Theory Of Equations Research Articles

A class of natural pinus koraiensis population system with time delay and diffusion term

In this paper, we consider the long-term sustainability of the northeast Korean pine. We propose a class of natural Korean pine population system with time delay and diffusion term. First, by analyzing the roots distribution of the characteristic equation, we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation. Second, we introduce lactation time delay into a population model with a diffusion term, based on stability theory of ordinary differential equation, norm form methods and center manifold theorem, the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given. Finally, some numerical simulations are given.

Blow-up data for a two-component Camassa-Holm system with high order nonlinearity

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H1-norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time.

Phase equilibrium of three-component liquid systems composed of water, alcohol, and sodium chloride studied by the reference interaction-site model integral equation theory.

A theoretical method for calculating the thermodynamic properties and phase equilibria of a binary liquid mixture using the reference interaction-site model (RISM) integral equation theory, which we had proposed recently, was extended to ternary liquid systems containing salt. A novel dielectric correction of the RISM theory for a mixture of solvents was also proposed. The theory was applied to mixtures composed of water, alcohol, and NaCl, where the alcohol was either methanol or ethanol. The decrease in NaCl solubility with increasing alcohol molar fractions in the solvent was calculated. In the ethanol system, the theory yielded salt-induced liquid-liquid phase separation, which was observed experimentally in a ternary mixture of water, 1-propanol, and NaCl. The phase diagram of the ternary system was determined theoretically.

Autoregressive conditional proportion: A multiplicative‐error model for (0,1)‐valued time series

We propose a multiplicative autoregressive conditional proportion (ARCP) model for (0,1)‐valued time series, in the spirit of GARCH (generalized autoregressive conditional heteroscedastic) and ACD (autoregressive conditional duration) models. In particular, our underlying process is defined as the product of a (0,1)‐valued independent and identically distributed (i.i.d.) sequence and the inverted conditional mean, which, in turn, depends on past reciprocal observations in such a way that is larger than unity. The probability structure of the model is studied in the context of the stochastic recurrence equation theory, while estimation of the model parameters is performed with the exponential quasi‐maximum likelihood estimator (EQMLE). The consistency and asymptotic normality of the EQMLE are both established under general regularity assumptions. Finally, the usefulness of our proposed model is illustrated with two real datasets.

О приложении качественной теории дифференциальных уравнений к одной задаче тепломассопереноса

О приложении качественной теории дифференциальных уравнений к одной задаче тепломассопереноса

Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study

Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations.

Parametric Analysis of Thick FGM Plates Based on 3D Thermo-Elasticity Theory: A Proper Generalized Decomposition Approach.

In the present work, the general and well-known model reduction technique, PGD (Proper Generalized Decomposition), is used for parametric analysis of thermo-elasticity of FGMs (Functionally Graded Materials). The FGMs have important applications in space technologies, especially when a part undergoes an extreme thermal environment. In the present work, material gradation is considered in one, two and three directions, and 3D heat transfer and theory of elasticity equations are solved to have an accurate temperature field and be able to consider all shear deformations. A parametric analysis of FGM materials is especially useful in material design and optimization. In the PGD technique, the field variables are separated to a set of univariate functions, and the high-dimensional governing equations reduce to a set of one-dimensional problems. Due to the curse of dimensionality, solving a high-dimensional parametric problem is considerably more computationally intensive than solving a set of one-dimensional problems. Therefore, the PGD makes it possible to handle high-dimensional problems efficiently. In the present work, some sample examples in 4D and 5D computational spaces are solved, and the results are presented.

Thermoelectric transport properties of metal phosphide XLiP (X = Sr,Ba)

Metal phosphides are stable and have excellent electrical characteristics, their high thermal conductivity has prevented them from being used as thermoelectric materials. In this paper, the thermoelectric transport properties of XLiP (X = Sr Ba) are investigated on the basis of first-principles calculations, Boltzmann transport equation and self-consistent phonon theory. In addition, we also consider the effect of quartic anharmonicity on the thermal transport properties and lattice dynamics of SrLiP and BaLiP. The strong anharmonicity of the two compounds make the lattice thermal conductivity decrease rapidly with the increase of temperature. At 300 K, the lattice thermal conductivity of SrLiP and BaLiP on the a(b)-axis is only 2.98 and 2.93 Wm−1K−1, respectively. Due to its excellent electron transport properties, it has greater conductivity in the a(b) axis. Finally, due to the energy pocket and anisotropy at the bottom of the conduction band, the n-type maximum ZT values of trapped SrLiP and BaLiP on the a(b) axis are 0.87 and 0.94 at 900 K, respectively. The high thermoelectric performance of both compounds encourages further studies on the thermoelectric properties of metal phosphides.

On the Solvability of a Periodic Problem for a System of Ordinary Differential Equations with the Main Positive Homogeneous Nonlinearity

We study the solvability of a periodic problem for a system of ordinary differential equations in which we separate the main nonlinear part that is positive homogeneous mapping (of order greater than unity), with the rest called a perturbation. It is proved that if the unperturbed system of equations has no nonzero bounded solutions, then the periodic problem is solvable under any perturbation if and only if the degree of the positive homogeneous mapping on the unit sphere is nonzero. The result obtained is of interest from the point of view of the application and development of methods of nonlinear analysis in the theory of differential and integral equations.

Non-equilibrium view of the amorphous solidification of liquids with competing interactions.

The interplay between short-range attractions and long-range repulsions (SALR) characterizes the so-called liquids with competing interactions, which are known to exhibit a variety of equilibrium and non-equilibrium phases. The theoretical description of the phenomenology associated with glassy or gel states in these systems has to take into account both the presence of thermodynamic instabilities (such as those defining the spinodal line and the so called λ line) and the limited capability to describe genuine non-equilibrium processes from first principles. Here, we report the first application of the non-equilibrium self-consistent generalized Langevin equation theory to the description of the dynamical arrest processes that occur in SALR systems after being instantaneously quenched into a state point in the regions of thermodynamic instability. The physical scenario predicted by this theory reveals an amazing interplay between the thermodynamically driven instabilities, favoring equilibrium macro- and micro-phase separation, and the kinetic arrest mechanisms, favoring non-equilibrium amorphous solidification of the liquid into an unexpected variety of glass and gel states.

A comparative study on surface tension, diffusion coefficient and shear viscosity coefficient of liquid transition metals

Surface tensions of liquid transition metals of the 3d, 4d and 5d series have been calculated using modified Mayer’s formula involving hard sphere (HS) interaction of liquids derived from first and second order approximations of Percus–Yevick (PY) solutions of isothermal compressibility. Effective HS diameter and packing fraction are key ingredients of this study. To determine them, we have used effective pair potential derived from both Bretonnet–Silbert (BS) pseudopotential model and many body potential obtained from embedded atom method (EAM) in conjunction with variational modified hypernetted chain (VMHNC) integral equation theory of liquid structure. Calculated values of surface tension are used to evaluate transport coefficients such as diffusion and shear viscosity coefficients. Comparing with experimental and simulated data, calculated results suggest that expression derived from second order approximation of PY solution with the combination of EAM potential and VMHNC theory works better than any other combinations for the concerned systems.

Solving the Schrödinger equation of a planar model H4 molecule

The Schrödinger equation of a planar model H4 molecule was solved with the free complement (FC) - local Schrödinger equation (LSE) theory using the direct local sampling (DLS) method. Although this molecule is often used to examine the intrinsic weakness of the Hartree-Fock related theory, the present FC-LSE-DLS theory successfully produced accurate solutions of the Schrödinger equation at any geometries of this molecule. We mapped the two-dimensional potential energy surfaces and revealed that the ground state has a dissociation channel to a couple of H2 molecules but the two excited states have local minimums constructing H4 molecule.

Three-Party Stochastic Evolutionary Game Analysis of Supply Chain Finance Based on Blockchain Technology

In the process of accounts receivable financing under supply chain finance, the phenomenon of accounts receivable forgery and default have caused great pressure on the supervision of financial institutions. We consider the integration of blockchain technology with a supply chain finance platform around the fraudulent default phenomenon in supply chain finance receivables financing and construct a three-party stochastic evolutionary game model among financial institutions, core enterprises, and Micro, Small, and Medium Enterprises (MSMEs). Firstly, we use Ito^’s stochastic differential equation theory to analyze the conditions for the stability of the behavior of game subjects. Secondly, we use numerical simulations to quantitatively analyze the impact of the regulatory strength of financial institutions, the information sharing of the blockchain platform, and the change of incentive parameters on the strategy choice of game subjects. Through the above analysis, we conclude that the information-sharing incentive coefficient promotes financial institutions to choose to connect to the blockchain platform, and the information-sharing risk coefficient and the regulatory intensity have the opposite effect on the blockchain platform construction. Meanwhile, the allocation of incentive shares has a significant influence on the core enterprises. Finally, we give priorities and directions for adjusting the relevant parameters to provide recommendations for financial institutions to regulate the financing process more effectively.

Mathematical problems of dynamical interaction of fluids and multiferroic solids

In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region is immersed in an inviscid fluid occupying an unbounded domain . In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain . The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

Quasilinear Parabolic Equations Associated with Semilinear Parabolic Equations

We formulate a quasilinear parabolic equation describing the behavior of the global-in-time solution to a semilinear parabolic equation. We study this equation in accordance with the blow-up and quenching patterns of the solution to the original semilinear parabolic equation. This quasilinear equation is new in the theory of partial differential equations and presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis and a viscosity solution.

THE CONNECTION BETWEEN SCHRÖDINGER EQUATION AND QUANTUM FIELD THEORY

The purpose of this research review is to examine the connection between the Schrödinger equation and quantum field theory. The method used in this review is a literature review of existing research on the topic. The results of the review indicate that the Schrödinger equation, which is a fundamental equation in quantum mechanics, can be derived from the principles of quantum field theory. This connection highlights the underlying unity of the two theories and helps to further our understanding of the behavior of subatomic particles. Additionally, it has been found that the Schrödinger equation is useful in describing the behavior of systems with a small number of particles, while quantum field theory is more appropriate for systems with a large number of particles. Research in this area has led to the development of new methods for solving the equation, such as the path integral approach, which provides a powerful tool for studying quantum systems with a large number of degrees of freedom. Additionally, the Schrödinger equation has led to the discovery of new symmetries and conservation laws, such as the AdS/CFT correspondence, which connects quantum field theory in anti-de Sitter space with a conformal field theory on the boundary of that space. Theoretical frameworks such as quantum field theory in curved spacetime and the holographic principle have also been developed to unify quantum mechanics and general relativity, and to understand the thermodynamic behavior of quantum systems and their behavior in extreme conditions. The study of quantum systems in non-equilibrium conditions is an active area of research and continues to yield new insights and discoveries. Overall, the research reviewed in this study suggests that the Schrödinger equation and quantum field theory are closely related, and that a deeper understanding of one can lead to a deeper understanding of the other. KEYWORDS: Schrödinger equation, quantum field theory, general relativity, quantum systems

Optimization method for a multi-parameters identification problem in degenerate parabolic equations

Abstract In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients.

Theoretical Kinetic Studies on Thermal Decomposition of Glycerol Trinitrate and Trimethylolethane Trinitrate in the Gas and Liquid Phases.

Glycerol trinitrate (NG) and trimethylolethane trinitrate (TMETN), as typical nitrate esters, are important energetic plasticizers in solid propellants. With the aid of high-precision quantum chemical calculations, the Rice-Ramsperger-Kassel-Marcus (RRKM)/master equation theory and the transition state theory have been employed to investigate the decomposition kinetics of NG and TMETN in the gas phase (over the temperature range of 300-1000 K and pressure range of 0.01-100 atm) and liquid phase (using water as the solvent). The continuum solvation model based on solute electron density (SMD) was used to describe the solvent effect. The thermal decomposition mechanism is closely relevant to the combustion properties of energetic materials. The results show that the RO-NO2 dissociation channel overwhelmingly favors other reaction pathways, including HONO elimination for the decomposition of NG and TMETN in both the gas phase and liquid phase. At 500 K and 1 atm, the rate coefficient of gas phase decomposition of TMETN is 5 times higher than that of NG. Nevertheless, the liquid phase decomposition of TMETN is a factor of 5835 slower than that of NG at 500 K. The solvation effect caused by vapor pressure and solubility can be used to justify such contradictions. Our calculations provide detailed mechanistic evidence for the initial kinetics of nitrate ester decomposition in both the gas phase and liquid phase, which is particularly valuable for understanding the multiphase decomposition behavior and building detailed kinetic models for nitrate ester.

Structures and freezing transitions in two-dimensional colloids with tunable repulsive interactions

We have used density functional theory of freezing and hypernetted chain(HNC) integral equation theory to investigate the effects of gradual softening of inverse-power repulsive interaction on the structure and fluid-triangular-solid freezing transition of a one component system of particles confined to a two-dimensional plane and interacting via core-softened potential. We have also performed Monte Carlo simulation to test the accuracy of the pair-structures obtained by HNC theory and to see the equilibrium configuration of particle in different thermodynamic conditions. The pair-structures are found to compare well with those obtained by MC results, particularly in high softness parameter region. The radial distribution given by HNC exhibits a non-monotonous behavior with the change in particle density in addition to the emergence of a new peak at shorter distance. In contrast to HNC, the RDF, obtained by MC simulation for high-softness parameter, display a splitting of the second peak at high density.

Novel Results on Persistence and Attractivity of Delayed Nicholson's Blowflies System with Patch Structure

This paper is concerned with the dynamic characteristics of a class of Nicholson's blowflies system with patch structure and multiple pairs of distinct time-varying delays. We aim to find the influence of the distinct time-varying delays in the same reproductive function on its asymptotic behavior. First, we derive the global existence, positiveness and uniform persistence of solutions for the addressed system. Then, by employing the theory of functional differential equations, the fluctuation lemma and the technique of differential inequalities, we build up some new delay-dependent criteria for the global attractivity of the positive equilibrium point vector, which does not possess the same components. In addition, we exam the effectiveness and feasibility of the theoretical achievements by some numerical simulations.