Trivial Boundary Conditions Research Articles

On an inverse boundary-value problem for the pseudohyperbolic equation with nonclassical boundary conditions

In this paper, we consider an inverse boundary-value problem for a fourth-order pseudohyperbolic equation with nonclassical boundary conditions. The primary purpose of the work is to study the existence and uniqueness of the classical solution of the considered inverse boundary-value problem. To investigate the solvability of the considered problem, we carried out a transformation from the original problem to some auxiliary equivalent problem with trivial boundary conditions. Furthermore, we prove the existence and uniqueness theorem for the auxiliary problem by the contraction mappings principle. Based on the equivalency of these problems, the existence and uniqueness of the classical solution of the original problem are shown.

РАСЧЕТ ОСАДОК ПРОТЯЖЕННОГО ПЛИТНОГО ФУНДАМЕНТА КОНЕЧНОЙ ЖЕСТКОСТИ НА ОСНОВЕ ДАННЫХ КОМПЬЮТЕРНОГО МОДЕЛИРОВАНИЯ

The paper presents a database for estimation the magnitude of the settlement of an extended slab foundation of finite stiffness in the initial phase of projecting. The offered solution uses only interpolation methods without involving powerful computing devices. The computer program FEA, which formalises the finite element method, was used to obtain the result. The maximum horizontal and vertical dimensions of the computational schemes were 25H3•46H3. The computational scheme consisted of 57 500 finite elements in the form of right isosceles triangles conjugated at 29 106 nodes. The width of the stiffness matrix of the linear equation system was 466. This made it possible to eliminate the influence of trivial boundary conditions on the results. 560 computational operations were performed. It corresponded to the number of possible combinations of numerical values of the variables calculated parameters adopted in computer modelling of process of the slab foundation settlement. As an upshot of the computations, the authors made a table of coefficients of approximating expressions for the curves constructed from the results of the calculated dependences of slab foundation settlement on the variable design parameters. The article provides a calculation of settlements of two extended slab foundations of different widths, using a constructed database and interpolation methods. The control calculation in the FEA showed the difference between the values of the control settlements and the values obtained using the proposed tables and the linear interpolation method for 13.45 and 22.08 %. Summing up the results, the offered database can be used for preliminary (estimation) calculations of extended slab foundations settlements.

Sound synthesis of a 3D nonlinear string using a covariant Lie group integrator of a geometrically exact beam model

We present a three dimensional nonlinear string model based on a geometrically exact beam. The beam model is obtained by applying a variational principle using a covariant Lagrangian formulation; in particular, the equations of motion and the boundary conditions are treated in an unified manner. Following an analogous discrete variational principle, a Lie group variational integrator is given. The energy and momentum conservation properties of the integrator are discussed and illustrated. This geometrically exact beam serves as a basis to formulate a prestressed damped string model with coupled non trivial boundary conditions. Simulation results are discussed and validated against analytical solutions obtained in the context of a small displacement hypothesis.

On the Decomposition Theorem for Gluons

Recently, the problem of spin and orbital angular momentum (AM) separation has widely been discussed. Nowadays, all discussions about the possibility to separate the spin AM from the orbital AM in the gauge invariant manner are based on the ansatz that the gluon field can be presented in form of the decomposition where the physical gluon components are additive to the pure gauge gluon components, i.e. $A_\mu = A_\mu^{\text{phys}}+A_\mu^{\text{pure}}$. In the present paper, we show that in the non-Abelian gauge theory this gluon decomposition has a strong mathematical evidence in the frame of the contour gauge conception. In other words, we reformulate the gluon decomposition ansatz as a theorem on decomposition and, then, we use the contour gauge to prove this theorem. In the first time, we also demonstrate that the contour gauge possesses the special kind of residual gauge related to the boundary field configurations and expressed in terms of the pure gauge fields. As a result, the trivial boundary conditions lead to the inference that the decomposition includes the physical gluon configurations only provided the contour gauge condition.

Numerical study of Davey–Stewartson I systems

AbstractAn efficient high precision hybrid numerical approach for integrable Davey–Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable against dispersion and blow‐up. A finite‐time blow‐up of initial data in the Schwartz class of smooth rapidly decreasing functions is discussed.

A pseudospectral method for investigating the stability of linear population models with two physiological structures.

The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

Mixed Problem for the Differential Equation of Parabolic Type with Measures

We propose a scheme for the solution of a mixed problem for a parabolic differential equation with coefficients that are generalized derivatives of functions of bounded variation. We seek the solution of this problem by the method of reduction. According to this method, the solution of the proposed problem is reduced to the solution of two problems: (i) a quasistationary boundary-value problem with input boundary conditions and (ii) a mixed problem with trivial boundary conditions. The first of these problems is solved by introducing the quasiderivative. For the solution of the second problem, we use the Fourier method and the expansion in eigenfunctions of a certain boundary-value problem for a quasidifferential equation of the second-order. The obtained results can be used, in particular, for the investigation of the processes of heat transfer in multilayer plates, hollow cylinders, and spheres.

Growth of accretion driven scalar hair around Kerr black holes

Scalar fields around compact objects are of interest for scalar-tensor theories of gravity and dark matter models consisting of a massive scalar, e.g. axions. We study the behaviour of a scalar field around a Kerr black hole with non trivial asymptotic boundary conditions - both non zero density and non zero angular momentum. Starting from an initial radially homogeneous configuration, a scalar cloud is accreted, which asymptotes to known stationary configurations over time. We study the cloud growth for different parameters including black hole spin, scalar field mass, and the scalar field density and angular momentum far from the black hole. We characterise the transient growth of the mass and angular momentum in the cloud, and the spatial profile of the scalar around the black hole, and relate the results of fully non-linear simulations to an analytic perturbative expansion. We also highlight the potential for these accreted clouds to create monochromatic gravitational wave signals - similar to the signals from superradiant clouds, although significantly weaker in amplitude.

On an inverse boundary-value problem for a second-order elliptic equation with non-classical boundary conditions

An inverse boundary value problem for a second-order elliptic equation with periodic and integral condition is investigated. The problem is considered in a rectangular domain. To investigate the solvability of the inverse problem, we perform a conversion from the original problem to some auxiliary inverse problem with trivial boundary conditions. By the contraction mapping principle we prove the existence and uniqueness of solutions of the auxiliary problem. Then we make a conversion to the stated problem again and, as a result, we obtain the solvability of the inverse problem.

Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition

This article studies a nonlocal inverse boundary-value problem for a two-dimensional second-order parabolic equation in a rectangular domain. The purpose of the article is to determine the unknown coefficient and the solution of the considered problem. To investigate the solvability of the inverse problem, we transform the original problem into some auxiliary problem with trivial boundary conditions. Using the contraction mappings principle, existence and uniqueness of the solution of an equivalent problem are proved. Further, using the equivalency, the existence and uniqueness theorem of the classical solution of the original problem is obtained.

Verification of the Numerical Algorithm for Parameter Analysis of the Tube Heat Receiver of the Solar Parabolic Trough System

This paper presents the results of the first stage of verification of the generalized computational algorithm for determining the geometrical, dynamical, and energy parameters of the “parabolic trough concentrator – tube heat receiver” solar receiver system. This algorithm is based on the dimensionless coupled mathematical model of heat and mass exchange in the “Sun – parabolic trough concentrator – tube receiver” system. The Monte Carlo ray tracing method was used to calculate irregular heat flux from the concentrator to the tube receiver surface. The 3D mathematical model of heat exchange in the tube receiver was solved using the finite volume method. The mathematical model took local climatic and geographical features into account, as well as micro- and macroscopic imperfections of the concentrator surface. The dimensional problem of heat exchange in the tube receiver of the solar parabolic trough system was solved at the first stage of verification. A simplified physical model was described. It was assumed that the glass envelope around the tube receiver was removed. The dimensional mathematical model of heat exchange in the tube receiver is based on the system of Navier–Stokes equations for viscous incompressible liquid with constant physical properties. The mathematical model with real boundary conditions was solved numerically. An analytical solution for the test problem with trivial boundary conditions was obtained. The results of the test numerical problem were compared with analytical solution results and were found to be in close agreement. Numerical experiments with real boundary conditions were conducted using the computational algorithm. The numerical data were compared with the results of field experimental studies and showed nearly total agreement, which proved the adequacy of the basic mathematical model and the resulting numerical algorithm.

On one nonlocal inverse boundary problem for the Benney – Luke equation with integral conditions

An inverse boundary value problem for the Benney-Luke equation with periodic and integral condition is investigated. The definition of a classical solution of the problem is introduced. The goal of this paper is to determine the unknown coefficient and to solve the problem of interest. The problem is considered in a rectangular domain. To investigate the solvability of the inverse problem, we perform a conversion from the original problem to some auxiliary inverse problem with trivial boundary conditions. By the contraction mapping principle we prove the existence and uniqueness of solutions of the auxiliary problem. Then we make a conversion to the stated problem again and, as a result, we obtain the solvability of the inverse problem.

Inverse boundary-value problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions

We study the inverse coefficient problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions. The main purpose of this paper is to prove the existence and uniqueness of the classical solutions of an inverse boundary-value problem. To investigate the solvability of the inverse problem, we carried out a transformation from the original problem to some equivalent auxiliary problem with trivial boundary conditions. Applying the Fourier method and contraction mappings principle, the solvability of the appropriate auxiliary inverse problem is proved. Furthermore, using the equivalency, the existence and uniqueness of the classical solution of the original problem are shown.

Entanglement entropy of topological orders with boundaries

In this paper we explore how non trivial boundary conditions could influence the entanglement entropy in a topological order in 2+1 dimensions. Specifically we consider the special class of topological orders describable by the quantum double. We will find very interesting dependence of the entanglement entropy on the boundary conditions particularly when the system is non-Abelian. Along the way, we demonstrate a streamlined procedure to compute the entanglement entropy, which is particularly efficient when dealing with systems with boundaries. We also show how this method efficiently reproduces all the known results in the presence of anyonic excitations.

A systematic and efficient method to compute multi-loop master integrals

We propose a novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions. Thus it can be systematically applied to problems with arbitrary kinematic configurations. Numerical tests show that our method can not only achieve results with high precision, but also be much faster than the only existing systematic method sector decomposition. As a by product, we find a new strategy to compute scalar one-loop integrals without reducing them to master integrals.

Determination of Plane Stress-Strain States of the Plates on the Basis of the Three-Dimensional Theory of Elasticity

We construct a theory of plates loaded only on their sides by loads parallel and symmetric to the median surface. We use the general representation of three-dimensional stress-strain state and exactly satisfy the trivial boundary conditions on the plane surfaces of the plate. The numerical analysis of the three-dimensional stressed state of the plate is reduced to the determination of its two-dimensional state under the assumption that the normal stresses perpendicular to the median surface are negligible. The displacements and stresses are expressed via two two-dimensional harmonic functions. We obtain the homogeneous solutions and develop a numerical-analytic algorithm for the solution of the boundary-value problems posed for rectangular plates.

Solution of the Multiparticle Scattering Problems in Configuration Space by Using Trivial Boundary Conditions

A formalism based on the complex-scaling method is used to solve a four-nucleon scattering problem above the breakup threshold using the realistic nuclear Hamiltonians. This method allows to solve diverse scattering problems based on very trivial boundary conditions and is compatible with the techniques used to solve bound state problems.

LEAVING THE DARK AGES WITH AMIGA

We present an Analytic Model of Intergalactic-medium and GAlaxy evolution since the dark ages. AMIGA is in the spirit of the popular semi-analytic models of galaxy formation, although it does not use halo merger trees but interpolates halo properties in grids that are progressively built. This strategy is less memory-demanding and allows one to start the modeling at redshifts high enough and halo masses low enough to have trivial boundary conditions. The number of free parameters is minimized by making the causal connection between physical processes usually treated as independent from each other, which leads to more reliable predictions. But the strongest points of AMIGA are: i) the inclusion of molecular cooling and metal-poor, population III (Pop III) stars, with the most dramatic feedback, and ii) the accurate follow-up of the temperature and volume filling factor of neutral, singly, and doubly ionized regions, taking into account the distinct halo mass functions in those environments. We find the following general results. Massive Pop III stars determine the IGM metallicity and temperature, and the growth of spheroids and disks is self-regulated by that of massive black holes developed from the remnants of those stars. Yet, the properties of normal galaxies and active galactic nuclei appear to be quite insensitive to Pop III star properties owing to the much higher yield of ordinary stars compared to Pop III stars and the dramatic growth of MBHs when normal galaxies begin to develop, which cause the memory loss of the initial conditions.

Application of the complex scaling method for few-body scattering problems including the breakup

A formalism based on the complex-scaling method is used to solve a few-particle scattering problem in configuration space. This method allows to use trivial boundary conditions and is compatible with most of the bound state techniques. In this contribution calculations of neutron scattering on triton is presented using realistic nuclear Hamiltonians for neutron energies above four-nucleon breakup threshold.

Improving the Validity of Squeeze Film Air-Damping Model of MEMS Devices with Border Effect

Evaluation of squeezed film air damping is critical in the design and control of dynamic MEMS devices. The published squeezed film air damping models are generally derived from the analytical solutions of Reynolds equation or its other modified forms under the supposition of trivial pressure boundary conditions on the peripheral borders. These treatments ignoring the border effect can not give faithful result for structure with smaller air venting gap or the double-gimbaled structure in which the inner frame and outer one affect the air venting. In this paper, we use Green’s function to solve the nonlinear Reynolds equation with inhomogeneous boundary conditions. For two typical normal motion cases of parallel plate, the analytical models of squeeze film damping force with border effect are established. The viscous and inertial losses with real values and image values acoustic impedance are all included in the model. These models reduced the time consumption while giving satisfactory result. Without multifield coupling analysis, the estimation of the dynamic behavior of MEMS device is also allowed, and the simulation of the system performance is more convenient.