System Of First-order Equations Research Articles

A first-order Stein characterization for absolutely continuous bivariate distributions

A random variable X has a standard normal distribution if and only if for any continuous and piecewise continuously differentiable function f such that the expectations exist. This first-order characterizing equation, called the Stein identity, has been extended to other univariate distributions. For the multivariate normal distribution, a number of Stein identities have already been developed, all of them second order equations. In this study, we developed a new Stein characterization for the bivariate normal distribution. Unlike many existing multivariate versions in the literature, ours is a system of first-order equations which has the univariate Stein identity as a special case. We also constructed a generalized Stein characterization for other absolutely continuous bivariate distributions. Finally, we illustrated how this Stein characterization looks like for some known absolutely continuous bivariate distributions.

The Discontinuous Galerkin Method and its Implementation in the RAMEG3D Software Package

Currently, the Discontinuous Galerkin Method (DGM) is widely used to solve complex multi-scale problems of mathematical physics that have important applied significance. When implementing it, the question of choosing a discrete approximation of flows for viscous terms of the Navier-Stokes equation is important.It is necessary to focus on the construction of limiting functions, on the selection of the best discrete approximations of diffusion flows, and on the use of implicit and iterative methods for solving the obtained differential-difference equations for the successful application of DGM on three-dimensional unstructured grids.First-order numerical schemes and second-order DGM schemes with Godunov, HLLC, Rusanov-Lax-Friedrichs numerical flows and hybrid flows are investigated. For high-order precision methods, it is necessary to use high-order time schemes.The Runge-Kutta scheme of the third order is used in the work. The equations are written as a system of first-order equations, when solving the Navier-Stokes equation by the discontinuous Galerkin method.

Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.

Mathematical analysis of an abstract model and its applications to structured populations (I)

<p style='text-indent:20px;'>The first part of this works deals with an integro–differential operator with boundary condition related to the interior solution. We prove that the model is governed by a strongly continuous semigroup and we precise its growth inequality. In the second part of this works, we model the proliferation-quiescence phases through a system of first order equations. We also prove that the proliferation-quiescence model is governed by a strongly continuous semigroup and we precise its growth inequality. In the last part, we give some applications in Demography and Biology.</p>

Решение уравнений конвекции-диффузии локальным разрывным методом Галеркина

In the work, the discontinuous Galerkin method (LDG) is applied to solving linear and nonlinear convection-diffusion problems. The idea of this method is to transform high-order equations into a system of first-order equations, and then apply the classical discontinuous Galerkin method (DG) to this system. An orthogonal system of Legendre polynomials is chosen as the system of basis functions. The cases of both continuous and discontinuous solutions of the problem are considered. The dependence of the accuracy and stability of the method on the step of the spatial grid and the number of basis functions is investigated. The application of parallel computing to the algorithm is investigated. In the future, it is proposed to apply the LRG method for solving the problems of modeling gas mixtures flows.

On the asymptotic behavior of solutions of the Sturm-Liouville equation with an oscillating potential

In the article we consider the Sturm-Liouville equation −y ′′+(q(x)+h(x))y = 0, 0 ≤ x < +∞, where q(x) satisfies the conditions of regularity of growth at infinity and h(x) is a fast-oscillating function. Sufficient conditions are found under which the asymptotics of solutions of the equation is determined only by the function q(x). A new method for obtaining asymptotic formulas for solutions is proposed, consisting in the standard replacement of the equation by a system of first-order equations followed by the application of the Hausdorff identity [1]

Exact analytical solution for static deflection of Timoshenko composite beams on two-parameter elastic foundations

New exact analytical solutions are presented for the static deflection of coupled Timoshenko composite beams resting on two-parameter elastic foundations subject to arbitrary boundary and loading conditions. Governing differential equations are obtained from the principle of virtual work in which four degrees of freedom, namely axial elongation, twist, bending and transverse shear are coupled. In order to obtain the analytical solutions describing the static deflection of coupled Timoshenko composite beams resting on elastic foundations, the governing equations are first rewritten in matrix form to enable decoupling of axial displacement and twist from bending and transverse shear. Applying a separation of variables technique, axial elongation and twist are eliminated and the matrix differential equation for bending and transverse shear is transformed into a system of first-order equations which is solved using a fundamental matrix approach. The results obtained from the analytical solutions are firstly verified by comparison with solutions from the literature for isotropic homogeneous beams on elastic foundations and then with the numerical results from the Chebyshev collocation method obtained for composite beams on elastic foundations, and found to be in excellent agreement. The influence of elastic foundation coefficients, coupling terms and length-to-thickness ratio on the static deflection of Timoshenko composite beams resting on elastic foundations are investigated and discussed.

Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

<p style='text-indent:20px;'>Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic equations on a metric graph <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of <inline-formula><tex-math id="M4">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>.</p>

The numerical solutions for the nonhomogeneous Burgers' equation with the generalized Hopf-Cole transformation

<abstract><p>In this paper, with the help of the generalized Hopf-Cole transformation, we first convert the nonhomogeneous Burgers' equation into an equivalent heat equation with the derivative boundary conditions, in which Neumann boundary conditions and Robin boundary conditions can be viewed as its special cases. For easy derivation and numerical analysis, the reduction order method is used to convert the problem into an equivalent first-order coupled system. Next, we establish a box scheme for this first-order system. By the technical energy analysis method, we obtain the prior estimate of the numerical solution for the box scheme. Furthermore, the solvability and convergence are obtained directly from the prior estimate. The extensive numerical examples are carried out, which verify the developed box scheme can achieve global second-order accuracy for both homogeneous and nonhomogeneous Burgers' equations.</p></abstract>

Optimal control of a hyperbolic system that describes Slutsky demand

A non-linear optimal control problem for a hyperbolic system of first order equations on a line in the case of degeneracy of the initial condition line is considered. This problem describes many natural, economic and social processes, in particular, the optimality of the Slutsky demand, the theory of bio-population, etc. The research is based on the method of characteristics and the use of nonstandard variations of the increment of target functional, which leads to the construction of efficient computational algorithms.

LDG approximation of a nonlinear fractional convection-diffusion equation using B-spline basis functions

This paper develops new numerical schemes for solution to nonlinear fractional convection-diffusion equations of order β∈[1,2]. We propose the local discontinuous Galerkin methods by adopting linear, quadratic, and cubic B-spline basis functions and prove stability and optimal order of convergence O(hk+1) for the fractional diffusion problem. This method transforms the equation into a system of first-order equations and approximates the solution of the equation by selecting the appropriate basis functions. The B-Spline functions significantly improve the accuracy and stability of the method. The performed numerical results demonstrate the efficiency and accuracy of the proposed scheme in different conditions and confirm the optimal order of convergence.

The Effect of Independent Parameter on Accuracy of Direct Block Method

Block methods that approximate the solution at several points in block form are commonly used to solve higher order differential equations. Inspired by the literature and ongoing research in this field, this paper intends to explore a new derivation of block backward differentiation formula that employs independent parameter to provide sufficient accuracy when solving second order ordinary differential equations directly. The use of three backward steps and five independent parameters are considered adequately in generating the variable coefficients of the formulas. To ascertain only one parameter exists in the derived formula, the order of the method is determined. Such independent parameter retains the favorable convergence properties although the values of parameter will affect the zero stability and truncation error. An ability of the method to compute the approximated solutions at two points concurrently is undeniable. Another advantage of the method is being able to solve the second order problems directly without recourse to the technique of reducing it to a system of first order equations. The essential of the error analysis is to observe the effect of independent parameter on the accuracy, in the sense that with certain appropriate values of parameter, the accuracy is improved. The performance of the method is tested with some initial value problems and the numerical results confirm that the maximum error and average error obtained by the proposed method are smaller at certain step size compared to the other conventional direct methods.

Stability and Anti-Chaos Control of Discrete Quadratic Maps

A dynamical system describes the consequence of the current state of an event or particle in future. The models expressed by functions in the dynamical systems are more often deterministic, but these functions might also be stochastic in some cases. The prediction of the system's behavior in future is studied with the analytical solution of the implicit relations (Differential, Difference equations) and simulations. A discrete-time first order system of equations with quadratic nonlinearity is considered for study in this work. Classical approach of stability analysis using Jury's condition is employed to analyze the system's stability. The chaotic nature of the dynamical system is illustrated by the bifurcation theory. The enhancement of chaos is performed using Cosine Chaotification Technique (CCT).
 Simulations are carried out for different parameter values.

ВРАЩЕНИЯ ТОРОВ В СТРУКТУРЕ ЖИДКОГО КРИСТАЛЛА

Liquid crystals are a collection of flattened molecules. On the one hand, they have a welldefined structure, on the other hand, this structure is deformable, since its elements can change their position in space. In columnar structures, consisting of disks, plates, and tori, the fluid lines and cords are distinguished. In the case of tori and circular disks, a hexagonal structure of the arrangement of molecules is observed in a combination of the cord. The aim of this work is to obtain the stability of tori positions in an elementary fragment of a liquid crystal and to analyze their rotations in a self-consistent field of the surrounding toroidal molecules. To solve this problem, the method of mathematical modeling was used, based on classical models of molecular dynamics. The calculation is carried out on the basis of the model of cross-atom-atom interactions for molecular tori. The minimal fragment of the cord is selected, which makes it possible to determine the characteristic dynamic state of the central torus in the fragment. To describe the motion of the molecular tori, the equations of motion for their centers of mass and the Euler equations for their angular displacements are used. The minimal fragment of the material contains twenty-nine tori. The equations for displacements of the centers of mass of the tori are initially represented as ordinary differential equations of the second order. However, by introducing fictitious points into the consideration of velocities, they can be reduced to a system of first-order equations with a doubled number of lower-order equations. The resulting system of the first-order differential equations is integrated numerically using a highorder accuracy step-by-step scheme. All calculations are performed with a constant time step. The accuracy of the obtained numerical results is verified in terms of the balance of total energy of the system. Calculations show that the central torus of the presented fragment executes angular oscillations around its main axis with amplitude of more than one revolution. Thus, the performed calculations show that a representative fragment of the liquid crystal structure of molecular tori can be used as a generator of high-frequency mechanical vibrations.

Global Stability of Systems with Distributed and Concentrated Parameters

The paper considers the problem of stability in the systems with distributed and lumped parameters defined by linear differential equations partially and ordinary derivatives. The issue is resolved through the process of Lyapunov functions (functionals). The major problem of the method used is the development of the corresponding Lyapunov procedures. In most applications, such functionals were regularly formed intuitively, based on the first integrals,whole energy, and other factors. to make the solution of this problem more manageable, it has been recommended to modify the original higher-order differential equations into a system of first-order equations. When the order of higherorder partial differential equations is reduced by the introduction of additional variables for derivatives concerningspatial coordinates, the equations without time derivatives appear. To take such equations into account when they calculate the derivative of the Lyapunov function in view of the system under consideration, a modified method of Lagrange multipliers is used. The transition to partial differential equations of the first order collectively with thewriting of ordinary differential equations in the standard Cauchy form provides to use specific equations appropriately to develop the Lyapunov function as a total of integral and regular quadratic patterns, the sign-definiteness of which is checked by the Sylvester criterion, and to create a universal method to study stability in general for a broad class ofsystems with distributed and lumped parameters. The conditions for asymptotic stability, in general, are obtained. The results obtained in the article make it possible to study various complex engineering objects with distributed and lumped parameters. As an example, they take into account the stability of a rotary-type wind turbine operation with a load (pump and generator) and the elasticity of the shaft transmitting torque from the wind turbine to the load. The use of environmentally friendly wind turbines that can decrease energy expenses is an assuring area. A rotary-type wind turbine does not require a wind orientation system, it is safe, quiet, and contrary to propeller wind turbines can be installed near buildings and structures.

Comparison for the Approximate Solution of the Second-Order Fuzzy Nonlinear Differential Equation with Fuzzy Initial Conditions

This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.

New Analytical Model Used in Finite Element Analysis of Solids Mechanics

In classical mechanics, determining the governing equations of motion using finite element analysis (FEA) of an elastic multibody system (MBS) leads to a system of second order differential equations. To integrate this, it must be transformed into a system of first-order equations. However, this can also be achieved directly and naturally if Hamilton’s equations are used. The paper presents this useful alternative formalism used in conjunction with the finite element method for MBSs. The motion equations in the very general case of a three-dimensional motion of an elastic solid are obtained. To illustrate the method, two examples are presented. A comparison between the integration times in the two cases presents another possible advantage of applying this method.

The Local Discontinuous Galerkin Method with Generalized Alternating Flux Applied to the Second-Order Wave Equations

In this paper, we propose the local discontinuous Galerkin method based on the generalized alternating numerical flux for solving the one-dimensional second-order wave equation with the periodic boundary conditions. Introducing two auxiliary variables, the second-order equation is rewritten into the first-order equation systems. We prove the stability and energy conservation of this method. By virtue of the generalized Gauss–Radau projection, we can obtain the optimal convergence order in L2-norm of Ohk+1 with polynomial of degree k and grid size h. Numerical experiments are given to verify the theoretical results.

A Neural Network Study of Blasius Equation

In this work we applied a feed forward neural network to solve Blasius equation which is a third-order nonlinear differential equation. Blasius equation is a kind of boundary layer flow. We solved Blasius equation without reducing it into a system of first order equation. Numerical results are presented and a comparison according to some studies is made in the form of their results. Obtained results are found to be in good agreement with the given studies.

Adaptive First-Order System Least-Squares Finite Element Methods for Second-Order Elliptic Equations in Nondivergence Form

This paper studies adaptive first-order system least-squares finite element methods (LSFEMs) for second-order elliptic partial differential equations in nondivergence form. Unlike the classical finite element methods which use weak formulations of PDEs that are not applicable for the nondivergence equation, the first-order least-squares formulations naturally have stable weak forms without using integration by parts, allow simple finite element approximation spaces, and have built-in a posteriori error estimators for adaptive mesh refinements. The nondivergence equation is first written as a system of first-order equations by introducing the gradient as a new variable. Then, two versions of least-squares finite element methods using simple $C^0$ finite elements are developed in the paper. One is the $L^2$-LSFEM which uses linear elements, and the other is the W-LSFEM with a mesh-dependent weight to ensure the optimal convergence. Under the assumption that the PDE has a unique solution, a priori and a posteriori error estimates are presented. With an extra assumption on the operator regularity, convergence in standard norms for the W-LSFEM is also discussed. $L^2$-error estimates are derived for both formulations. Extensive numerical experiments for continuous, discontinuous, and even degenerate coefficients on smooth and singular solutions are performed to test the accuracy and efficiency of the proposed methods.