Multidimensional Equations Research Articles

The Use of Multidimensional Nomial Logistic Model and Structural Equation Model in the Validation of the 14-Item Health-Literacy Scale in Chinese Patients Living with Type 2 Diabetes.

To evaluate the psychometric properties of the 14-item health literacy scale (HL-14) in patients living with type 2 diabetes mellitus (T2DM) in clinical setting. Cross-sectional study using item response theory and structural equation modeling (SEM) for testing the item difficulty and three dimensional-HL configurations was adopted in this study. Chinese patients living with T2DM admitted to endocrinology department of Huadong hospital were evaluated by the HL-14 including communication, functional and critical health literacy from August to December 2021. The multidimensional random coefficients multinomial logistic model indicated the difficulty settings of the scale are appropriate for the study populations, and differential item functioning was not observed for sex in the study. SEM demonstrated that the three-dimensional configuration of the scale is good in the study population (x2/df=2.698, Comparative Fit Index = 0.965, Root Mean Square Error of Approximation = 0.076, standard residual mean root = 0.042). The HL-14 scale is a reliable and valid measurement, which can perform equitably across sex in evaluating the health literacy in Chinese patients living with T2DM. Moreover, the scale may help fill the gaps of multidimensional health literacy assessment and rapid screening of health literacy ability for clinical practice.

An advanced numerical scheme for multi-dimensional stochastic Kolmogorov equations with superlinear coefficients

This work develops a novel approximation for a class of superlinear stochastic Kolmogorov equations with positive global solutions. On the one hand, most existing explicit methods that work for the superlinear stochastic differential equations (SDEs), e.g. various modified Euler–Maruyama (EM) methods, fail to preserve positivity of the solution. On the other hand, methods that preserve positivity are mostly implicit, or fail to cope with the multi-dimensional scenario. This work aims to construct an advanced numerical method which is not only naturally structure preserving but also cost effective. A strong convergence framework is then developed with an almost optimal convergence rate of order arbitrarily close to 1/2. To make the arguments concise, we elaborate our theory with the generalised stochastic Lotka–Volterra model, though the method is applicable to a wide bunch of multi-dimensional superlinear stochastic Kolmogorov systems in various fields including finance and epidemiology.

Using Homotopy Perturbation and Analysis Methods for Solving Different-dimensions Fractional Analytical Equations

The aim of the research, we extended the one-dimensional to multi-dimensional, we applied the homotopy perturbation and analysis methods to solve Volterra integral equations and to obtain approximate analytical solutions of systems of the second kind multi-dimensional Volterra integral equations. We proved the convergence of the homotopy analysis method (HAM). The HAM solutions contained an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method. The results indicate that this method is efficient for the linear and non - linear models with the dissipative terms.

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).

Efficient energy structure-preserving schemes for three-dimensional Maxwell's equations

Two energy structure-preserving schemes are proposed for Maxwell's equations in three dimensions. The Maxwell's equations are split into several local one-dimensional subproblems which successfully reduces the scale of algebraic equations to be solved. To improve the convergence rate in space and to keep the sparsity of the resulting algebraic equations, the spatial derivatives are approximated by high order compact method. Some key indicators, such as stability, energy structure-preserving and convergence of the schemes are investigated. To make the theoretical more persuasive, some numerical examples are shown. Numerical results are accord with the theoretical results. This provides a practical approach to construct efficient structure-preserving algorithms multidimensional Maxwell's equations.

Enthalpy effect on the kinetics of concurrent nucleation and chemical aging of aqueous organic aerosols: The stage of thermal relaxation

The size and composition distribution of an ensemble of aqueous organic droplets, evolving via nucleation and concomitant chemical aging, may be affected by the latent heat of condensation and enthalpy of heterogeneous chemical reactions, so the temperature of the droplet may deviate from the air temperature and thus become an independent variable of its state (additional to its size and composition variables). Using the formalism of the classical nucleation theory, we derive a partial differential equation for the temporal evolution of the distribution of an ensemble of such droplets with respect to all their variables of state via Taylor series expansions of the corresponding multidimensional discrete equation of balance, describing the material and heat exchange between droplets and air. The resulting kinetic equation goes beyond the framework of the Fokker-Planck approximation with respect to the temperature variable. A hierarchy of time scales of nonisothermal nucleation and concomitant chemical aging of aqueous organic aerosols is established and an analytical description of their thermal relaxation stage is developed, allowing one to estimate the characteristic time of the establishment of the equilibrium distribution of aerosol particles with respect to their temperatures. Theoretical results are illustrated with numerical calculations for the concurrent nucleation and chemical aging of model aqueous hydrophilic-hydrophobic organic aerosols in air.

New Aspects on the Solvability of a Multidimensional Functional Integral Equation with Multivalued Feedback Control

The current study demonstrates the existence of solutions to a multidimensional functional integral equation with multivalued feedback. We seek solutions for the multidimensional functional problem that is defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in BC(R+). Also, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions to that multidimensional functional equation. Additionally, we give an example and some particular cases to illustrate our outcomes.

Modified Boundary Knot Method for Multi-Dimensional Harmonic-Type Equations

This paper presents the modified boundary knot method (MBKM) for solving the homogeneous harmonic type boundary value problems (BVPs). Since no non-singular general solutions are applicable for harmonic-type equations, the general solutions of Helmholtz-type operator with a free parameter [Formula: see text] can be used to approximate the solutions of these problems by adjusting the [Formula: see text]. Compared with the classical boundary knot method (BKM) where the source nodes are the same as the boundary collocation nodes, the MBKM employs the ghost points method, which resets the source points to a disk-like region covering the primary problem area. This modification results in better accuracy without any increase in the computational cost. On the other hand, as the accuracy of the MBKM depends heavily on the parameter [Formula: see text], the effective condition number (ECN) is first employed to find a proper [Formula: see text] in MBKM. Several 2D and 3D numerical examples are listed to illustrate the superior performance of the MBKM in solving harmonic-type BVPs. The accuracy of MBKM is improved by one to two orders of magnitude compared to the classical BKM. Meanwhile, the validity of the ECN for obtaining a suitable [Formula: see text] for problems under complex geometric regions is also demonstrated.

On the sticky particle solutions to the multi-dimensional pressureless Euler equations

In this paper we consider the multi-dimensional pressureless Euler system and we tackle the problem of existence and uniqueness of sticky particle solutions for general measure-type initial data. Although explicit counterexamples to both existence and uniqueness are known since [6], the problem of whether one can still find sticky particle solutions for a large set of data and of how one can select them was up to our knowledge still completely open.In this paper we prove that for a comeager set of initial data in the weak topology the pressureless Euler system admits a unique sticky particle solution given by a free flow where trajectories are disjoint straight lines.Indeed, such an existence and uniqueness result holds for a broader class of solutions decreasing their kinetic energy, which we call dissipative solutions, and which turns out to be the compact weak closure of the classical sticky particle solutions. Therefore any scheme for which the energy is l.s.c. and is dissipated will converge, for a comeager set of data, to our solution, i.e. the free flow.

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators: The general result

This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator g(t,y,z), by relaxing the assumptions of Hu and Tang [15] on the generator and terminal value. More precisely, the generator g(t,y,z) can have more general growth and continuity in the first unknown variable y in the local solution; while in the global solution, the generator g(t,y,z) can have a skew sub-quadratic but in addition “strictly and diagonally” quadratic growth in the second unknown variable z, or the terminal value can be unbounded but the generator g(t,y,z) is “diagonally dependent” on the variable z (i.e., the i-th component gi of the generator g only depends on the i-th row zi of z for each i=1,⋯,n). Three new results are established on the local and global solutions when the terminal value is bounded and the generator g is subject to some general assumptions. When the terminal value is unbounded but is of exponential moments of arbitrary order, an existence and uniqueness result is given under the assumptions that the generator g(t,y,z) is Lipschitz continuous in y, and varies with z in a “diagonal”, “component-wisely convex or concave”, and “quadratically growing” way, which seems to be the first general solvability of system of quadratic BSDEs with unbounded terminal values. This generalizes and strengthens some existing results via some new ideas.

Numerical solutions to the Sharma–Tasso–Olver equation using Lattice Boltzmann method

AbstractA novel numerical scheme depending on Lattice Boltzmann method is proposed to solve the Sharma–Tasso–Olver equation in one, two and three dimension. The local equilibrium distribution functions and modified functions are obtained via Taylor expansion and Chapman–Enskog multiscale expansion techniques. The macro equation is recovered accurately based on the above functions, and the stability conditions of the equations are deduced. In addition, through simulating numerically some of the initial‐boundary value problems of multidimensional STO equations, the results demonstrate that the numerical solutions and the exact solutions tend to be identical. It also indicates that the model is feasible within a specific range and provides a reliable the approach for solving the multidimensional STO equations.

The Tricomi Problem for a Class of Multidimensional Hyperbolic-Elliptic Equations

The Tricomi Problem for a Class of Multidimensional Hyperbolic-Elliptic Equations

Finite-Time Stabilization and Impulse Control of Heat Equation with Dynamic Boundary Conditions

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms.

Numerical Method for Multi-Dimensional Coupled Forward-Backward Stochastic Differential Equations Based on Fractional Fourier Fast Transform

Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified.

A Legendre spectral method for multidimensional partial Volterra integro-differential equations

We present a Legendre spectral method for solving multidimensional partial Volterra integro-differential equations. The main idea of our approach is first to employ some function transformations and variable transformations to transform the equations into new partial Volterra integro-differential equations, and then the Legendre spectral method is used to solve the equivalent equations. We derive error bounds in L∞- and L2-norms, which indicate that the errors of solution and the first order partial derivative decay exponentially. Three numerical examples are displayed to confirm the reliability of the Legendre spectral analysis.

Asymptotic behavior of solutions to multi-dimensional semilinear parabolic equations with general diffusion coefficients

In this paper, we study asymptotic behavior of solutions to the Cauchy problem of a class of multi-dimensional semilinear parabolic equations with general diffusion coefficients. It is shown that the critical Fujita exponent is determined by the spatial dimension and the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is proved to belong to the blowing-up case.

Does copyright understand intangible heritage? The case of flamenco in Spain

ABSTRACT This paper illustrates the gap between the perceived value of performer input in intangible heritage and how it is considered in the market, drawing on a large sample of flamenco in Spain. The method used is constructed from four different components: an analysis of the intellectual property legal framework to determine the scope of copyright and neighbouring rights, interviews with experts to explain in detail the flamenco work creation process, double system surveys to determine the perceived evaluation of its agents and contributions, and finally – through a multidimensional structural equations model – flamenco work’s perceived value-scale is validated through three dimensions: composition elements, feelings, and virtuosity. The results show how the perceived value of flamenco production critically depends on the externalisation accomplished by the performer through virtuosity. Nevertheless, copyright or royalties by no means ensure a level of protection that aligns with the key creative role of flamenco performers.

Asymptotic stability and sharp decay rates to the linearly stratified Boussinesq equations in horizontally periodic strip domain

We consider an initial boundary value problem of the multi-dimensional Boussinesq equations in the absence of thermal diffusion with velocity damping or velocity diffusion under the stress free boundary condition in horizontally periodic strip domain. We prove the global-in-time existence of classical solutions in high order Sobolev spaces satisfying high order compatibility conditions around the linearly stratified equilibrium, the convergence of the temperature to the asymptotic profile, and sharp decay rates of the velocity field and temperature fluctuation in all intermediate norms based on spectral analysis combined with energy estimates. To the best of our knowledge, our results provide first sharp decay rates for the temperature fluctuation and the vertical velocity to the linearly stratified Boussinesq equations in all intermediate norms.

RESEARCH OF COMPUTATIONAL MATHEMATICAL MODELS FOR TECHNICAL SYSTEMS

In the theory of analysis and synthesis of technical systems, mathematical modelling and optimization of multilayer systems containing sources of physical fields occupy an important place. This is due to the fact that their state is described by means of boundary value problems with multidimensional differential equations. To solve the boundary value problems and implement the process of optimizing the technical parameters of the modelled systems, it is necessary to conduct interdisciplinary studies of computational and applied optimization mathematical models. Fulfilment of the conditions for the existence of a single solution to boundary value problems by default is possible only when the object of study is a single-layer material under the action of load sources. If it is necessary to calculate and optimize the technical parameters of a multilayer material subjected to load sources, then it is impossible to immediately guarantee the correctness of the calculated and applied optimization mathematical models, since it is necessary to obtain the conditions for the existence and uniqueness of solutions to boundary value problems with systems of differential equations. Maximizing the technical parameters of load sources and averaging the characteristics of material layers will lead to approximate values of the objective function and technical parameters of the modelled system, which leads to irrational consumption of energy and heat resources and uncontrolled losses, and useless losses of the test material in the technological process. The article presents the conditions for the correctness of multipoint boundary value problems with multidimensional differential equations describing the state of a multilayer material under thermal action. It is advisable to use these studies to substantiate the correctness of other technical and biotechnological systems, which will increase the accuracy of the implementation of applied optimization problems of economic and mathematical modelling.

Optimal Decay Rates of the Compressible Euler Equations with Time-Dependent Damping in \({\mathbb {R}}^n\): (II) Overdamping Case

This paper is concerned with the large time behavior of the multidimensional compressible Euler equations with time-dependent overdamping of the form in , where , , and . This continues our previous work dealing with the underdamping case for . We show the optimal decay estimates of the solutions such that for and , and , which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of , we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that and for . The overdamping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.