Mathematical Physics Research Articles

Efficient Method for Solving Fourth Order PDEs

The aim of the article is to implement the LA Transform with HPM to solve the 4th order non-linear PDEs arising in mathematical physics and astrophysics. This method is based on a combination of LA Transformation with HPM without any discretization or restrictive assumptions. Some well-known 4th order non-linear PDEs are solved as illustrative examples to demonstrate the accuracy, efficiency and easy implementation of this technique.

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.

Applications of New Double Integral Transform (Laplace–Sumudu Transform) in Mathematical Physics

The primary purpose of this research is to demonstrate an efficient replacement of double transform called the double Laplace–Sumudu transform (DLST) and prove some related theorems of the new double transform. Also, we will discuss the fundamental properties of the double Laplace–Sumudu transform of some basic functions. Then, by utilizing those outcomes, we will apply it to the partial differential equations to show its simplicity, efficiency, and high accuracy.

The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis

The Estevez-Mansfield-Clarkson (EMC) equation and the (2+1)-dimensional Riemann wave (RW) equation are important mathematical models in nonlinear science, engineering and mathematical physics which have remarkable applications in the field of plasma physics, fluid dynamics, optics, image processing etc. Generally, through the sine-Gordon expansion (SGE) method only the lower-dimensional nonlinear evolution equations (NLEEs) are examined. However, the method has not yet been extended of finding solutions to the higher-dimensional NLEEs. In this article, the SGE method has been developed to rummage the higher-dimensional NLEEs and established steady soliton solutions to the earlier stated NLEEs by putting in use the extended higher-dimensional sine-Gordon expansion method. Scores of soliton solutions are figure out which confirms the compatibility of the extended SGE method. The solutions are analyzed for both lower and higher-dimensional nonlinear evolution equations through sketching graphs for alternative values of the associated parameters. From the figures it is notable to perceive that the characteristic of the solutions depend upon the choice of the parameters. This study might play an impactful role in analyzing higher-dimensional NLEEs through the extended SGE approach.

Kahoot quiz: interactive media in knowing the increase in understanding of the numerical method of oceanographic physics based on Matlab programming

This study aims to improve the practical understanding of the numerical method of oceanographic physics based on Mathlab software to calculate the roots of the Airy wave equation through the Kahoot application as an interactive media. This study used a population of 3 classes totaling 105 people. The determination of the increase in quiz scores with and without the Kahoot application was analyzed using the basic statistical comparison method, Independent T-Test Comparison (Intecomp) testing, as well as analysis of the distribution of the increase in quiz scores per class and per, mean predicate with a weighted scale of 4. The average score of the quiz scores without Kahoot 76.28; with Kahoot 84.96; the Intecomp P-Value test results 6,1.10-6 (α = 0.05), a significant increase in quiz scores from the use of the Kahoot application. The distribution of data for the increase in weighted quiz scores is 9.63%; As for the distribution of the rise in the average quiz score per class was 5.42%. Based on the statistical analysis test, it can be concluded that the use of the Kahoot application as an interactive media can increase the practitioner's understanding of the mathematical physics numerical method of oceanography based on Matlab.

Hessian metric via transport information geometry

We propose to study the Hessian metric of a functional on the space of probability measures endowed with the Wasserstein-2 metric. We name it transport Hessian metric, which contains and extends the classical Wasserstein-2 metric. We formulate several dynamical systems associated with transport Hessian metrics. Several connections between transport Hessian metrics and mathematical physics equations are discovered. For example, the transport Hessian gradient flow, including Newton’s flow, formulates a mean-field kernel Stein variational gradient flow; the transport Hessian Hamiltonian flow of Boltzmann–Shannon entropy forms the shallow water equation; and the transport Hessian gradient flow of Fisher information leads to the heat equation. Several examples and closed-form solutions for transport Hessian distances are presented.

PARALLEL CUDA IMPLEMENTATION OF THE ALGORITHM FOR SOLVING THE NAVIER-STOKES EQUATIONS USING THE FICTITIOUS DOMAIN METHOD

An important direction in the development of numerical simulation methods is the study of approximate methods for solving problems of mathematical physics in complex multidimensional fields. To solve many applied problems in irregular areas, the method of fictitious areas is widely used, which is characterized by a high degree of automation of programming. The main idea of the method of fictitious domains is that the problem is solved not in the original complex domain, but in some other, simpler domain. This allows you to create software immediately for a fairly wide class of problems with arbitrary computational domains. The possibilities of applying the method of fictitious domains to the problems of hydrodynamics in the variables “stream function, velocity vortex” have been considered in many works. In this paper, we study a numerical method for solving the Navier-Stokes equations in doubly connected domains. To solve the two-dimensional Navier-Stokes equations in irregular domains, an approximate method based on the method of fictitious domains is proposed. A computational finite difference algorithm for solving an auxiliary problem of the method of fictitious domains has been developed. The results of numerical simulation of two-dimensional Navier-Stokes equations by the method of fictitious domains with continuation by the lowest coefficient are presented. For this problem, a parallel algorithm was developed using CUDA technologies, which was tested on various mesh dimensions

DIGITAL ONLINE PLATFORMS «GRAPHICS» AND «INTEGRALS» IN THE DISCIPLINE «METHODS OF MATHEMATICAL PHYSICS»

The examples of practical tasks implementation in the discipline "Methods of Mathematical Physics" are considered, in which students get acquainted with the digital online platforms "Graphics" and "Integrals" for self-fulfillment of control tasks in this discipline. The skills acquired by students in mastering these digital platforms can be useful when studying courses in special disciplines, completing course projects and performing research projects.

Сходимость метода экстраполяции из прошлого и метода операторной экстраполяции

Одним из популярных направлений современного прикладного нелинейного анализа является исследование вариационных неравенств. Многие актуальные проблемы исследования операций и математической физики можно записать в форме вариационных неравенств. С появлением генерирующих соревновательных нейронных сетей интерес к алгоритмам решения вариационных неравенств возник и в среде специалистов машинного обучения. Данная работа посвящена исследованию трех новых алгоритмов с брегмановской проекцией для решения вариационных неравенств в гильбертовом пространстве. Первый алгоритм — результат модификации двухэтапного брегмановского метода с помощью экономного регулирования величины шага, не требующего знания лепшицевой константы оператора. Второй алгоритм — алгоритм операторной экстраполяции, полученный заменой в методе Малицкого-Тама евклидовой метрики на дивергенцию Брегмана. Привлекательная черта алгоритма — всего одно вычисление на итерационном шаге проекции Брэгмана на допустимое множество. Третий алгоритм — адаптивный вариант второго, где используется правило обновления величины шага, не требующее знания лепшицевых констант и вычислений значений оператора в дополнительных точках. Для вариационных неравенств с псевдомонотонными, лепшицевыми и секвенционно слабо непрерывными операторами, действующими в гильбертовом пространстве, доказаны теоремы о сходимости методов.

Analytical behavior of the fractional Bogoyavlenskii equations with conformable derivative using two distinct reliable methods

The nonlinear fractional model, videlicet, the space–time fractional (2+1)-dimensional fractional Bogoyavlenskii equations is an important nonlinear mathematical model to elucidate the hydrodynamic model of shallow water wave, plasma physics, the wave of leading fluid flow etc. This paper is devoted to studying the dynamics of the traveling wave with fractional conformable nonlinear evaluation equations (NLEEs) arising in nonlinear wave mechanics. By utilizing of oncoming exp(−K(ρ))-expansion technique and rational tan(K(ρ))-expansion technique where are the series of novel exact solutions in terms of rational, periodic and hyperbolic functions for the fractional cases are derived. These types of long wave propagation phenomena play a dynamic role to interpret the water waves as well as mathematical physics. It has been demonstrated that our proposed methods are further efficient, general, succinct, power full, straight forward and can be asserted to install the new exact solutions of different kinds of fractional equations in engineering and nonlinear dynamics.

Analytical Interpretation of the Early Literature on the Theory of Vibrations

A discussion is given of early literature pertaining to the theory of vibration from the time of Pythagoras up through 1750. The paper attempts to give an analytical interpretation to early anecdotal works concerning Pythagoras and to publications of Galileo, Huygens, Hooke, Taylor, John Bernoulli, Leibniz and Euler. To bridge the “culture gap,” mathematical developments by the latter cited authors are, whenever appropriate, rephrased in modern notation, using, for the most part, only those techniques that should have been well known to the authors at the time. The emphasis is on what might be loosely called the physics (or the mathematical physics) of vibration.

Lie symmetry analysis, abundant exact solutions and dynamics of multisolitons to the $$(2+1)$$-dimensional KP-BBM equation

In the present article, our main aim is to construct abundant exact solutions for the $$(2+1)$$ -dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM) equation by using two powerful techniques, the Lie symmetry method and the generalised exponential rational function (GERF) method with the help of symbolic computations via Mathematica. Firstly, we have derived infinitesimals, geometric vector fields, commutation relations and optimal system. Therefore, the KP-BBM equation is reduced into several nonlinear ODEs under two stages of symmetry reductions. Furthermore, abundant solutions are obtained in different shapes of single solitons, solitary wave solutions, quasiperiodic wave solitons, elastic multisolitons, dark solitons and bright solitons, which are more relevant, meaningful and useful to describe physical phenomena due to the existence of free parameters and constants. All these generated exact soliton solutions are new and completely different from the previous findings. Moreover, the dynamical behaviour of the obtained exact closed-form solutions is analysed graphically by their 3D, 2D-wave profiles and the corresponding density plots by using the mathematical software, which will be comprehensively used to explain complex physical phenomena in the fields of nonlinear physics, plasma physics, optical physics, mathematical physics, nonlinear dynamics, etc.

ФРАКТАЛЬНАЯ МАСШТАБНО-ИНВАРИАНТНАЯ СТРУКТУРА ВРЕМЕННОЙ ИЕРАРХИИ В ПРОЦЕССАХ РЕЛАКСАЦИИ

The phenomena of elastic aftereffects during loading/unloading of viscoelastic and capillary-porous bodies, relaxation of their stresses is accompanied by the energy accumulation and dissipation to be taken into account in the theory of oscillations which also considers the behavior of materials when the force is applied to them, the elastic aftereffect and stress relaxation forms ostensibly opposite energy processes that’s why the main problem to one is to understand and discovery laws for such aftereffects. The goal of the research to show that the distribution of relaxation time in viscoelastic and capillary-porous media may have a scale-invariant structure and that the indirect confirmation of the scale invariance of relaxation time hierarchy can be the principle of temperature-time superposition according to which the experimental relaxation functions obtained for different temperatures can be combined with each other using the appropriate coordinate axes stretching. We used methods of viscoelastic theory, fractal analysis and methods of mathematical physics. So, in this paper, an attempt has been made to harmonize both these theories and numerous experiments on the destruction of materials described in the academic literature. It is shown that the hierarchy of times determining shear and bulk relaxation in viscoelastic/capillary-porous medium has a fractal structure and it was observed that the presence of time fractality eases the modeling of viscoelastic/capillary-porous bodies resulting in the universal relaxation function of a rather simple kind.

New Perspectives and Computational Challenges in High Dimensions

High-dimensional systems are frequent in mathematics and applied sciences, and the understanding of high-dimensional phenomena has become increasingly important. The mathematical subdisciplines most strongly related to such phenomena are functional analysis, convex geometry, and probability theory. In fact, a new area emerged, called asymptotic geometric analysis, which is at the very core of these disciplines and bears a number of deep connections to mathematical physics, numerical analysis, and theoretical computer science. The last two decades have seen a tremendous growth in this area. Far reaching results were obtained and various powerful techniques have been developed, which rather often have a probabilistic flavor. The purpose of this workshop was to explored these new perspectives, to reach out to other areas concerned with high-dimensional problems, and to bring together researchers having different angles on high-dimensional phenomena.

ФРАКТАЛЬНАЯ МАСШТАБНО-ИНВАРИАНТНАЯ СТРУКТУРА ВРЕМЕННОЙ ИЕРАРХИИ В ПРОЦЕССАХ РЕЛАКСАЦИИ

The phenomena of elastic aftereffects during loading/unloading of viscoelastic and capillary-porous bodies, relaxation of their stresses is accompanied by the energy accumulation and dissipation to be taken into account in the theory of oscillations which also considers the behavior of materials when the force is applied to them, the elastic aftereffect and stress relaxation forms ostensibly opposite energy processes that’s why the main problem to one is to understand and discovery laws for such aftereffects. The goal of the research to show that the distribution of relaxation time in viscoelastic and capillary-porous media may have a scale-invariant structure and that the indirect confirmation of the scale invariance of relaxation time hierarchy can be the principle of temperature-time superposition according to which the experimental relaxation functions obtained for different temperatures can be combined with each other using the appropriate coordinate axes stretching. We used methods of viscoelastic theory, fractal analysis and methods of mathematical physics. So, in this paper, an attempt has been made to harmonize both these theories and numerous experiments on the destruction of materials described in the academic literature. It is shown that the hierarchy of times determining shear and bulk relaxation in viscoelastic/capillary-porous medium has a fractal structure and it was observed that the presence of time fractality eases the modeling of viscoelastic/capillary-porous bodies resulting in the universal relaxation function of a rather simple kind.

A mathematical model of the velocity of a cyclist riding over uneven terrain

Derived in this article is a mathematical model of a cyclist riding over uneven terrain. The model is very general; it is expressed in terms of arbitrary functions that describe how the elevation changes with horizontal distance and how the cyclist’s power output varies, and also accounts for road friction and air resistance. The model was validated by considering special cases such as horizontal and inclined surfaces. Although the governing equation is too complicated to solve exactly for the general case, a numerical solution procedure is proposed and tested. In addition, an approximate analytical solution procedure is outlined. As an illustration some numerical results are presented for the case of a cyclist riding over rolling hills. The adopted approach is well suited for undergraduate students in mathematical physics. The formulation equally applies to a vehicle driving over a variable landscape. This work also unites concepts from physics with mathematical modelling and coding. Lastly, a video abstract is also provided.

Abundant wave solutions to two novel KP-like equations using an effective integration method

In this work, two new nonlinear evolution equations arising from the B-type Kadomtsev-Petviashvili equation, called BKP-like equations, are investigated. The integration technique that used in this paper to determine the exact solutions of the equation is the generalized exponential rational function method. The examined models may be extended to diversify problems in natural phenomena, such as ocean waves. After applying the aforesaid method, abundant wave solutions are formally generated with some free parameters to exhibit various versions of propagations of traveling solitary waves. Notably, upon choosing appropriate values to free parameters, some kink and periodic waves are demonstrated in 3D figures and 2D contour plots. Most of all, the results show that free parameters drastically influence the existence of all kinds of traveling waves, including nature, profile, and stability. The method used in this paper can be easily adopted in other similar equations in mathematical physics.

Semi-algebraic sets method in PDE and mathematical physics

This paper surveys recent progress in the analysis of nonlinear partial differential equations using Anderson localization and semi-algebraic sets method. We discuss the application of these tools from linear analysis to nonlinear equations such as the nonlinear Schrödinger equations, the nonlinear Klein–Gordon equations (nonlinear wave equations), and the nonlinear random Schrödinger equations on the lattice. We also review the related linear time-dependent problems.

Computer modelling of physical processes described by parabolic type equations

The article considers the application of modern methods of mathematical modelling in the process of solving problems of mathematical physics in the aerospace industry. To describe the temperature distribution in a given area of space and its change in time, the heat conduction equation is used. This equation is a special case of the diffusion equation and is a second order partial differential equation. The tools of the mathematical package MathCad are considered, which can be used to study physical processes described by the heat conduction equation under various initial and boundary conditions.

Variational method for solving the boundary value problem of hydrodynamics

The analysis of the application of the variational method for solving the boundary value problem of hydrodynamics is carried out. From the point of view of numerical research of mathematical physics problems, these variation formulations are considered as the basis of projection methods (the Ritz method). The article presents the main techniques that allow reducing the cost of machine time and speed up the convergence of the computational process when calculating the hydrodynamic characteristics of cavities of various configurations. Using the Trefts method allows you to reduce the calculation time of the boundary value problem. The green transformation allows us to reduce the three dimensional integral to a one-dimensional. This creates a universal method for determining hydrodynamic coefficients for rotation cavities with an arbitrary contour of the Meridian section. However, for most configurations of cavities, the convergence rate is satisfactory and provides numerical values with a high degree of accuracy.