System Of Second-order Equations Research Articles

Векторная частица с аномальным магнитным моментом во внешнем однородном электрическом поле

In the paper, spin 1 particle with an anomalous magnetic moment
 is examined in presence of an external uniform electric
 field. The generalized ten-dimensional Duffin-Kemmer equation
 is specified in Cartesian coordinates (t, x, y, z) . On its
 solutions there are diagonalized operators of energy and linear
 momentums Px and Py. The external electric field is oriented
 along the axes z. The system of ten differential equations
 in the variable z is derived. With the use of the generator
 j03 for ten-component field we introduce three projective
 operators which permit us to divide the complete ten-component
 wave function into three projective constituents. One of
 them is taken as the primary constituent, it depends on four
 functions; the two remaining projective constituents are defined
 by the primary one. For these four functions we derive
 one linear constraint and the system of second order equations
 for three functions. This system after linear transformation
 is reduced to three separated equations for three new
 variables. Their solutions are constructed in terms of confluent
 hypergeometric functions. Properties of the obtained
 solutions are studied.

Method of own forms for determination of separate bands

The problem of determining the separate bands of the phosphor photoluminescence spectrum described by the Gaussian function is solved by the method of decomposition into its own forms. This allows you to calculate two parameters of each band: amplitude and variance. An example is the photoluminescence spectrum of a ZnS:Mn single crystal. It consists of five separate bands. Two systems of fifth-order equations linear for one parameter of each band are obtained. This parameter is generalized and is equal to the product of the amplitude and the root of the variance. The coefficients of these systems depend on the second parameter. This non-linear parameter is related to variance. In order to simplify the two systems, the problem is solved in several stages. Based on the analysis of literary sources, the two largest bands are selected. In addition, the origin of coordinates is well chosen. This allows us to simplify both systems and completely determine the parameters of each of the largest bands from the systems of second-order equations. As a result of these simplifications, the order of the two systems and the parameters of the other three bands are reduced. Thus, the problem is reduced to solving a system of the second and third order nonlinear equations.

A convergent evolving finite element algorithm for Willmore flow of closed surfaces

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

Mass-conservative and positivity preserving second-order semi-implicit methods for high-order parabolic equations

We consider a class of finite element approximations for fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. In our approach, we first solve a variational problem and then an optimization problem to satisfy the desired physical properties of the solution such as conservation of mass, positivity (non-negativity) of solution and dissipation of energy. Furthermore, we show existence and uniqueness of the solution to the optimization problem and we prove that the methods converge to the truncation schemes [10]. We also propose new conservative truncation methods for high-order parabolic equations. A numerical convergence study is performed and a series of numerical tests are presented to show and compare the efficiency and robustness of the different schemes.

Higher order approximation of biharmonic problem using the WEB-Spline based mesh-free method

Abstract In this paper, we consider a biharmonic equation with Navier boundary conditions and we decoupled this fourth order biharmonic equation into a system of second order equations. In numerics, there is an advantage because of this decoupling, i.e., one can use C 0 finite elements to solve the decoupled system instead of C 1 finite elements. For solving this decoupled system of second order equations, we use the weighted extended b-spline (WEB-Spline) based mesh-free finite element method. The WEB-Spline method does not require any mesh generation and eliminates the difficult, time-consuming preprocessing step. Also, by the WEB-Spline method higher order approximations are possible with relatively low dimensional spaces. Numerical results based on the WEB-Spline method are compared with the usual finite element method in order to demonstrate the efficiency of the proposed method.

Solution of the Rodriguez Equation in Modeling Volumetric Geometric Accuracy of Multi-Coordinate Systems

The article describes the solution to the Rodrigues equation for determining the volumetric accuracy of multi-axis CNC-controlled systems. An algorithm for calculating the position of the axis of a rotary kinematic pair in problems of volumetric accuracy of mechanical motion of a portal-type system with an additional pair of rotation. The algorithm is based on the analytical solution of the Rodrigues equation in the inverse problem of finding the vector of the final rotation of the known modulus from the known initial and final values of the characteristic vector of the rotated rigid body. In contrast to the well-known direct problem, where based on a finite rotation vector known in direction and magnitude, and the initial value of the characteristic vector of a body, its final value is found, the inverse problem of the Rodrigues equation is not that common due to the nonlinearity and need to solve a nonlinear coupled system of second order equations. The results of this work make it possible to expand the dimension of the space of generalized coordinates of the system analyzed for the volumetric accuracy from three to four. This is expected contribute to the development of ultra-precise systems of controlled mechanical movement. The analytical results of this study were verified by comparing with numerical solutions of the inverse problem in Maple.

Efficient second-order semi-implicit finite element method for fourth-order nonlinear diffusion equations

We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to approximate these equations. In our approach, we propose new techniques using the second-order backward differentiation formula for the time derivative and a special technique for the approximation of nonlinear terms. The use of the proposed technique for nonlinear terms makes the developed numerical scheme efficient in terms of computational cost since the proposed method only deals with a linear system at each time step and no iterative resolution is needed. A numerical convergence study is performed using the method of manufactured and analytical solutions of the system where we investigate different boundary conditions. With respect to the spatial discretization, convergence rates are found to at least match a priori error estimates available for linear problems. The convergence analysis is completed with an investigation of the temporal discretization where we numerically demonstrate the second-order time-accuracy of the proposed scheme using the method of reference solution. We present a series of numerical tests to demonstrate the efficiency and robustness of the proposed scheme.

Two-Frequency Propagation Mode of Acoustic−Gravity Waves in the Earth’s Atmosphere

As is known from multiyear theoretical and experimental studies, acoustic−gravity waves (AGW) determine the dynamics and energy balance of the atmospheres of planets and the Sun to a considerable extent. Linear wave perturbations in the atmosphere can be described with a system of second-order equations for the vertical and horizontal components of the perturbed velocity. It follows from this system that small perturbations in the atmosphere may be considered as oscillations of coupled oscillators with two degrees of freedom. This suggests that it is worth studying more thoroughly the linear acoustic−gravitational wave modes in the atmosphere with well-developed methods of the theory of oscillations. To study small perturbations in the Earth’s atmosphere, the methods of the theory of coupled oscillatory systems were used. It has been shown that acoustic−gravity waves in an isothermal atmosphere can be considered as a superposition of oscillations that occur simultaneously at two natural frequencies—acoustic and gravitational. The equations for the natural frequencies of oscillations, as well as for the components of the perturbed velocity under specified initial conditions, were derived. From the analysis of temporal changes in the components of the perturbed velocity, new features in their behavior were found. All solutions are presented with the use of real quantities only. This representation is more convenient for comparison with observational data than the complex one commonly used in the theory of acoustic−gravity waves. The conditions under which the usual single-frequency oscillation mode can be realized in the atmosphere are analyzed. The results of the study can be used to explain some features in space-borne observations of wave perturbations in the Earth’s atmosphere that do not fit the framework of the known theoretical concepts.

On the smoothness of the solution of a nonlocal boundary value problem for the multidimensional second-order equation of the mixed type of the second kind in Sobolev space

In this paper we prove the unique solvability and smoothness of the solution of a nonlocal boundary-value problem for a multidimensional mixed type second-order equation of the second kind in Sobolev space Wℓ2(Q), (2≤ℓ is an integer). First, we have studied the unique solvability of the problems in the space W22(Q). Solution uniqueness for a nonlocal boundary-value problem for a mixed-type equation of the second kind is proved by the methods of a priori estimates.Further, to prove the solution existence in the space W22(Q), the Fourier method is used. In other words, the problem under consideration is reduced to the study of unique solvability of a nonlocal boundary value problem for an infinite number of systems of second-order equations of mixed type of the second kind. For the unique solvability of the problems obtained, the ``ε-regularization'' method is used, i.e, the unique solvability of a nonlocal boundary-value problem for an infinite number of systems of composite-type equations with a small parameter was studied by the methods of functional analysis. The necessary a priori estimates were obtained for the problems under consideration. Basing on these estimates and using the theorem on weak compactness as well as the limit transition, solutions for an infinite number of systems of second-order equations of mixed type of the second kind are obtained. Then, using Steklov-Parseval equality for solving an infinite number of systems of second-order equations of mixed type of the second kind, the unique solvability of original problem was obtained. At the end of the paper, the smoothness of the problem's solution is studied.

Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.

ОБ ОДНОЙ ЗАДАЧЕ ДЛЯ СТАЦИОНАРНОЙ СИСТЕМЫ В ДВУХЖИДКОСТНОЙ СРЕДЕ В ПОЛУПРОСТРАНСТВЕ

This paper considers a classical solution in the half-space of the second boundary value problem for an overdetermined stationary system of second order equations arising in a two-fluid medium with a single pressure. The solution of the considered boundary value problem using the Fourier transform apparatus has been obtained. The effect of thermodynamic and kinetic parameters of the medium is shown on the solution to the system in question.

A new error analysis of a mixed finite element method for the quad-curl problem

In this paper, we study a new numerical approach for a quad-curl model problem which arises in the inverse electromagnetic scattering problems and magnetohydrodynamics (MHD). We first split the quad-curl problem with homogeneous boundary conditions into a system of second order equations, and then apply a mixed finite element method to solve the resulting system. The perturbed mixed finite element method is constructed by using edge elements. The well posedness of the numerical scheme is derived. The optimal error estimates in H(curl) and L2 norms for the primal and auxiliary variables are obtained, respectively. The theoretical results are verified by numerical experiments.

The Dirichlet Problem for an Elliptic System of Second-Order Equations with Constant Real Coefficients in the Plane

A solution of the Dirichlet problem for an elliptic systemof equations with constant coefficients and simple complex characteristics in the plane is expressed as a double-layer potential. The boundary-value problem is solved in a bounded simply connected domain with Lyapunov boundary under the assumption that the Lopatinskii condition holds. It is shown how this representation is modified in the case of multiple roots of the characteristic equation. The boundary-value problem is reduced to a system of Fredholm equations of the second kind. For a Holder boundary, the differential properties of the solution are studied.

Twice Periodic Solutions of a Nonlinear Elliptic Second-Order Systems

The problem of the existence of twice periodic solutions of a nonlinear elliptic system of second-order equations on a plane whose main periods are given is studied. Conditions for the existence of solutions with prescribed poles, as well as with prescribed zeros and poles, are obtained.

The Poisson integral and Green’s function for one strongly elliptic system of equations in a circle and an ellipse

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green’s function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series.

Regularity of Weak Solutions to a Model Problem with Conjugation Conditions for Quasilinear Parabolic Systems of Equations

We consider a parabolic quasilinear second order system of equations in divergence form in a model parabolic cylinder. We prove the Holder continuity of a weak solution on a set of full measure in the cylinder. It is shown that the linear system has no singular set. We use a modified method of A-caloric approximation which takes into account the conjugation conditions on the interface between media.

Fast and accurate analyses of spacecraft dynamics using implicit time integration techniques

This paper deals with the implementation techniques of an implicit integrator to achieve fast and accurate analyses of spacecraft dynamics. For this purpose, the pseudospectral method is adopted to directly integrate the second-order system of equations for both the spacecraft dynamics and corresponding state transition matrix. Various implementation techniques are proposed to enhance the numerical efficiency and integration accuracy, which include a moving horizon approach, the decoupled integration of the second-order dynamics for the state transition matrix, and the grid adaptation method. The numerical features of the proposed techniques are investigated through their applications to a spacecraft’s motion around highly eccentric elliptic orbits, and the resultant numerical errors and computing times are compared with those from the Runge-Kutta method to show the relative efficiency and accuracy of the presented methods. In addition, an optimal two-impulse orbit transfer from the Earth to the Moon is analyzed by implementing the proposed methods using a multiple-shooting framework. The results show that the proposed techniques are extremely effective for dynamical problems requiring intensive and accurate time integrations, and can provide much better accuracy and efficiency than the explicit Runge-Kutta integrator.

A staggered approach for the coupling of Cahn–Hilliard type diffusion and finite strain elasticity

We develop an algorithm and computational implementation for simulation of problems that combine Cahn---Hilliard type diffusion with finite strain elasticity. We have in mind applications such as the electro-chemo-mechanics of lithium ion (Li-ion) batteries. We concentrate on basic computational aspects. A staggered algorithm is proposed for the coupled multi-field model. For the diffusion problem, the fourth order differential equation is replaced by a system of second order equations to deal with the issue of the regularity required for the approximation spaces. Low order finite elements are used for discretization in space of the involved fields (displacement, concentration, nonlocal concentration). Three (both 2D and 3D) extensively worked numerical examples show the capabilities of our approach for the representation of (i) phase separation, (ii) the effect of concentration in deformation and stress, (iii) the effect of strain in concentration, and (iv) lithiation. We analyze convergence with respect to spatial and time discretization and found that very good results are achievable using both a staggered scheme and approximated strain interpolation.

Efecto del oleaje no lineal en tanques de almacenamiento sometidos a sismos intensos

A numerical method is proposed for the seismic response analysis of rectangular and cylindrical tanks accounting for non-linear hydrodynamic effects. The equations of motion are mapped from the physical to the computational domain by means of a change of variable. The numerical solution is then obtained by means of finite differences. The system of second order equations obtained from the finite difference discretization are linearized using the Newton method and solved numerically by means of the biconjugate gradient method. The numerical formulation was applied to study the response of tanks subjected to ground motion records from the Mexico earthquake of 19 September 1985, obtained in Mexico City. Response results are given in terms of wave height on the liquid surface, and shear force and overturning moment at the base wall, and are compared with those obtained for the linear solution.

DISCONTINUOUS OBLIQUE DERIVATIVE PROBLEM FOR NONLINEAR ELLIPTIC SYSTEM OF SECOND ORDER EQUATIONS IN MULTIPLY CONNECTED DOMAINS

In this article, we discuss that the discontinuous oblique derivative boundary value problem for nonlinear uniformly elliptic system of second order equations in multiply connected domains. We first propose the discontinuous oblique derivative problem and its new modified well-posedness. Next we give a priori estimates of solutions of the modified discontinuous boundary value problem for corresponding elliptic system of first order complex equations and verify its solvability by the above estimates of solutions and the Leray-Schauder theorem. Finally the solvability results of the original discontinuous oblique derivative problem can be derived. Here we mention the discontinuous boundary value problems possess many applications in mechanics and physics etc.