Topological structures of boundary value problems in block elements

Block structures are considered; a boundary value problem for a system of inhomogeneous partial differential equations with constant coefficients is formulated in each block of a structure. The problem of matching solutions to boundary value problems in blocks with each other by topological study of the properties of solutions in the block structure is examined in the conditions of correct solvability of boundary value problems in blocks of the block structure. Some new properties of solutions to boundary value problems in block structures are found that are important for applications.     

一种不规范的斯特姆-刘维本征值问题

由于端点系有集中质量的弹性杆的振动问题的边界条件不是斯特姆－刘维本征值问题所要求的边界条件，分离变数所得到的函数系{Xn(x)}不是正交函数系，故求解十分困难，本文提出一种变换，将其边界条件化为第一类，第二类或第三类边界条件，此时{Xn(x)}为正交完备函数系，问题大为简化，最后给出了几种不同端点情况下此类问题的解。     

一类奇摄动薄板弯曲问题的匹配渐近解

研究了一类薄板弯曲问题. 对四阶奇摄动边值问题, 引入伸长变量, 构造边界附近的内层解, 然后与外部解匹配. 最后用合成展开式理论, 得到了原问题的的渐近解. 关键词： 薄板弯曲 挠度 渐近解     

Some solutions of the 3D Laplace equation in a layer with oscillating boundary describing an array of nanotubes and an application to cold field emission. I. Regular array

The aim of this paper is to construct solutions of the Dirichlet problem for the 3D Laplace equation in a layer with highly oscillating boundary. The boundary simulates the surface of a nanotube array, and the solutions are applied to compute the cold field electron emission. We suggest a family of exact solutions that solve the problem for a boundary with appropriate geometry. These solutions, along with the Fowler-Nordheim formula, allow one to present explicit asymptotic formulas for the electric field and the emission current. In this part of the paper, we consider the main mathematical aspects, restricting ourselves to the analysis of properties of the potential created by a single tube and a regular array of tubes. In the next part, we shall consider some cases corresponding to nonregular arrays of tubes and concrete physical examples.     

Existence of positive solutions for nonlocal boundary value problem of fractional differential equation

In this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower solutions method and fixed point theorems, some results for the existence of positive solutions for the boundary value problem are established. Some examples are also given to illustrate our results.     

Spontaneous soliton generation in the higher order Korteweg-de Vries equations on the half-line

Some new effects in the soliton dynamics governed by higher order Korteweg-de Vries (KdV) equations are discussed based on the exact explicit solutions of the equations on the positive half-line. The solutions describe the process of generation of a soliton that occurs without boundary forcing and on the steady state background: the boundary conditions remain constant and the initial distribution is a steady state solution of the problem. The time moment when the soliton generation starts is not determined by the parameters present in the problem formulation, the additional parameters imbedded into the solution are needed to determine that moment. The equations found capable of describing those effects are the integrable Sawada-Kotera equation and the KdV-Kaup-Kupershmidt (KdV-KK) equation which, albeit not proven to be integrable, possesses multi-soliton solutions.     

Fractional diffusion equation in half-space with Robin boundary condition

The initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.     

Boundary Singularity of Moments for the Linearized Boltzmann Equation

We study the boundary singularity for stationary solutions of the linearized Boltzmann equation with hard-sphere potential. An asymptotic estimate for the gradient of the moments is established, which shows the logarithmic singularity near the boundary. Our formula holds for the solutions of the Milne and Kramers problems obtained by Bardos-Caflish-Nicolaenko (Commun. Pure Appl. Math. 49:323–452, 1986). Our theorem requires the Hölder continuity of the boundary data. In particular, it applies to the complete condensation problem for half space.     

Accurate Solutions of the Navier-Stokes Equations Despite Unknown Outflow Boundary Data

A very common procedure when constructing boundary conditions for the time-dependent Navier-Stokes equations at artificial boundaries is to extrapolate the solution from grid points near the boundary to the boundary itself. For supersonic outflow, where all the characteristic variables leave the computational domain, this leads to accurate results. In the case of subsonic outflow, where one characteristic variable enters the computational domain, one cannot in general expect accurate solutions by this procedure. The problem with outflow boundary operators of extrapolation type at artificial boundaries with errors in the boundary data of order one will be investigated. Both the problem when the artificial outflow boundary is located in essentially uniform flow and the situation when the artificial outflow boundary is located in a flow field with large gradients are discussed. It will be shown, that in the special case when there are large gradients tangential to the boundary, extrapolation methods can be used even in the subsonic case.     

Boundary conditions in the radiative relaxation problem

The problem of the dissipation of temperature perturbations in a finite homogeneous atmosphere is solved for the situation in which the temperature at one boundary is maintained constant (that is, the temperature perturbation is zero for all times) while energy can be freely radiated to space through the other boundary. Exact solutions are shown for the exponential-sum fit to the kernel of the basic integral equation. These solutions constitute the set of radiative eigenfunctions. Also, approximate solutions in terms of the radiative eigenfunctions in the diffusion approximation (one exponential term in the expansion of the kernel) are obtained. These, in turn, are used in the solution of an initial value problem. The constant temperature boundary condition simulates the interface between two regions in one of which the relaxation processes are much more rapid than the purely radiative relaxation of the other.     

On some boundary value problems on a strip in the complex plane

The Sommerfeld integral inversion method for the Helmholtz equation in an angular region with different boundary values leads to boundary value problems in an infinite strip of the complex plane. We investigate a generic system for such boundary value problems and give the existence and uniqueness results with optimal growth estimates on the solution. We also give the solutions of the Dirichlet problem in a strip when the boundary functions grow exponentially.     

On the weakly damped vibrations of a vertical beam with a tip-mass

In this paper the wind-induced, horizontal vibrations of a weakly damped vertical Euler-Bernoulli beam with and without a tip-mass will be studied. The damping is assumed to be boundary damping and global Kelvin-Voigt damping. The boundary damping is assumed to be proportional to the velocity of the beam at the top. The horizontal vibrations of the beam can be described by an initial-boundary value problem. In this paper, the multiple-timescales perturbation method will be applied to construct approximations of the solutions of the problem. Also it will be shown that a combination of boundary damping and Kelvin-Voigt damping can be used to damp the wind-induced vibrations of a vertical beam with tip-mass uniformly.     

Superpositions of the diffusion-vortex solutions and flow control inside infinite domains

Superpositions of self-similar solutions of diffusion-vortex type are constructed and analyzed. These solutions simulate a one-dimensional acceleration from the quiescent state of a Newtonian medium in a half-plane (a planar problem) or in the space without a straight line (an axially symmetric problem) under the action of a combination of force factors, namely, of the pressure drop along the entire infinite thickness of the domain and of the tangent stress at the boundary. For the existence of a classical parabolic self-similar variable, these loadings must change, each according to its own temporal law, which is singular as t → 0. It is proved that, when prescribing, for example, the pressure drop, one can control the tangent stress along the boundary in such a way that the points inside the domain at which the stress either vanishes or has a local extremum move from the boundary deep down with a prescribed rate coefficient.     

奇异微分方程边值问题的数值解法

本文给出求解奇异微分方程边值问题的正则化方法。解在奇点邻域内展开成级数形式,在余下区间上推导出正则边值问题,应用差分方法求解,并给出收敛结果和数值算例。     

The formulation of spontaneous breakdown in the path-integral method

We examine the problem of formulating spontaneous breakdown of symmetries in the path-integral framework; since different solutions of the field equations correspond to different boundary conditions, the possibility of different boundary conditions must be incorporated in the generating functional itself. We propose a modification of the generating functional that provides the required flexibility that can lead to different solutions. We apply our formalism to a model of scalar complex fields with invariance under phase transformations of the first kind and derive Ward-Takahashi identities and the Goldstone theorem.     

Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

We study the integrable structure of the Dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply-connected domains. The solution to the Dirichlet boundary problem in the multiply-connected case is given through a quasiclassical tau-function, which generalizes the tau-function of the dispersionless Toda hierarchy. It is shown to obey an infinite hierarchy of Hirota-like equations which directly follow from properties of the Dirichlet Green function and from the Fay identities. The relation to multi-support solutions of matrix models is briefly discussed.     

一类非线性方程激波解的Sinc-Galerkin方法

研究了一类非线性奇摄动方程的激波问题.利用Sinc-Galerkin方法,构造出边值问题的激波解,并由Newton法得到其近似解. 关键词： 非线性方程 激波 Sinc-Galerkin方法 Newton法     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.     

Inverse scattering transforms of the inhomogeneous fifth-order nonlinear Schrödinger equation with zero/nonzero boundary conditions

The present work studies the inverse scattering transforms (IST) of the inhomogeneous fifth-order nonlinear Schrödinger (NLS) equation with zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs). Firstly, the bound-state solitons of the inhomogeneous fifth-order NLS equation with ZBCs are derived by the residue theorem and the Laurent’s series for the first time. Then, by combining with the robust IST, the Riemann-Hilbert (RH) problem of the inhomogeneous fifth-order NLS equation with NZBCs is revealed. Furthermore, based on the resulting RH problem, some new rogue wave solutions of the inhomogeneous fifth-order NLS equation are found by the Darboux transformation. Finally, some corresponding graphs are given by selecting appropriate parameters to further analyze the unreported dynamic characteristics of the corresponding solutions.     

Investigation of the boundary value problem corresponding to a generalized Skyrme lagrangian

A detailed numerical analysis of the boundary value problem resulting from the most general Skyrme type lagrangian containing up to quartic terms in field gradients is presented. The additional parameters in the lagrangian can be related to pion-pion scattering lengths. It is found that solutions to the boundary value problem does not exist for all values of the parameters and in particular, for the values predicted from pion-pion scattering data. Physical quantities of the nucleon are calculated for the highest possible values of the parameters admitting a solution and are compared with the corresponding values for the Skyrme model and experimental values.