Problems Of Mathematical Physics Research Articles

New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied in many different problems of mathematical physics and engineering. In this note, a new development of homotopy analysis method (ND-HAM) is demonstrated for non-linear integro-differential equation (NIDEs) with initial conditions. Practical investigations revealed that ND-HAM leads an easy way how to find initial guess and it approaches the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM). Uniqueness solution of the problem and convergence of ND-HAM are proved in the Banach space. Finally, two examples are illustrated to show the accuracy and validity of the proposed method. Five methods are compared in each example.

Некоторые формулы дробного интегрирования от одной функции Фокса с четырьмя параметрами

Solutions to many problems of mathematical physics, engineering and economics arc expressed through the so-called special functions. In the theory of special functions an important place is occupied by functions of the hypergeometric type. Many of them can be written in terms of the Meyer G-function. A generalization of the Meyer function is the Fox H-function. Some properties of this function can be obtained from its representation using the Mellin – Barnes integral. When deriving some formulas for this function for particular values of its parameters, due to the cumbersome writing of the Fox function, it is more convenient to use simplified notation. In this paper, we consider a special case of such a Fox function containing four parameters. For this function, Riemann-Liouville and Erdelyi-Kober fractional integration formulas arc obtained. An integral representation of the considered function h through the Mellin – Burns integral, we write out the conditions under which it converges absolutely, and the asymptotic expansions for this function for large and small values of the argument. The formulas proved in the paper are obtained using the indicated Mellin – Burns integral representation and the well-known integration formulas from power functions. For particular values of the parameters, the function under consideration yields some well-known elementary and special functions, and from the obtained formulas of fractional integration – the known integral values of these functions.

Soliton solution of fractional Sharma-Tasso-Olever equation via an efficient (G′/G)-expansion method

This paper investigates the solutions of the conformable fractional Sharma-Tasso-Olever equation with the help of Atangana’s conformable differential operator. In this respect, an efficient (G′/G)-expansion method is used for obtaining the new solitary wave solutions. This work addresses the physical and dynamic behavior of some new exact trigonometric, hyperbolic, and rational solitary wave solutions in the form of 3D-plots, contour plots, and 2D- plots using different measures of parameters. The graphical representation of these results is very helpful for understanding the real physical importance of the studied model equation. The obtained results are new and more general, and show the efficiency of the proposed method for the analytical treatment of nonlinear problems in mathematical physics and engineering and helpful in better understanding the propagating wave dynamics in diverse situations.

Математичне моделювання полів температури антенних рефлекторів космічних апаратів за допомогою нового інтегрального перетворення

Prediction of the mechanical properties of the reflector, and above all, the deviation of the highly accurate shape of the reflecting surface (RSS) from the given one is the main goal of designing spacecraft antennas. Distortion of the RSS is determined by the stress-deformed state of the elements of the reflector structures under the conditions of orbital operation. At the same time, the main factor determining the distortion of reflectors with RSS in open space is temperature deformation due to the uneven distribution of solar heat fluxes among structural elements. Therefore, the development of methods and models for calculating temperature fields in reflectors during heat flows on the surface is relevant. In the article, for the first time, a new finite integral transformation for the Laplace equation in a cylindrical coordinate system is constructed for a region bounded by several closed piecewise smooth contours. The inverse transformation formula is given. In the article, for the first time, a mathematical model for the calculation of temperature fields in a paraboloid rotating with a constant angular velocity is constructed, taking into account the finite speed of heat propagation in the form of a boundary value problem of mathematical physics for the hyperbolic equation of heat conduction with Dirichlet boundary conditions. With the help of the developed integral transformation, the temperature fields in the paraboloid were found in the form of convergent series according to the Fourier functions. The found solution to the generalized boundary value problem of the heat transfer of the paraboloid of rotation can find application in modulating the temperature fields that arise in the antenna reflectors of space vehicles. The developed integral transformation makes it possible to obtain solutions to complex boundary value problems of mathematical physics.

The embedding theorems for anisotropic Nikol’skii-Besov spaces with generalized mixed smoothness

The theory of embedding of spaces of differentiable functions studies the important relations of differential (smoothness) properties of functions in various metrics and has a wide application in the theory of boundary value problems of mathematical physics, approximation theory, and other fields of mathematics. In this article, we prove the embedding theorems for anisotropic spaces Nikol’skii-Besov with a generalized mixed smoothness and mixed metric, and anisotropic Lorentz spaces. The proofs of the obtained results are based on the inequality of different metrics for trigonometric polynomials in Lebesgue spaces with mixed metrics and interpolation properties of the corresponding spaces.

The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder

А new closed-loop solution for the coupled nonstationary problem of thermoelectric elasticity is designed for a long piezoceramic radially polarized cylinder. The case of the nonstationary load acting on its inner cylindrical surface is considered as a function of temperature change at a given law of the convection heat exchange on the outer face wall (boundary conditions of heat conductivity of the 1st and 3rd types). Electrodynamic cylinder surfaces are connected to a measuring device with a high input resistance (electric idling). We investigate the problem where the rate of the temperature load changes does not affect the inertial characteristics of the elastic system. It makes it possible to expand the initial linear computational relations with the equilibrium, electrostatics and heat conductivity equations with respect to the radial component of the displacement vector, electric potential as well as the function of temperature field changes. Hyperbolic LS-theory of the thermal conductivity is used in the computations. The problem is solved with a generalized method of biorthogonal finite integral transformation based on a multicomponent ratio of eigen functions of two homogeneous boundary value problems. The structural algorithm of this approach allows identifying a conjugated operator, without which it is impossible to solve non-self-conjugated linear problems in mathematical physics. The resulted computational relations make it possible to determine the stress-strain state, temperature and electric fields induced in the piezoceramic element under an arbitrary external temperature effect. By connecting the electroelastic system to the measuring tool, we can find voltage. Firstly, the analysis of the numerical results allows identifying the rate of the temperature load changes, at which it is necessary to use the hyperbolic theory of thermal conductivity. Secondly, it allows determining the physical characteristics of the piezoceramic material for the case when the rate of changing the body volume leads to a redistribution of the temperature field. The developed computational algorithm can be used to design non-resonant piezoelectric temperature sensors.

Combinatorics of KP hierarchy structural constants

We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on tau -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study.

Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

Mathematical Physics Problem Solving Skills for Secondary School physics Teachers

The aim of the current research is to identify the extent to which secondary school physics teachers possess the skills to solve physical problems and to identify the differences in their physical problem solving skills according to the gender variables. The research identified physics teachers in secondary schools affiliated with the Directorate of Education in Al-Diwaniyah, for the academic year 2020-2021, and the sample size was (160) teachers who were randomly selected, and by reviewing some studies that dealt with physical problem solving skills, such as the study of (Islami et al., 2018). and study (Prasetyo, 2020). It was found that these studies share a number of points with the current research, but they did not provide a tool for measuring physical problem solving skills. Therefore, the researcher designed a scale of physical problem solving skills, which consisted of (32) items. After the exploratory application, the validity and stability of the study tool was confirmed by appropriate methods, and the correlation of the item’s degree with the total score was calculated, and after applying the test to the sample, the data was processed statistically, as the researcher used the equation of the t-test for one sample to verify that physics teachers possess the skills of solving Physical problems, and the results showed that teachers of physics possess the skills to solve physical problems, and the statistical method was used to verify the differences in the skills of solving physical problems according to gender variables (male and female), and the results showed that there are differences in favor of (male) and the researcher recommended in the light of Results Preparing special training programs by the Ministry of Education to develop physical problem solving skills among physics teachers, and suggested the need to focus on the diversity of solving processes based on physical problem solving skills.

On the number of equilibria balancing Newtonian point masses with a central force

We consider the critical points (equilibria) of a planar potential generated by n Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered previously in studies of the circular-restricted n-body problem. We show that the number of equilibria is finite for a generic set of parameters, and we establish estimates for the number of equilibria. We prove that the number of equilibria is bounded below by n + 1, and we provide examples to show that this lower bound is sharp. We prove an upper bound on the number of equilibria that grows exponentially in n. In order to establish a lower bound on the maximum number of equilibria, we analyze a class of examples, referred to as “ring configurations,” consisting of n − 1 equal masses positioned at vertices of a regular polygon with an additional mass located at the center. Previous numerical observations indicate that these configurations can produce as many as 5n − 5 equilibria. We verify analytically that the ring configuration has at least 5n − 5 equilibria when the central mass is sufficiently small. We conjecture that the maximum number of equilibria grows linearly with the number of point masses. We also discuss some mathematical similarities to other equilibrium problems in mathematical physics, namely, Maxwell’s problem from electrostatics and the image counting problem from gravitational lensing.

The automated construction of a scheme for solving compute-intensive problems based on the ontological approach and Semantic Web technologies

This paper describes the tools for supporting researchers in the development of a parallel code. The tools are based on the ontology of the knowledge area “Support for solving compute-intensive problems of mathematical physics on supercomputers”. The main result of these tools operation is a scheme for solving the problem, built according to its specification provided by the user. The scheme includes the most suitable mathematical models for solving the problem, numerical methods, algorithms, and parallel architectures, links to available fragments of a parallel code that the user can use when developing his own code. The scheme construction is carried out on the basis of ontology and expert rules built using the Semantic Web technology.

Construction of mathematical models of the stressed-strained state of a material with a porous water-saturated base under dynamic load

Materials of beams, plates, slabs, strips have been commonly applied in various fields of industry and agriculture as flat elements in the structures for machinery and construction. They are associated with the design of numerous engineering structures and facilities, such as the foundations of various buildings, airfield and road surfaces, floodgates, including underground structures. This paper reports a study into the interaction of the material (of beams, plates, slabs, strips) with the deformable base as a three-dimensional body and in the exact statement of a three-dimensional problem of mathematical physics under dynamic loads. The tasks of studying the interaction of a material (beams, plates, slabs, strips) with a deformable base have been set. A material lying on a porous water-saturated viscoelastic base is considered as a viscoelastic layer of the same geometry. It is assumed that the lower surface of the layer is flat while the upper surface, in a general case, is not flat and is given by some equation. Classical approximate theories of the interaction of a layer with a deformable base, based on the Kirchhoff hypothesis, have been considered. Using the well-known hypothesis by Timoshenko and others, the general three-dimensional problem is reduced to a two-dimensional one relative to the displacement of points of the median plane of the layer, which imposes restrictions on external efforts. In the examined problem, there is no median plane. Therefore, as the desired values, displacements and deformations of the points in the plane have been considered, which, under certain conditions, pass into the median plane of the layer. It is not possible to find a closed analytical solution for most problems while experimental studies often turn out to be time-consuming and dangerous processes

Difficulties of Solving Physics Mathematical Problems of the Students of the Department of Physics, College of Education

The current study aimed to answer the following questions: 1. What's the level of difficulty of solving physics mathematical problems for the students of Physics Department, college of Education? 2. What is the level of difficulty of solving physics mathematical problems for students of Physics Department, College of Education as attributed to sex variable? In order to achieve the study objectives, the researchers used the descriptive approach because it is appropriate for the study nature and objectives. The researchers constructed a quadruple scale to measure the difficulties of solving physics mathematical problems, which includes (36) items (applicable to me: high, medium, low or does not apply to me). The study sample consisted of (200) students from Depts. of physics, Colleges of Education at the Middle Euphrates Universities. The study tools are applied in the first semester of the academic year (2020 - 2021). The researchers used a number of statistical methods as t-test for one sample and two samples, Chi square, items discrimination coefficient, the Pearson correlation coefficient, the Spearman-Brown equation, and the Alpha-Cronbach equation. The researchers has reached for the following findings: 1. There are difficulties when students solve physics mathematical problems in the dept. of Physics, College of Education. 2. There are no differences in the difficulties of solving physics mathematical problems for the students of the dept. of Physics, College of Education as attributed to sex variable. Based on the researchers findings, the researchers made several recommendations and proposals. Key words: Difficulty, mathematical physics problems, solving physics mathematical problems.

Efficient Implementation of Liquid Crystal Simulation Software on Modern HPC Platforms

In this paper we demonstrate the process of efficient porting a software package for Markov chain Monte Carlo (MCMC) simulations on a finite cubic lattice on multiple modern architectures: Pascal, Volta and Turing NVIDIA GPUs, NEC SX-Aurora TSUBASA vector engines and Intel Xeon Gold processors. In the studied software, MCMC methodology is used for simulations of liquid crystal structures, but it can be as well employed in a wide range of problems of mathematical physics and numerical methods. The main goals of this work are to determine the best software optimization strategy for this class of algorithms and to examine the speed and the efficiency of such simulations on modern HPC platforms. We evaluate the effects of various optimizations, such as using more suitable memory access patterns, multitasking for efficient utilization of massive parallelism on the target architectures, improved cache hit-rates, parallel workload balancing, etc. We perform a detailed performance analysis for each target platform using software tools such as nvprof, Ftrace and VTune. On this basis, we evaluate and compare the efficiency of the developed computational kernels on different platforms and subsequently rank these platforms by their performance. The results show that NVIDIA GPU and NEC SX-Aurora TSUBASA platforms, although at first glance seem very different, require similar optimization approaches in many cases due to similarities in data processing principles.

Адаптивный метод операторной экстраполяции для вариационных неравенств в банаховых пространствах

Многие актуальные задачи исследования операций и математической физики могут быть записаны в форме вариационных неравенств. Разработка и исследование алгоритмов решения вариационных неравенств является направлением прикладного активно развивающегося нелинейного анализа. Отметим, что часто негладкие задачи оптимизации могут эффективно решаться, если переформулировать их в виде седловых задач и применить алгоритмы решения вариационных неравенств. В последнее время наметился прогресс в изучении алгоритмов для задач в банаховых пространствах. Это обусловлено широким привлечением результатов и конструкций геометрии банаховых пространств. В работе предложен и исследован новый алгоритм для разрешения вариационных неравенств в банаховом пространстве. Предлагаемый алгоритм является адаптивным вариантом «forward-reflected-backward algorithm», где используется правило обновления величины шага, не требующее знания лепшицевой константы оператора. Кроме того, вместо метрической проекции на допустимое множество используется обобщенная проекция Альбера. Преимуществом использования алгоритма является лишь одно вычисление на итерационном шаге проекции на допустимое множество. Для вариационных неравенств с монотонными, лепшицевыми операторами, действующими в 2-равномерно выпуклом и равномерно гладком банаховом пространстве, доказана теорема о слабой сходимости метода.

IMPLEMENTATION OF A FINITE ELEMENT CLASS LIBRARY USING GENERALIZED PROGRAMMING

Context. For computer modeling of complex objects and phenomena of various nature, in practice, the numerical finite element method is often used. Its software implementation (especially for the study of new classes of problems) is a rather laborious process. The high cost of software development makes the development of new approaches to improving the efficiency of programming and maintenance (including the addition of new functions) urgent. Objective. The aim of the work is to create a new effective architecture of programs for finite element analysis of problems in mathematical physics, which makes it easy to expand their functionality to solve new classes of problems. Method. A method for developing programs for finite element analysis using generalized programming is proposed, which makes it possible to significantly simplify the architecture of the software and make it more convenient for maintenance and modification by separating algorithms and data structures. A new architecture of classes that implement finite element calculation is proposed, which makes it possible to easily expand the functionality of programs by adding new types of finite elements, methods for solving systems of linear algebraic equations, parallel computations, etc. Results. The proposed approach was implemented in software as a class library in C ++. A number of computational experiments have been carried out, which have confirmed its efficiency in solving practical problems. Conclusions. The developed approach can be used both to create general-purpose finite element analysis systems with an open architecture, and to implement specialized software packages focused on solving specific classes of problems (fracture mechanics, elastomers, contact interaction, etc.).

Use of the Method of Integral Relations for the Analytic Solutions of Hyperbolic Models of Thermal Conductivity

The study of the processes of unsteady heat conduction, the calculation of the parameters of media under conditions of unsteady heat conduction of the latter is an important direction, which is used in applied problems of heat and mass transfer. When solving a mathematical model under various boundary conditions, there is a problem of the reliability of numerical calculations, therefore there is a need to solve the mathematical model by an analytical method. For example, a mathematical model of heat and mass transfer processes in a heat accumulator during its charging and discharging is solved analytically by the Green's function method, similarly, a mathematical model of heat carrier heating processes in solar collectors is solved. The specific definition of the Green's function corresponds to a specific problem in mathematical physics. Green's function contains complete information about the studied equation, and with its help one can construct a solution for any inhomogeneity. The development of the method of Green's functions for solving boundary value problems of unsteady heat conduction of generalized type on the basis of the Maxwell-Cattaneo-Lykov law is proposed. On the basis of the introduced Green's function of the differential equation, the Green's function of the boundary value problem is determined. Green's function of a boundary value problem is considered as an element of the set of Green's functions of an equation or a system of equations. Boundary conditions are formulated in accordance with the specified law. When considering specific problems, in a number of cases, it is expedient to transfer the integral form of writing boundary conditions of the second or third kind into a differential form equivalent to the integral one. The proposed integral relations for analytical solutions of boundary value problems of unsteady heat conduction for equations of hyperbolic type. Necessary and sufficient conditions for the existence and uniqueness of the Green's function of the boundary value problem are given and its analytical representation is given in terms of the fundamental system of solutions and boundary conditions. Boundary conditions are formulated for hyperbolic models of heat conduction in integral and differential forms. Boundary value problems for a semi-infinite region are considered, analytical solutions are obtained, their analysis is carried out, and temperature jumps at the heat wave front are calculated. Illustrative problems for a semi-infinite region are considered, and the heat wake region and the unperturbed region are described.

Development and comparison of the techniques for solving the inverse problem in photoacoustic characterization of semiconductors

In this work, theoretically/mathematically simulated (TMS) model is presented for the photoacoustic (PA) frequency response of a semiconductor in a minimum volume PA cell. By analyzing of the TMS model, the influences of thermal diffusivity and linear coefficient of thermal expansion on silicon sample PA frequency response were investigated and two methods were developed for their estimation. The first one is a self consistent inverse procedure for solving the exponential problems of mathematical physics, based on regression. The second one, a well trained three-layer perceptron with back propagation, based upon theory of artificial neural networks, is developed and presented. These two inverse problem solving concepts are applied to thermo-elastic characterization of silicon, compared and discussed in the domain of semiconductor characterization.

The uniform convergence of Fourier series expansions of a Sturm-Liouville problem with boundary condition which contains the eigenparameter

This paper is devoted to investigating the uniform convergence conditions of Fourier series expansions of continuous functions in terms of eigenfunctions of a Sturm-Liouville problem with eigenparameter in one of the boundary conditions on a closed interval. Such problems are quite common in mathematical physics problems.

SPECIAL SPLINE APPROXIMATION FOR THE SOLUTION OF THE NON-STATIONARY 3-D MASS TRANSFER PROBLEM

In this paper we consider the conservative averaging method (CAM) with special spline approximation for solving the non-stationary 3-D mass transfer problem. The special hyperbolic type spline, which interpolates the middle integral values of piece-wise smooth function is used. With the help of these splines the initial-boundary value problem (IBVP) of mathematical physics in 3-D domain with respect to one coordinate is reduced to problems for system of equations in 2-D domain. This procedure allows reduce also the 2-D problem to a 1-D problem and thus the solution of the approximated problem can be obtained analytically. The accuracy of the approximated solution for the special 1-D IBVP is compared with the exact solution of the studied problem obtained with the Fourier series method. The numerical solution is compared with the spline solution. The above-mentioned method has extensive physical applications, related to mass and heat transfer problems in 3-D domains.