, most of applications of these theorems are suitably justified even for such singular cases. This work is motivated by various interests with different backgrounds from Frobenius structure in mathematical physics to Deep Learning in data science.

ПРО КРАЙОВІ ЗАДАЧІ ДЛЯ РІВНЯННЯ ПУАССОНА У БАГАТОЛИСТІЙ ОБЛАСТІ, СКЛАДЕНІЙ ІЗ РІЗНИХ КРУГОВИХ СЕГМЕНТІВ

The non-classical boundary problem of the mathematical physics for the two-dimensional Poisson equation is considered. As the area, in which the solution is sought, the area, made up of different circular segments, folded into a multi-sheet plate of a book structure, is taken. All sheets are different from each other, both in their physical properties and in geometric dimensions, and are interconnected by a chord common to all sheets. The problem statement is given and its exact solution is obtained.The solution to the problem is considered in bipolar coordinate systems, each of which is associated with one of the segments. In this case, all coordinate systems have a common parameter - the length of the rectilinear segment boundary. As a method for solving the problem, the classical method of separation of variables is used – the Fourier method. Although the Dirichlet problem is considered as a basic one, however, the proposed method can be applied in the case when conditions of other types are given on the arcs of separate circles: Neumann or the third main problem.The statement of the considered problem differs from the classical one in that the conjugation conditions of fields on the line of connection of segments are added to the traditional boundary conditions. These conditions represent the equality of the values of the functions and the equality to zero of the sum of linear combinations of their normal derivatives. The solution is constructed (selected) in such a way that the first of the field conjugation conditions is fulfilled automatically for any choice of unknown functions. The boundary conditions on the segments and the second conjugation condition make it possible to determine all the unknown functions of the problem. To apply the Fourier method, it is necessary that all boundary functions are equal to zero at the corner points of the segments. If this condition is violated, a modification of the method that allows one to obtain an exact solution in this case is proposed. As an application, such problems are considered: a) on the torsion of a composite rod, the cross-section of which is two different segments; b) the stationary heat conductivity problem for two glued half-segment with sources of heat inside the area. Exact analytical solutions to these new problems have been obtained.

Why All the Shapes on the Universe Tend To Be Circular and Spherical – To Dimensional Space (This Universe)

This paper is based on the concept of mathematical axiomatic physics. This means that through the structure of the things critical conclusions could be extracted about the spherical and the cyclic shapes in the universe. The concept has as mainstream the theoretical background of the theory “Count Theory – The Role of Zero in the Existence of Numbers and Things”, about the role of one and zero to the structure of the things and the paper “Time travel - According to Mathematical Axiomatic Physics”, which includes also the role one and zero, for the function of dimensions to time travel and teleportation.

On the Hilbert boundary-value problem for Beltrami equations with singularities

We investigate the Hilbert boundary-value problem for Beltrami equations \( \overline{\partial}f \) = μ∂f with singularities in generalized quasidisks D whose Jordan boundary ∂D consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves.Generally speaking, generalized quasidisks do not satisfy the standard (A)–condition in PDE by Ladyzhenskaya–Ural’tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring–Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations.In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain D of a general nature. As usual, a point in \( \overline{D} \) is called a singularity of the Beltrami equation, if the dilatation quotient Kμ := (1 + |μ|)/(1 – |μ|) is not essentially bounded in all its neighborhoods.Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincaré boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

Applications of noncommutative geometry in function theory and mathematical physics

Applications of noncommutative geometry in function theory and mathematical physics

Abundant exact solutions to a generalized nonlinear Schrödinger equation with local fractional derivative

The paper aims to employ a new effective methodology to build exact fractional solutions to the generalized nonlinear Schrödinger equation with a local fractional operator defined on Cantor sets. The equation contains group velocity dispersion and second‐order spatiotemporal dispersion coefficients. We obtain exact solutions of the equation via the generalized version of the exponential rational function method. This new version of the method uses a set of elementary functions that are adopted on the contour set. To study the dynamic behavior of the obtained results, extensive numerical simulations are provided. We observe that the employed method is simple but quite efficient for determining the exact solutions of the problem in the local sense. Moreover, they are practically compatible with solving various classes of nonlinear problems arising in mathematical physics. All computations are carried out using the Maple package.

ДОСЛІДЖЕННЯ ТЕМПЕРАТУРНИХ РЕЖИМІВ В ПРИСТРОЯХ ЕЛЕКТРОПОБУТОВОЇ ТЕХНІКИ

The aim of the work is to improve the work of hair styling devices, which is aimed at increasing the efficiency of hair styling under the action of sufficiently high temperatures with the least damage to it. To achieve this goal, the following tasks were solved: the relationship between the physical properties of hair and temperature regimes of curling was analyzed; experimentally investigated the distribution of curling temperature during heating and cooling; the distribution of the temperature of the curling iron when heated with dry hair and with moist hair was experimentally studied. Methodology. The methods of mathematical modeling, methods of mathematical physics, methods of experimental researches with processing of results of experiment by means of a modern software product and computer equipment are used in the work. Findings. Analytical and experimental studies have shown the need for a uniform temperature field on the surface of the work surface to achieve the desired result. The necessity of temperature regulation in devices for work with hair is proven. Originality. The possibility of using the installation for experimental evaluation of temperature control in devices for working with hair is substantiated, which will significantly increase the time of continuous operation of the device and reduce electricity consumption. Practical value. A stand for research and analysis of different temperature regimes of electrical appliances has been developed. The thermostat circuit for temperature control in devices for work with hair is applied. It has been proven that you need to choose the temperature in relation to the condition and type of hair, so as not to cause great harm. Due to its small size, economic and technical indicators, this stand is easy to use when studying the temperature conditions in devices for working with hair in the work space, which contributes to more modern and demonstrative training. It is proposed to use the two-position law of temperature regulation for the surfaces of the working bodies.

Logarithmic potential and generalized analytic functions

The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Neural-network-based multistate solver for a static Schrödinger equation

Solving a multivariable static Schr\"odinger equation for a quantum system, to produce multiple excited-state energy eigenvalues and wave functions, is one of the basic tasks in mathematical and computational physics. Here we propose a neural-network-based solver, which enables us to cover the high-dimensional variable space for this purpose. The efficiency of the solver is analyzed by examples aimed at demonstrating the concept and various aspects of the task: the simultaneous finding of multiple excited states of lowest energies, the computation of energy-degenerate states with orthogonalized wave functions, the scalability to handle a multivariable problem, and the self-consistent determination and automatic adjustment of the imbedded Monte Carlo procedure. The solver adheres to the computational techniques developed in machine learning and is vastly different from traditional numerical methods.

GLOBAL WARMING – PHYSICS AND GEOPOLITICS (Review). 1. Anthropogenic and natural concepts of climate change

Introduction. Global warming and so-called the “greenhouse effect” is one of the most discussed problems of physics and geopolitics, which has caused protest environmental movements in the world.
 Problem Statement. Widely discussed are both anthropogenic (the emission of greenhouse gases into the atmosphere) and the natural concepts of global warming with the dramatic effects of climate change on the planet and individual regions.
 Purpose. Evaluate the state of the problem, to present the results of research and measures to possible reduction of the greenhouse effect.
 Materials and methods. Authoritative literature sources with the analysis of anthropogenic and natural factors of global warming are used, including with the explanation of the physical processes which are shown at solar-geomagnetic activity and other natural phenomena.
 Results. The main causes and mechanisms of natural and man-made nature for the actual growth of greenhouse gas emissions and global temperature in the 20th century are identified and their estimates for the period up to 2100 are given.
 Conclusions. Most likely, the effect of global warming on the planet's climate is totally due to natural causes of cosmogenic origin and somewhat exacerbated by the effects of industrial development of mankind. For the final solution of this problem it is necessary to fully and adequately conduct more in-depth complex experimental and theoretical-model studies taking into account various factors of geological, climatic, meteorological, thermophysical, astrophysical nature and application of modern methods of mathematical physics, statistics and probability theory. reasonable measures to adapt the economy and life to climate change.

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.

Situation analysis of mathematical physics learning with online learning during the COVID-19 pandemic

This research was conduct to determine the situation of learning mathematics physics online during the COVID-19 pandemic. Participants in this study were physics semester 4th students of Universitas Muhammadiyah Makassar with 21 students. Data collection techniques include online interviews and questionnaires. The collected data analyzed by reduction, display, and conclusion drawing. The results of the study describe the learning process of mathematical physics not conducive during the COVID-19 pandemic outbreak. The problem that arises in the form of difficulty lectures in teaching explicit physics material that has many mathematical formulas and calculations. The difficulty on the part of students is that students experience limited costs to buy internet quota in online learning. The lecturer provides the solution by creating a mathematical physics module that is given in stages. This module is presented with language and examples of questions that are easily understood by students through the google classroom application. Lectures also occasionally use the zoom application, and Google meets to explain the material that is difficult to understand.

Genişletilerek Düzenlenmiş Kudryashov Yöntemi ile İki Fiziksel Modelinin Hareketli Dalga Çözümleri

In this paper, we propose the extended modified Kudryashov method (EMKM) for solving the Biswas-Milovic equation and Gerdjikov-Ivanov equation which are commonly special cases of Schrödinger equation in mathematical physics. We received many new extended traveling wave solutions when the special values of the parameters are taken for these equations which are pointed out by rational function, exponential function and hyperbolic function forms. The results show that EMKM is advantageous mathematical technique for solving nonlinear partial differential equations.

Non-asymptotic behavior of the spectrum of the sinc-kernel operator and related applications

Prolate spheroidal wave functions have recently attracted much attention in applied harmonic analysis, signal processing, and mathematical physics. They are eigenvectors of the sinc-kernel operator Qc: the time- and band-limiting operator. The corresponding eigenvalues play a key role, and the aim of this paper is to obtain precise non-asymptotic estimates with explicit constants related to the spectrum of Qc. This issue is rarely studied in the literature, while the asymptotic behavior of the spectrum of Qc has been well established from the 1960s. However, many recent applications require such non-asymptotic behavior. As applications of our non-asymptotic estimates, we first provide estimates for the constants appearing in the Remez- and Turàn–Nazarov-type concentration inequalities. Then, we give a non-asymptotic upper bound for the gap probability of the sinc determinantal point process. Consequently, one gets a non-asymptotic estimate for the hole probability, associated with bulk scaled asymptotics of a random matrix from the Gaussian unitary ensemble. This last result can be considered as a complement of the various and more involved asymptotic counterparts of this estimate.

Energy conservation for weak solutions of a surface growth model

We are concerned with one-dimensional scalar surface growth model arising from the physical process of molecular epitaxy. The mathematical theory of the surface growth model is known to share a number of striking similarities with the Navier-Stokes equations, including the results regarding existence and uniqueness of solutions. In this paper, we shall investigate an important subject in mathematical physics: the energy conservation for weak solutions of the surface growth model. As an analogue of the Navier-Stokes equations, we find some sufficient integral conditions that guarantee the validity of energy equality. As far as we know, this is the first result in this aspect.

Bright-dark and multi wave novel solitons structures of Kaup-Newell Schrödinger equations and their applications

In this paper, the soliton solutions which indicate long wave parallel to the magnetic fields of Kaup-Newell models are argued via described method. Modified extended Simple equation is suggested to explore the novel solitons and other wave structures of two different types of Kaup-Nawell equations, which have never been constructed before. As a consequence, bright-dark solitons, singular solitons, multi-wave solitons, breather type wave of strange structures and other waves solutions of two Kaup-Newell (K-N) Schrödinger equations are achieved in different form. The obtained novel solitons and other exact wave solutions have key applications in engineering and applied physics. Novel wave structures of solitons are explained graphically by providing suitable values to parameters that help for understanding the physical phenomena of these models. The constructed solutions are evaluated with available results in the literature. This technique can be productively utilized to more equations that occur in mathematical physics.

A Move Recent Review of The Integral Equations and Their Applications

In many branches of pure analysis, Integral Equations are one of the most useful techniques, such as functional analysis theories and stochastic processes. It is one of the most significant branches of mathematical analysis, in many fields of mechanics and mathematical physics,. In this research, we will address the integral equations in many physical issues and their applications. They are also associated with mechanical vibration problems, analytic function theory, orthogonal systems, quadratic form theory of infinitely many variables.