Mathematical Physics Research Articles

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

A variety of novel closed‐form soliton solutions to the family of Boussinesq‐like equations with different types

This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion. The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary, kink, anti-kink, combo, and singular-periodic wave solutions. The attained solutions are expressed by the trigonometric and hyperbolic functions including tan, sec, cot, csc, tanh, sech, coth, csch, and of their combination. In addition, the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions. Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family. For the bright wave solutions in terms of Atangana’s conformable derivative, the amplitudes of the bright wave gradually decrease, but the smoothness increases when the fractional parameters α and β increase. On the other hand, the periodicities of periodic waves increase. The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases. The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.

Lower Bounds on the Chromatic Number of Random Graphs

We prove that a formula predicted on the basis of non-rigorous physics arguments [Zdeborová and Krzakala: Phys. Rev. E (2007)] provides a lower bound on the chromatic number of sparse random graphs. The proof is based on the interpolation method from mathematical physics. In the case of random regular graphs the lower bound can be expressed algebraically, while in the case of the binomial random we obtain a variational formula. As an application we calculate improved explicit lower bounds on the chromatic number of random graphs for small (average) degrees. Additionally, we show how asymptotic formulas for large degrees that were previously obtained by lengthy and complicated combinatorial arguments can be re-derived easily from these new results.

Functional Inequalities: Nonlinear Flows and Entropy Methods as a Tool for Obtaining Sharp and Constructive Results

Interpolation inequalities play an essential role in analysis with fundamental consequences in mathematical physics, nonlinear partial differential equations (PDEs), Markov processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo–Nirenberg–Sobolev inequalities and symmetry results in Caffarelli–Kohn–Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.

Dirichlet problem with measurable data for quasilinear Poisson equations

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb D$ goes to the famous dissertation of Luzin, see e.g. its reprint \cite{L}. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. Following this way, we proved in \cite{GNR1} Theorem 7 on the solvability of the Dirichlet problem for the Poisson equations $\triangle\, U=G$ with sources in classes $G\in L^p,$ $ p>1$, in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. There we assumed that the domains satisfy the quasihyperbolic boundary condition by Gehring--Martio, generally speaking, without the known $(A)-$condition by Ladyzhenskaya--Ural'tseva and, in particular, without the outer cone condition that were standard for boundary-value problems in the PDE theory. Note that such Jordan domains cannot be even locally rectifiable. With a view to further development of the theory of boundary value problems for semi-linear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is first constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations $\triangle\, U=G$ with the sources $G\in L^p,$ $ p>1$. The latter makes it possible to apply the Leray-Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form $\triangle\, U(z)=H(z)\cdot Q(U(z))$ for multipliers $H\in L^p$ with $ p>1$ and continuous functions $Q: \mathbb R\to\mathbb R$ with $Q(t)/t\to 0$ as $t\to \infty$. As consequences, we give applications to some concrete quasilinear equations of mathematical physics, arising under modelling various phy\-si\-cal processes such as diffusion with absorption, plasma states, stationary burning etc. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

On non-homogeneous Robin reflection for harmonic functions

This paper concerns the reflection of harmonic functions, , defined in a neighborhood of a real-analytic curve in the plane subject to the Robin condition, , on that curve. Here a and b are constants, and is the restriction of a holomorphic function onto the curve. For the case, when , while a and b are real-analytic functions, a reflection formula was derived in Belinskiy and Savina [The Schwarz reflection principle for harmonic functions in subject to the Robin condition. J Math Anal Appl. 2008;348:685–691], using the reflected fundamental solution method. Here, we construct a Robin-to-Neumann mapping and use it for obtaining the reflection operator. Since the two formulae look different, we show their equivalence when a and b are constants and . As examples, we show reflection formulae for non-homogeneous Neumann and Robin conditions on the common within mathematical physics curves, such as circles and lines.

Study of S-Function in Relationship with Natural Transform

The objective of this paper is to investigate natural transform of the S -function defined and studied by Saxena and Daiya [4] in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties in terms of Mittag-Leffler functions. Further, we show some applications of these natural transform in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

Universes as big data

In this paper, we briefly overview how, historically, string theory led theoretical physics first to precise problems in algebraic and differential geometry, and thence to computational geometry in the last decade or so, and now, in the last few years, to data science. Using the Calabi–Yau landscape — accumulated by the collaboration of physicists, mathematicians and computer scientists over the last four decades — as a starting-point and concrete playground, we review some recent progress in machine-learning applied to the sifting through of possible universes from compactification, as well as wider problems in geometrical engineering of quantum field theories. In parallel, we discuss the program in machine-learning mathematical structures and address the tantalizing question of how it helps doing mathematics, ranging from mathematical physics, to geometry, to representation theory, to combinatorics and to number theory.

On boundary-value problems for semi-linear equations in the plane

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Soliton Solutions for (3+1)-Dimensional Nonlinear Evolution Equations by Tanh Method

In this paper, the Tanh method is employed to compute the exact solutions for nonlinear PDEs. These equations are (3+1)-dimensional Jimbo-Miwa , potential-YTSF, and generalized KP equations. The solutions obtained by this method are compared with the solutions obtained through other methods. It is shown that the Tanh method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear dispersive equations via deep learning

In the field of mathematical physics, there exist many physically interesting nonlinear dispersive equations with peakon solutions, which are solitary waves with discontinuous first-order derivative at the wave peak. In this paper, we apply the multi-layer physics-informed neural networks (PINNs) deep learning to successfully study the data-driven peakon and periodic peakon solutions of some well-known nonlinear dispersion equations with initial–boundary value conditions such as the Camassa–Holm (CH) equation, Degasperis–Procesi equation, modified CH equation with cubic nonlinearity, Novikov equation with cubic nonlinearity, mCH-Novikov equation, b-family equation with quartic nonlinearity, generalized modified CH equation with quintic nonlinearity, and etc. Moreover, we also study the data-driven parameter discovery of the CH equation with the aid of the single peakon These results will be useful to further study the peakon solutions and corresponding experimental design of nonlinear dispersive equations.

Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

The Ξ -expansion method for several types of nonlinear PDEs and traveling wave solutions

In this article, we applied the improved and generalized method to some nonlinear problems such as: the (2+1) dimensional Bogoyavliskii equation and the Ablowitz-Kuap-Newell-Segur equation (AKNSE), but distinctively and simple, we called it the Ξ -expansion method. This method is characterized by the addition of an arbitrary constant d to the general solution, which leads to the presentation of new exact solutions. This method is simple but powerful for extracting exact solutions of partial differential equation, which habitually occur in mathematical physics, engineering sciences and applied mathematics.

Abundant analytical soliton solutions and Evolutionary behaviors of various wave profiles to the Chaffee–Infante equation with gas diffusion in a homogeneous medium

The generalized exponential rational function method is used in this work to obtain a variety of rich families of analytical soliton solutions and to exemplify the dynamics of solitonic structures of the (2+1)-dimensional ChaffeeInfante (CI) equation. This CI equation is a well-known reaction–diffusion equation that can describe the physical process of mass transport and particle diffusion and has been widely used in fluid dynamics, electromagnetic wave fields, high-energy physics, fluid mechanics, coastal engineering, and ion-acoustic waves in plasma physics, optical fibers, and other disciplines. The GERF method is successfully used to derive closed-form analytic solutions to the CI equation, including rational, exponential, trigonometric, and hyperbolic function solutions. The CI equation has indeed been observed to have bright and dark solitons, singular and combined singular soliton profiles, periodic oscillating nonlinear waves, and kink-wave profiles. Furthermore, three-dimensional graphical representations are presented to depict the dynamical behavior of the achieved solutions. We can better understand the dynamical properties and structures of these solutions by using these three-dimensional postures. This technique could be used to solve a wide variety of higher-dimensional nonlinear equations confronted in mathematical physics and applied sciences.

Preface PT Seminars 2020

Non-Hermitian theories that are explicitly PT-symmetric and/or quasi/pseudo-Hermitian have become the subject of an active research field for the last two decades. They constitute well-defined classical and quantum theories possessing real eigenvalue spectra and unitary time-evolution. Since the underlying concepts are generic, the general principles have been applied in essentially all areas of physics and have unravelled new theoretical insights and experimental realisations especially in optical applications.Beginning in 2003 a regular conference series entitled “Pseudo-Hermitian Hamiltonians in Quantum Physics” has taken place annually to bring together researchers in order to take stock of the current state of the field and to assist participants to collaborate in developing new ideas. Unfortunately, the 20th meeting in this series, which was scheduled originally for 2020 to be held in Los Alamos had to be postponed to 2022 due to the outbreak of the Covid 19 pandemic. To bridge the gap and to provide a platform to allow for a continued exchange of ideas and novelties a virtual seminar series via Zoom with a dedicated website (https://vphhqp.com) was organised by Andreas Fring and Francisco Correa. With currently more than 50 seminars scheduled and well over 200 registered participants, the series is still ongoing. The main motivation behind this special issue has been to allow speakers and participants in this series to present their results in a coherent written in the form of mini-reviews of certain subtopics in the field and new research results.We hope that this special issue will become a valuable reference and inspiration for a broad scientific community, experts and newcomers alike, working in mathematical and theoretical physics. Carl Bender, Department of Physics, Washington University, St. Louis, MO 63130, USA;cmb@wuphys. wustl. edu Francisco Correa, Institute de e Ciencias Físicas y Matemáticas , Universidad Austral de ChileCasilla 567, Valdivia, Chile; francisco.correa@uach.cl Andreas Fring, Department of Mathematics, City, University of London, Northampton SquareLondon EC1V 0HB, UK; a.fring@city.ac.uk

Modeling of the physical phenomenon of heat generation by using partial differential equations

The equations of mathematical physics are a natural environment for modeling physical phenomena, an example of the above is evidenced by the heat equation in relation to its use in a variety of applications; directly related to the equations of mathematical physics are the solution methods that are used to construct the predictive models. This paper describes step by step the analytical method of separation of variables to perform a complete description of the heat conduction phenomenon in the presence of a heat generation source. The investigation by using mathematical arguments allowed to calculate the temperature function as the addition of a Fourier series and a function which represents the steady state; by performing a computational simulation, it was possible to demonstrate the accuracy of the results achieved.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

Exceptional Hahn and Jacobi polynomials with an arbitrary number of continuous parameters

AbstractWe construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second‐order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum‐mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

ДИНАМИКА ГАЗОВЫДЕЛЕНИЯ С ПОВЕРХНОСТИ ОБНАЖЕНИЯ ГАЗОНОСНЫХ УГОЛЬНЫХ ПЛАСТОВ СРЕДНЕЙ МОЩНОСТИ

The goal of the research was to clarify the regularities of the dynamics of gas release from the surface of the outcrop of the developed coal seam. The main research methods were theoretical methods of mathematical physics and non-equilibrium thermodynamics. Gas-bearing coal seams are usually mined underground. When driving development workings, outcropping surfaces of gas-bearing coal seams appear and gases in the seams under excessive pressure are released into the atmosphere of the mine workings. Gas-bearing coal seams are usually mined underground. When driving preparatory workings, surfaces of outcropping of gas-bearing coal seams arise and gases that are in the seams under excessive pressure are released into the atmosphere of the mine workings. The most important gas-dynamic characteristic of this process is the rate of gas release, which represents the volume of gases released from a unit area of exposure of a coal seam per unit of time. A generalized law of resistance for gas filtration in a rock mass is recommended, and a fairly rigorous thermodynamic substantiation is given. It is shown that the densities of gas mass flows in accordance with the postulate of their linear relationship with the driving forces are determined by the Onsager relation. The results obtained and their discussion is presented. Mathematical models are proposed for engineering calculations of the dynamics of methane release from the outcropping surface of medium-thick coal seams. The error of the adopted approximations does not exceed 3%. The intensity of methane release is directly related to the planogram of work in the working face. Analysis of this dependence indicates that during the extraction cycle, methane release increases due to an increase in the area of the gas-release surface. The main conclusions are as follows: mathematical modeling of the processes of gas movement in a porous sorbing medium using approximate mathematical models representing linearized equations of mathematical physics; the regularities of the dynamics of the rate of gas release from the surface of the outcrop of a gas-bearing coal seam is the theoretical basis for the mathematical description of the process of gas release; the use of a linearized hyperbolic filtration equation most accurately describes the processes of methane release from the outcropping surface of mined coal seams.

Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.