System Of Equations Research Articles

Lie symmetries for the shallow water magnetohydrodynamics equations in a rotating reference frame

Abstract We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition ∇ h B ≠ 0 for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential g and the Coriolis term f 0, related to the constant rotation of the reference frame. For four different cases, namely g = 0, f 0 = 0; g ≠ 0 , f 0 = 0; g = 0, f 0 ≠ 0; and g ≠ 0, f 0 ≠ 0 the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the L 10 = A 3 , 3 ⋊ A 2 , 1 ⨂ s A 5 , 34 a ; L 8 = A 2,1 ⋊ A 6,22; L 7 = A 3 , 5 ⋊ A 2 , 1 ⋊ A 2 , 1 ; and L 6 = A 3,5 ⋊ A 3,3respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where f 0 g ≠ 0, we derive all the invariants for the Adjoint action of the Lie algebra L 6 and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system.

О пондеромоторном перераспределении тяжелых ионов вдоль линии магнитного поля

We set up the problem of ponderomotive separation and acceleration of ions with different charge-to-mass ratios under the influence of Alfvén waves, which constantly exist in the magnetosphere in the form of geomagnetic pulsations. Formulas for partial ponderomotive forces acting on light and heavy (metallic) ions are derived. In the quasi-hydrodynamic approximation, a system of equations is obtained which describes the distribution of ions along magnetic field lines in Earth’s magnetosphere. The Clarke number, which characterizes plasma metallicity, is found to be maximum at a minimum magnetic field on the field line along which the Alfvén wave propagates. This leads to the accumulation of heavy ions at the top of the field line at the point of its intersection with the magnetic equator. The theoretical conclusions agree with satellite measurements of the distribution of heavy ions along field lines in Earth’s magnetosphere.

Ponderomotive redistribution of heavy ions along a magnetic field line

We set up the problem of ponderomotive separation and acceleration of ions with different charge-to-mass ratios under the influence of Alfvén waves, which constantly exist in the magnetosphere in the form of geomagnetic pulsations. Formulas for partial ponderomotive forces acting on light and heavy (metallic) ions are derived. In the quasi-hydrodynamic approximation, a system of equations is obtained which describes the distribution of ions along magnetic field lines in Earth’s magnetosphere. The Clarke number, which characterizes plasma metallicity, is found to be maximum at a minimum magnetic field on the field line along which the Alfvén wave propagates. This leads to the accumulation of heavy ions at the top of the field line at the point of its intersection with the magnetic equator. The theoretical conclusions agree with satellite measurements of the distribution of heavy ions along field lines in Earth’s magnetosphere.

Model-based conceptualization of thyroid hormone equilibrium via set point and stability behavior

The HPT complex, consisting of the hypothalamus, pituitary and thyroid, functions as a regulated system controlled by the respective hormones. This system maintains an intrinsic equilibrium, called the set point, which is unique to each individual. In order to optimize the treatment of thyroid patients and understand the dynamics of the system, a validated theoretical representation of this set point is required. Therefore, the research field of mathematical modeling of the HPT complex is significant as it provides insights into the interactions between hormones and the determination of this endogenous equilibrium. In literature, two mathematical approaches are presented for the theoretical determination of the set point in addition to a time-dependent model. The two approaches are based on the maximum curvature of the pituitary response function and the optimal gain factor in the representation of the HPT complex as a closed feedback system. This paper demonstrates that all hormone curves described by the model converge to the derived set point with increasing time. This result establishes a clear correlation between the physiological equilibrium described by the set point and the mathematical equilibrium with respect to autonomous systems of differential equations. It thus substantiates the validity of the theoretical set point approaches.

Quasi-Classical Models of Nonlinear Relaxation Polarization and Conductivity in Electric, Optoelectric, and Fiber Optic Elements Based on Materials with Ionic–Molecular Chemical Bonds

A generalized scientific review with elements of additions and clarifications has been carried out on the methods of theoretical research on the electrophysical properties of crystals with ionic–molecular chemical bonds (CIMBs). The main theoretical tools adopted are the methods of quasi-classical kinetic theory as applied to ionic subsystems relaxing in layered dielectrics (natural silicates, crystal hydrates, various types of ceramics, and perovskites) in an electric field. A universal (applicable for any CIMBs class crystals) nonlinear quasi-classical kinetic equation of theoretical and practical importance has been constructed. This equation describes, in complex with the Poisson equation, the mechanism of ion-relaxation polarization and conductivity in a wide range of polarizing field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). The physical model is based on a system of non-interacting ions (due to the low concentration in the crystal) moving in a one-dimensional, spatially periodic crystalline potential field, perturbed by an external electric field. The energy spectrum of ions is assumed to be continuous. Elements of quantum mechanical theory in a quasi-classical model are used to mathematically describe the influence of tunnel transitions of hydrogen ions (protons) during the interaction of proton and anion subsystems in hydrogen-bonded crystals (HBC) on the polarization of the dielectric in the region of nitrogen (50–100 K) and helium (1–10 K) temperatures. The mathematical model is based on the solution of a system of nonlinear Fokker-Planck and Poisson equations, solved by perturbation theory methods (via expanding solutions into infinite power series in a small dimensionless parameter). Theoretical frequency and temperature spectra of the dielectric loss tangent were constructed and analyzed, the molecular parameters of relaxers were calculated, and the physical nature of the maxima of the experimental temperature spectra of dielectric losses for a number of HBC crystals was discovered. The low-temperature maximum, which is caused by the quantum tunneling of protons and is absent in the experimental spectra, was theoretically calculated and investigated. The most effective areas of scientific and technical application of the theoretical results obtained were identified. The application of the equations and recurrent formulas of the constructed model to the study of nonlinear optical effects in elements of laser technologies and nonlinear radio wave effects in elements of microwave signal control systems is of the greatest interest.

Functional Central Limits Theorem for epidemic model with varying infectivity and waning immunity

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection (generalized SIS model). Speci cally, after each infection, the infectivity is a random function of the time elapsed since the infection, and each recovered individual loses immunity gradually (equivalently, becomes gradually susceptible) after some time according to a random susceptibility function. The epidemic dynamics is described by the average infectivity and susceptibility processes in the population together with the numbers of infected and susceptible/uninfected individuals. In [12], a functional law of large numbers (FLLN) is proved as the population size goes to in nity, and asymptotic endemic behaviors are also studied. In this paper, we prove a functional central limit theorem (FCLT) for the stochastic ™uctuations of the epidemic dynamics around the FLLN limit. The FCLT limit for the aggregate infectivity and susceptibility processes is given by a system of stochastic non-linear integral equation driven by a two-dimensional Gaussian process.

Legendre wavelet based numerical method for the solution of marangoni convective flow of dusty ree-erying fluid over a riga plate with activation energy

AbstractLegendre wavelet based numerical method has been developed for the solution of the system of non-linear differential equations arising in the study of a Marangoni convective flow of dusty Ree-Erying fluid over a Riga plate in Darcy-Forchheimer medium with Soret, Dufour, non-linear radiation, activation energy effects and entropy generation analysis. The proposed method is validated by comparing the solutions of the test problem with the Bernoulli wavelet based numerical method and the exact solution and by comparing the results of the considered dusty fluid problem for some fixed parameters with the previously published results. The effects of various factors on the velocity, thermal, concentration attributes are studied by using the proposed method. It is observed that the Ree-Erying fluid parameter and Marangoni ratio parameter escalates the velocity and declines the temperature and concentration. The opposite behaviour is observed for Deborah number. The concentration declines for higher chemical reaction parameter and improves for activation energy. Both entropy generation and Bejan number escalates for radiation parameter and diffusion parameter. The proposed method is reliable, fast computable and powerful tool to solve the differential equations.

Exploring New Algorithms for Molecular Vibrational Spectroscopy Using Physics-Informed Program Synthesis.

Inductive program synthesis (PS) has recently begun to emerge as a useful new approach to automatically generate algorithms in quantum chemistry, as demonstrated in recent applications to the vibrational Schrödinger equation for simple model systems with one or two degrees-of-freedom. Here, we report a new physics-informed approach to inductive PS that is more conducive to the generation of discrete variable representation algorithms for real molecular systems. The new framework ensures separability of the kinetic and potential operators and does not require an exact solution to compare synthesized algorithmic predictions with. Algorithms with a tridiagonal matrix structure are generated via a variational-based stochastic optimization procedure. Crucially, through an extensive testing procedure, we demonstrate that variationally synthesized algorithms perform just as well as those generated using a target function. Assuming a direct product representation of normal coordinates, these algorithms are applied to three triatomic molecules. In total, we identify a set of seven PS algorithms that accurately reproduce the vibrational spectra of H2O, NO2, and SO2, as predicted by Colbert-Miller and sine-DVR algorithms.

Numerical Model of Temperature-Filtration Regime of Earth Dam in Harsh Climatic Conditions

The article addresses the issue of numerical modeling of the process of forming the temperature regime of earth dams, along with their foundations, built and operated in permafrost conditions. A large number of such structures have been constructed in the permafrost regions of the Earth to meet the needs of industry and population. The paper outlines the key principles of designing and constructing such structures. These principles were developed based on years of experience in hydrotechnical construction. Failure to follow these principles leads to structural failures, as confirmed by the presented statistics on accidents. It is essential to ensure the appropriate thermal condition of the structure and its foundation, either frozen or thawed. An unplanned transition of soils from one state to another may lead to an emergency situation. Temperature changes can cause phase transitions of water from liquid to solid (ice), which also affects the formation of the structure’s regime. Numerical methods of calculation allow for the most comprehensive consideration of the influencing factors and processes. The article presents the results of numerical modeling of the filtration-temperature regime of an earth dam with a foundation in permafrost conditions, using two computational programs. The first is based on a locally variational approach (Termic, authored by the researchers), while the second uses a classical linear equation system solution (PLAXIS 2D 2022 software). A comparison of the results obtained from both programs showed good qualitative and quantitative consistency. Under the influence of seepage flow, the zone of frozen ground degradation is spreading in the lower part of the earth dam and its foundation. By September of the 27th year of operation, the thawed ground zone reaches approximately the middle of the structure at the base. The temperature values along the screen axis at the base of the structure are +1.2 °C (according to the Termic program—ver. 1.1) and +1.06 °C (according to PLAXIS 2D PC). Recommendations and future research directions on this topic are also formulated.

Numerical Analysis of the Influence of 2D Dispersion Parameters on the Spread of Pollutants in the Coastal Zone

The transport of pollutants with flowing waters is one of the most common processes in the natural environment. In general, this process is described by a system of differential equations, including the continuity equation, dynamic equations, pollutant transport equations and equations of state. For the analyzed problem of pollutant migration in wide rivers and the coastal zone, a two-dimensional model is particularly useful because the velocity and mass concentration profile is vertically averaged. In this model, taking into account the dispersion flux leads to appropriate equations, and the dispersion process is described by the dispersion tensor. Due to the transverse isotropy of the dispersion process, the coordinates of this tensor are expressed in terms of local dispersion coefficients along the direction of the velocity and in the direction perpendicular to it. Commonly used methods for determining mass dispersion coefficients refer to a gradient velocity profile, typical for rivers. However, in the coastal zone, the velocity profile changes from gradient to drift when shear stresses on the surface caused by the wind begin to dominate. The drift profile also occurs in estuaries, where there is a difference in the density of fresh and salt water. This paper analyzes the numerical solution of the two-dimensional dispersion equations in the coastal zone for the dispersion coefficients adopted for the gradient and drift velocity profiles and then assesses how this affects the final result. Four typical scenarios of pollutant migration in the coastal zone of the Bay of Puck are presented. The calculated dispersion coefficients differ significantly depending on the adopted velocity profile: for the gradient, DLG = 0.17 [m2/s], and for the drift, DLD = 89.94 [m2/s].

On generalized asymmetric harmonic oscillator with quadratic nonlinearity within fractional variational principles

This work studies the nonlinear fractional dynamics of asymmetric harmonic oscillators. The classical description of the physical system is generalized using the principles of fractional variational analysis. As a system of two-coupled fractional differential equations with a quadratic nonlinear component, the fractional Euler–Lagrange equations of the motion of the corresponding system are obtained. The Adams–Bashforth predictor-corrector numerical approach is used to approximate the system’s outcomes, which are then simulated comparatively with respect to various model parameter values, including mass, linear and quadratic nonlinear stiffness, and the order of the fractional derivative. The simulations provided the possibility of investigating various dynamical behaviours within the same physical model that is generalized by the use of fractional operators.

Propagation of Non-stationary Skew-Symmetric Waves from a Spherical Cavity in a Porous-elastic Half-space

Problems of propagation and diffraction of non-stationary waves in porous-elastic mediums are of great theoretical and practical importance in such fields of science and technology as geophysics, seismic exploration of minerals, seismic resistance of structures, and many others. The work considers the problem of propagation of non-stationary skew-symmetric waves from a spherical cavity in a porous-elastic half-space saturated with liquid. To solve the problem, the integral Laplace transform in dimensionless time and the method of incomplete separation of variables were used. In the space of Laplace images in time, known and unknown functions are expanded into Gegenbauer polynomials. The problem is reduced to solving an infinite system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential series. Recurrence relations for the coefficients of the series and initial conditions for them are obtained, which makes it possible to obtain a solution to the infinite system without using the reduction method. Recurrence relations make it possible to determine the coefficients of a series in the form of rational functions, which makes it possible to calculate their originals using the theory of residues. In image space, formulas are obtained for the coefficients of the series of components of the displacement vector and stress tensor. Numerical experiments were carried out, the results of which are presented in the form of graphs. The results obtained can be used in geophysics, seismology, and design organizations during the construction of structures, as well as in the design of underground reservoirs.

Accurate Dynamic Analysis Method of Cable-Damper System Based on Dynamic Stiffness Method

To suppress large vibrations of the cable in cable-stayed bridges, it is common to install transverse dampers near the end of the cable. This paper focuses on the cable-damper system; based on the dynamic stiffness method, an accurate dynamic analysis method considering cable parameters, damper parameters, and cable forces is proposed. First, a mechanical analysis model is established which is closer to the cable with a transverse damper installed in the bridge. The model considers the cable bending stiffness, sag, inclination angle, cable force, damper stiffness, damping coefficient, and damper installation height. Then, the characteristic frequency equation of the cable-damper system is established, and a solution method that combines the initial value method and Newton–Raphson method is proposed. This method is confirmed to provide more accurate frequency analysis for the cable-damper system. Finally, using this method, the effect of the damper parameters on the dynamic characteristics of the system is investigated.

Research on the characteristics and protection of water hammer in long-distance dual-pipe water supply systems.

Hydraulic transients in long-distance pressurized water pipelines significantly impact their normal operation. This study develops a one-dimensional mathematical model for pressurized water pipelines using the method of characteristics and incorporates water hammer equations for dual-pipeline systems. The model is validated with experimental data, and simulations are conducted under real engineering conditions, focusing on valve closure operations. The analysis examines the transient responses for varying valve closure times (T) and the effect of installing surge tanks. Results show that increasing valve closure time and installing surge tanks both mitigate water hammer impacts. Specifically, when valve closure time exceeds 300 seconds, surge tanks reduce maximum pressure below the pipeline's tolerance (Pmax) and decrease the number of nodes experiencing damaging negative pressures. This model effectively simulates hydraulic transients in dual-pipeline systems and provides a foundation for developing protective measures for pipeline operations.

Analytical Investigation of a Dirac Particle Dynamics in a Hyperbolic Magnetic Field with Position-Dependent Mass: Exact Solutions and Ladder Operators

Abstract This study involves analytically scrutinizing a system consisting of a Dirac particle in a certain external magnetic field. In the case where the external magnetic field depends on the position in a hyperbolic form and the particle has a hyperbolic effective mass (position dependent mass), the two-component Dirac equation is used to algebraically obtain the polynomial eigenfunctions and exact energy-eigenvalues of the system. By means of these exact solutions, the ladder operators of the system are obtained. Thus, those who handle the DKP equation for the same system may benefit from these operators.

Investigation on the effect of probe stiffness on surface roughness in single crystal copper nanomachining process

Surface roughness is one of the most important features in nanoscale machining. In this paper, molecular dynamics simulations have been performed to investigate the combination effect of the tool stiffness, machining parameters and the tool tip geometry on the surface roughness of the single-crystal copper workpiece in the nano machining process by diamond tool. To consider the effects of probe beam dynamics in calculating the workpiece surface roughness, the probe cantilever is modeled as an equivalent spring-mass system that the tip mass is added to its end. The differential equation of the probe motion is solved simultaneously with the molecular dynamics equations for the tool tip and the workpiece and the effect of probe stiffness on the surface roughness is obtained. The result shows that a higher cutting speed and depth of cut and lower tool radius lead to a larger effect of probe stiffness on the surface roughness of the workpiece. Also, for the negative rake angles, considering the tool as rigid creates a large error in determining the surface roughness.

A fractional-order model for the effects of expansions in hospitals on the applicability of performance certificate programme

This study explores the impact of the Leadership in Energy and Environmental Design (LEED) certification system on healthcare services in private hospitals in North Cyprus using a fractional-order system of equations. The Near East Hospital (NEH) earned Gold-level certification with 79 points, while Burhan Nalbantoğlu Hospital (BNH) obtained 31 points. Through sensitivity analysis of indoor air quality and sustainable land criteria, the study demonstrates that hospitals adopting LEED certification provide superior service quality. The findings suggest that government hospitals could benefit from aligning with green certification criteria, potentially improving patient outcomes. Data collected from NEH and BNH support a positive relationship between green certification and increased patient and employee satisfaction, with significant evidence ( p < 0.05 ) distinguishing LEED-certified hospitals from non-certified ones. The study also evaluates how hospital expansions affect the applicability of certification programs and the management of COVID-19, focusing on criteria such as sustainable sites, water efficiency, and indoor environmental quality. The analysis provides insights for healthcare administrators and policymakers seeking to enhance hospital infrastructure to meet certification standards, improve healthcare services, and better manage future pandemics. The research underscores the importance of green certification in improving patient care, employee satisfaction, and overall hospital performance.

VARIATIONAL-DIFFERENCE SOLUTION OF DEFORMATION AND BUCKLING PROBLEMS OF ELASTOPLASTIC SHELLS OF REVOLUTION WITH ELASTIC FILLER UNDER COMBINED QUASI-STATIC AND DYNAMIC AXISYMMETRIC LOADINGS

The paper suggests a formulation and method for a numerical solution of deformation and buckling of elastoplastic shells of revolution with elastic filler under quasi-static and dynamic loadings. The problem is solved in a two-dimensional plane or generalized axisymmetric formulation with torsion. The governing system of equations is written in a Cartesian or cylindrical coordinate system. Modeling of deformation of an elastic-plastic shell is carried out based on the hypotheses of the theory of shells of the Timoshenko type, taking into account geometric nonlinearities. Kinematic relations are written in velocities and formulated in the metric of the current state. The elastoplastic properties of the shell are described by the flow theory with nonlinear isotropic hardening. Filler modeling is based on continuum mechanics hypotheses. The filler material is assumed to be linearly elastic. The variational equations of motion of structural elements (both shells and filler) are reduced from the three-dimensional equation of the balance of virtual powers of the work of continuum mechanics taking into account the accepted hypotheses of the theory of shells or a flat deformed state or generalized axisymmetric deformation with torsion. The modeling of the contact interaction between the shell and the filler is based on the condition of nonpenetration along the normal and slippage along the tangential. The finite-difference method and an explicit time integration scheme of the cross type are used to solve the defining system of equations. Approbation of the technique was carried out on the problem of buckling of a steel cylindrical shell with an elastic filler under quasi-static and dynamic compression by an external pressure that linearly increases with time. The results of the numerical study are compared with calculations performed using two other approaches developed earlier by the authors. The first approach is based on full-scale modeling of the process of deformation of the shell and filler within the framework of continuum mechanics. In the second approach, a simplified formulation is used, in which the deformation of the shell is modeled according to the hypotheses of the theory of non-sloping shells of the Timoshenko type taking into account geometric nonlinearities, and the filler is modeled according to the Winkler foundation hypothesis. The developed approaches make it possible to model the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates taking into account geometric shape imperfections, to study buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static combined loadings in plane and axisymmetric deformations.

From Collective Mind to Communication

"Collective mind" is introduced as a set of simple intelligent units (say, neurons, or interacting agents) that can communicate by exchanging information without explicit global control. Incomplete information is compensated for by a sequence of random guesses symmetrically distributed around expectations with prescribed variances. Both the expectations and variances are the invariants characterizing the whole class of agents. These invariants are stored as parameters of the collective mind, while they contribute to dynamical formalism of the agents' evolution, and in particular, to the reflective chains of their nested abstract images of the selves and nonselves. The proposed model consists of the system of stochastic differential equations in the Langevin form to represent motor dynamics, and the corresponding Fokker-Planck equation to represent mental dynamics. The main departure of this model from newtonian and statistical physics is due to feedback from the mental to the motor dynamics, which makes the Fokker-Planck equation nonlinear. Interpretations of this model from mathematical, physical, biological, and psychological viewpoints are discussed. The model is illustrated by the dynamics of a dialog.

VOLUME SINGULAR INTEGRAL EQUATIONS FOR PROBLEMS OF LOW-FREQUENCY SCATTERING OF ELECTROMAGNETIC WAVES IN ANISOTROPIC STRUCTURES

This paper deals with volume singular integral equations describing the problems of low-frequency scattering of electromagnetic waves in bounded three-dimensional anisotropic structures. The spectrum of integral operators is studied. The domain of the operator spectrum on the complex plane for the low-frequency case is presented explicitly. The generalized method of simple iteration is described, for application of which it is necessary to know the area of the operator spectrum on the complex plane. The collocation method on a uniform grid is used to discretize the integral equations. This allows, using a fast discrete Fourier transform, to construct an algorithm for fast multiplication of the matrix of a system of linear equations by a vector. The results of numerical solution of the considered problems are given.