Equations Of Mathematical Physics Research Articles

Soil Temperature Controls the Month‐To‐Month Lead‐Lag Correlations of Near‐Surface Air Temperatures in the Middle and Lower Reaches of the Yangtze River Basin

AbstractThe relationships between the variables at different times are crucial in climate prediction. This study explores the lead‐lag correlations between monthly near‐surface air temperatures, and analyzes the associated influencing factors and processes using mathematical physics equations and statistical methods, based on monthly observations and ERA5 reanalysis data spanning from 1979 to 2013. The results reveal high lead‐lag correlations of near‐surface air temperatures in the middle and lower reaches of the Yangtze River basin, characterized by high frequency and intensity of heatwaves. The lead‐lag correlations between near‐surface temperatures are mainly due to the lead‐lag relationships between soil temperatures (STs). Current and accumulated net solar radiation, current latent heat flux and antecedent ST of several or many months ago are the main factors affecting the lead‐lag relationships between STs. The processes associated with lead‐lag correlations between STs are triggered by antecedent ST, concurrent and accumulated influence factors, respectively. The anomalies of antecedent ST can lead to the anomalies of current one by the persistence of anomaly signals in soil. The effects of accumulated or concurrent influential variables on the lead‐lag correlations between STs are attributed to the signals similar to antecedent STs in these influential variables, and the signals may arise from the response of atmosphere to land or ocean forcing. Moreover, the use of reanalysis data increases the uncertainty of the results. The study reveals influential factors and possible physical processes associated with lead‐lag correlations between near‐surface temperatures, which is helpful for understanding lead‐lag relationship between land surface and the atmosphere.

Analytical study of nonlinear models using a modified Schrödinger’s equation and logarithmic transformation

Optical solitons are special waves that maintain their shape while traveling. Hence, solitary optical waves have great significance when it comes to representing pulse propagation through nonlinear partial differential equations. In our research, we have uncovered unique solutions for the formation of solitary wave patterns in the equation of the Heisenberg ferromagnetic spin chain (HFSC). This equation aptly describes the behavior of electromagnetic waves in contemporary magnetism. By employing innovative techniques based on logarithmic transformations and analytical methods, we have obtained various solution forms expressed in a concise manner using elementary functions. We have tested the correctness of these solutions by substituting them directly into the main equation. These recent discoveries provide novel perspectives on the complex realm of solitons as depicted by this model. Moreover, since the model finds applications in fiber optic communications, fluid dynamics, and other fields, the implications of these discoveries are broad. The utilized methods stand out for being simple, reliable, and capable of creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. The research findings presented here showcase the effectiveness of the applied methods in reliably studying nonlinear phenomena in the HFSC equation and other nonlinear problems in mathematical physics. By utilizing these tools, scholars have the opportunity to expand and enhance their understanding of the complex mathematical frameworks underlying real-world problems.

Analytic and algebraic properties of dispersion relations (Bloch varieties) and Fermi surfaces. What is known and unknown

The article surveys the known results and conjectures about the analytic properties of dispersion relations and Fermi surfaces for periodic equations of mathematical physics and their spectral incarnations.

Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics

Abstract In this article, we investigated the Landau–Ginzburg–Higgs (LGH) equation, focusing on the analysis of isolated soliton solutions and their stability. To compute the isolated soliton solutions, we used the advanced auxiliary equation (AAE) approach, which has proven to be a powerful and efficient method for extracting soliton solutions in various nonlinear partial differential equations (NLPDEs). We provided a detailed explanation, both graphically and physically, of the obtained soliton solutions in this article. Furthermore, we used the linear stability technique to conduct a stability analysis of the LGH equation. Additionally, we studied the bifurcation and stability of the equilibria and performed phase plane analysis of the model. We also provided a discussion on the comparisons between the AAE method and two other well-known approaches: the generalized Kudryashov method and the improved Bernoulli sub-equation function method. The application of the AAE approach in this study demonstrates its effectiveness and capability in analysing and extracting soliton solutions in NLPDEs.

Implementation of the HHL Algorithm for Solving the Poisson Equation on Quantum Simulators

The Poisson equation is a fundamental equation of mathematical physics that describes the potential distribution in static fields. Solving the Poisson equation on a grid is computationally intensive and can be challenging for large grids. In recent years, quantum computing has emerged as a potential approach to solving the Poisson equation more efficiently. This article uses quantum algorithms, particularly the Harrow–Hassidim–Lloyd (HHL) algorithm, to solve the 2D Poisson equation. This algorithm can solve systems of equations faster than classical algorithms when the matrix A is sparse. The main idea is to use a quantum algorithm to transform the state vector encoding the solution of a system of equations into a superposition of states corresponding to the significant components of this solution. This superposition is measured to obtain the solution of the system of equations. The article also presents the materials and methods used to solve the Poisson equation using the HHL algorithm and provides a quantum circuit diagram. The results demonstrate the low error rate of the quantum algorithm when solving the Poisson equation.

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Viscoelastic rod using the generalized finite difference method

The finite difference method is quite extensively used to obtain the approximate solutions of many equations of mathematical physics. In this study, the precise algorithm in the time domain is combined with the generalized finite difference method to solve dynamic viscoelasticity problems. The numerical results obtained are satisfactory, and they are presented together with finite difference and finite element solutions.

“Computational Mathematics and Mathematical Physics”—Editorial I (2021–2023)

Based on the papers published in the Special Issue of the scientific journal Axioms, here we present the Editorial Article “Computational Mathematics and Mathematical Physics”, the main topics of which include both fundamental and applied research in computational mathematics and differential equations of mathematical physics [...]

Study of mechanical properties of H-shaped beam-column joints considering panel zone

In order to clarify the influence of panel zone (PZ) strength on the mechanical properties of each component (column, beam and PZ) of H-shaped steel beam-column joint, the equations of mathematical physics for the mechanical relationship of framing member end considering panel zone were established, and five groups of T-shaped beam-column joint specimens with different strength are tested to study the influence of panel zone to beam strength ratio (PBSR) on the deformation performance, bearing capacity and stiffness of the specimens. Through the finite element analysis, the effect of PBSR and weld access hole on the deformation of joints, the shear and flexural capacity of beams is discussed. Finally, based on the analysis of deformation and bearing capacity of the specimens and models, the reasonable value of PBSR is proposed.

KdV-type equations in projective Gevrey spaces

We prove well-posedness of the Cauchy problem for a class of third order quasilinear evolution equations with variable coefficients in projective Gevrey spaces. The class considered is connected with several equations in Mathematical Physics as the KdV and KdVB equation and some of their many generalizations.

To the Problem of Discontinuous Solutions in Applied Mathematics

This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data.

Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. The relevance of these operators and the possibility to be involved in the modeling of an important class of real phenomena is once again revealed by their applications. The use of certain variational methods facilitates the complete solution of the problem using appropriate numerical methods and computational algorithms. Some theoretical results are involved to complete the solutions for a sequence of models issued from real phenomena drawing.

Exact traveling wave solutions for the (2+1)-dimensional double sine-Gordon equation using direct integral method

In this paper, the direct integral method and complete discrimination system for quartic polynomial in standard form are applied to the (2+1)-dimensional double sine-Gordon equation. All possible exact traveling wave solutions of different types are successfully obtained from nine subcases. These exact solutions include kink solutions, and periodic solutions. Different from traditional methods (the Jacobi elliptic function method, the Exp-function method, extended Tanh method and the algebraic mapping method, etc.), the direct integral method described in this work is very natural and simple, and gives a new idea to obtain exact solutions for the mathematical physics equations considered.

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER

The physical model of many electrical objects is a hollow cylinder of finite length. As a basis for constructing analytical models of electrical machines, linear equations of mathematical physics are used. They are the solution of the Laplace three-dimensional partial differential equation, which is widely used in analytical calculations.
 
 The purpose of the study is to solve the Laplace three-dimensional differential equation for a hollow cylinder of finite length, which can be adapted for the electromagnetic calculation of electromechanical devices with cylindrical active parts.
 
 Materials and methods. To solve the Laplace equation in a cylindrical coordinate system, the Fourier variable separation method was used. To obtain a non-trivial solution of the equation, eigenfunctions of the Sturm–Liouville problem were used. Dirichlet, Neumann boundary value problems of the mixed type for hollow cylinders are adapted to the electromagnetic calculation of electromechanical devices having cylindrical active parts.
 
 The results of the study. The Laplace equation given in a cylindrical coordinate system is considered, on the basis of which the Sturm–Liouville equation with zero initial values is compiled to find eigenfunctions. The complete solution of the Lapalace equation with given boundary conditions is obtained as the sum of the solutions of two separate Dirichlet problems with different boundary conditions.
 
 Findings. The obtained analytical expression can be used as a mathematical basis for constructing three-dimensional analytical models of electrical machines with cylindrical active parts and carrying out electromagnetic calculations of the corresponding electromechanical devices.

Generalized (G’ / G) – expansion method for the loaded shallow water wave equation

This article is devoted to finding solutions for the traveling wave of the loaded wave equation in shallow water. One of the approaches to finding solutions by the expansion method (G / G) is given, which is one of the most effective ways to obtain solutions. When parameters are taken as special values, solitary waves are also derived from traveling waves. Solutions for the traveling wave are expressed by hyperbolic and trigonometric functions. This method is easy to implement using well-known software packages that allow solving complex nonlinear evolutionary equations of mathematical physics.

Solutions for Some Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 1

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. This article details three ways to use surjectivity results for a special type of operator involving the duality mapping and a Nemytskii operator, three methods starting from Ekeland’s variational principle and, lastly, one with a generalized variational principle to solve or describe the above-mentioned solutions. The relevance of these operators and the possibility of their involvement in the modeling of an important class of real phenomena determined the author to group these seven procedures together, presented in detail, followed by many applications, accompanied by a general overview of specialty domains. The use of certain variational methods facilitates the complete solution of the problem via appropriate numerical methods and computational algorithms. The exposure of the sequence of theoretical results, together with their demonstration in as much detail as possible has been fulfilled as an opportunity for the complete development of these topics.

Compatible extension of the [formula omitted]-expansion approach for equations with conformable derivative

In this study, the compatible extensions of the (G′/G)-expansion approach and the generalized (G′/G)-expansion scheme are proposed to generate scores of radical closed-form solutions of nonlinear fractional evolution equations. The originality and improvements of the extensions are confirmed by their application to the fractional space-time paired Burgers equations. The application of the proposed extensions highlights their effectiveness by providing dissimilar solutions for assorted physical forms in nonlinear science. In order to explain some of the wave solutions geometrically, we represent them as two- and three-dimensional graphs. The results demonstrate that the techniques presented in this study are effective and straightforward ways to address a variety of equations in mathematical physics with conformable derivative.

THERMAL PROCESSES OF FIRE DETECTORS’ HEAT-RESISTANT SENSITIVE ELEMENTS

The purpose of the article. The article deals with thermal processes in the heat-resistant sensitive element of fire detectors when an electric current flows through it. Such processes are described by an inhomogeneous equation of mathematical physics with boundary conditions of the third kind. Methodology. The Hankel integral transformation is used to solve the differential equation that describes the thermal processes in the heat-resistant sensitive element of fire detectors. As a result of the transformations, a general expression was obtained for the temperature of the heat-resistant sensitive element of fire detectors under the condition that an electric current flows through it without restrictions on its parameters. This expression is represented as a series of Bessel functions. Results. The results of scientific research confirmed that due to the small size of the fire detectors’ sensitive element, the expression for its temperature is presented in the form of an average value over the sensitive element volume. Scientific novelty. For the first time, a general expression for the temperature of the heat-resistant sensitive element of fire detectors is obtained, which allows obtaining its main technical characteristics, and also serves as a basis for moving to other mathematical descriptions. Practical value. Also, using the integral Laplace transformation, an expression for the transfer function of the fire detectors’ heat-resistant sensitive element was obtained and it was confirmed that with an error of no more than 4,6 %, it is described by the transfer function of the inertial link.

Blow-up criteria of the simplified Ericksen–Leslie system

In this paper, we establish scaling invariant blow-up criteria for a classical solution to the simplified Ericksen–Leslie system in terms of the positive part of the second eigenvalue of the strain matrix and orientation field in mixed-norm Lebesgue spaces. Our result may be also regarded as an extension or improvement of the corresponding results obatined by Neustupa and Penel (Trends in Partial Differential Equations of Mathematical Physics, pp. 197–212, 2005), Miller (Arch. Ration. Mech. Anal. 235(1):99–139, 2020) and Huang and Wang (Commun. Partial Differ. Equ. 37(5):875–884, 2012).