几类特定边界值条件下的常微分方程特征值问题

在文献[1]的基础上,借助于图形分析方法,讨论了几类特定边界值条件下的特征值问题.同时,本文对文献[1]中的错误结论作出了更正.本文给出的方法对讨论非线性边值问题的可解性提供了一种新的途径.

一类带有Dirichlet边界值条件的椭圆型方程正解的存在性

利用欧拉变分原理,证明了一类带有Dirichlet边界值条件的椭圆型方程正解的存在性。

反应条件指数及其与饱和指数的关系初步探讨

反应条件指数(RCI)和饱和指数(SI)都是建立在热力学基础上说明水中物质溶解与沉淀作用的水文地球化学参数。饱和指数能够客观地反映水文地球化学作用发生的方向和状态,反应条件指数综合了水文地球化学状态条件和水文地球化学作用边界条件的信息,它不仅能反映水文地球化学作用发生的方向和状态,而且可以反映其发生的条件及原因。根据反应条件指数和反应条件边界值的概念,运用热力学基本原理和质量作用定律对反应条件指数与饱和指数的关系进行了论证,结果表明两者呈线性函数关系。这对认识和解决生产、科研中存在的水文地球化学问题具有重要指导意义。

带Navier边值条件的 (p_(1),…,p_(n))双调和方程组多解性研究

利用Ricceri的三临界点理论研究了带有Navier边值条件的(p_(1),…,p_(n))双调和方程组{Δ(|Δu_(i)|^( p i-2)Δu_(i))-α_(i)Δ_(p i) |u_(i)|+β_(i) u_(i) |^(p i-2) u_(i)=λF _(ui)(x,u_(1),…,u_(n)),x∈Ω,u_(i)=Δu_(i)=0,x∈∂Ω,弱解的存在性,得到了问题至少存在3个弱解的充分条件.

电流变方程解的稳定性

考虑一类电流变方程,如果方程的扩散系数在边界上退化,一般只能引入局部边界值条件。当α<p--1时,在局部边界值条件下,研究了解的稳定性;当α>p^++1时,在没有边界值条件的情况下,得到了解的稳定性。

具有变指数的退化抛物方程解的唯一性

考虑具有变指数的退化抛物方程u t=div(ραa(u)p(x)-2 a(u))+g(x)div(b(u))弱解的存在唯一性问题,其中ρ(x)=dist(x,∂Ω)是其到边界的距离函数,a(s)是一个严格单调上升的函数.通过选取合适的检验函数证明在无边界值条件情形下该方程弱解的唯一性成立.

Application of the double absorbing boundary condition in seismic modeling

We apply the newly proposed double absorbing boundary condition（DABC）（Hagstrom et al., 2014） to solve the boundary reflection problem in seismic finite-difference（FD） modeling. In the DABC scheme, the local high-order absorbing boundary condition is used on two parallel artificial boundaries, and thus double absorption is achieved. Using the general 2D acoustic wave propagation equations as an example, we use the DABC in seismic FD modeling, and discuss the derivation and implementation steps in detail. Compared with the perfectly matched layer（PML）, the complexity decreases, and the stability and fl exibility improve. A homogeneous model and the SEG salt model are selected for numerical experiments. The results show that absorption using the DABC is considerably improved relative to the Clayton–Engquist boundary condition and nearly the same as that in the PML.

一类拟线性椭圆型方程的正解

利用欧拉变分原理,说明一类带有Dirichlet边界值条件的拟线性椭圆型方程正解的存在性问题。

Improved absorbing boundary condition based on linear interpolation for ADI-FDTD method

With the linear interpolation method, an improved absorbing boundary condition（ABC）is introduced and derived, which is suitable for the alternating-direction-implicit finite- difference time-domain （ADI-FDTD） method. The reflection of the ABC caused by both the truncated error and the phase velocity error is analyzed. Based on the phase velocity estimation and the nonuniform cell, two methods are studied and then adopted to improve the performance of the ABC. A calculation case of a rectangular waveguide which is a typical dispersive transmission line is carried out using the ADI-FDTD method with the improved ABC for evaluation. According to the calculated case, the comparison is given between the reflection coefficients of the ABC with and without the velocity estimation and also the comparison between the reflection coefficients of the ABC with and without the nonuniform processing. The reflection variation of the ABC under different time steps is also analyzed and the acceptable worsening will not obscure the improvement on the absorption. Numerical results obviously show that efficient improvement on the absorbing performance of the ABC is achieved based on these methods for the ADI-FDTD.

Numerical simulation of an extreme haze pollution event over the North China Plain based on initial and boundary condition ensembles

The North China Plain often su ers heavy haze pollution events in the cold season due to the rapid industrial development and urbanization in recent decades.In the winter of 2015,the megacity cluster of Beijing Tianjin Hebei experienced a seven-day extreme haze pollution episode with peak PM2.5(particulate matter(PM)with an aerodynamic diameter≤2.5μm)concentration of 727μg m 3.Considering the in uence of meteorological conditions on pollu-tant evolution,the e ects of varying initial conditions and lateral boundary conditions(LBCs)of the WRF-Chem model on PM2.5 concentration variation were investigated through ensemble methods.A control run(CTRL)and three groups of ensemble experiments(INDE,BDDE,INBDDE)were carried out based on difierent initial conditions and LBCs derived from ERA5 reanalysis data and its 10 ensemble members.The CTRL run reproduced the meteorological conditions and the overall life cycle of the haze event reasonably well,but failed to capture the intense oscillation of the instantaneous PM2.5 concentration.However,the ensemble forecasting showed a considerable advantage to some extent.Compared with the CTRL run,the root-mean-square error(RMSE)of PM2.5 concentration decreased by 4.33%,6.91%,and 8.44%in INDE,BDDE and INBDDE,respectively,and the RMSE decreases of wind direction(5.19%,8.89%and 9.61%)were the dominant reason for the improvement of PM2.5 concentration in the three ensemble experiments.Based on this case,the ensemble scheme seems an e ective method to improve the prediction skill of wind direction and PM2.5 concentration by using the WRF-Chem model.

A truncated implicit high-order finite-difference scheme combined with boundary conditions

In this paper, first we calculate finite-difference coefficients of implicit finite- difference methods （IFDM） for the first and second-order derivatives on normal grids and first- order derivatives on staggered grids and find that small coefficients of high-order IFDMs exist. Dispersion analysis demonstrates that omitting these small coefficients can retain approximately the same order accuracy but greatly reduce computational costs. Then, we introduce a mirrorimage symmetric boundary condition to improve IFDMs accuracy and stability and adopt the hybrid absorbing boundary condition （ABC） to reduce unwanted reflections from the model boundary. Last, we give elastic wave modeling examples for homogeneous and heterogeneous models to demonstrate the advantages of the proposed scheme.

Solution of Terzaghi one-dimensional consolidation equation with general boundary conditions

Boundary conditions for the classical solution of the Terzaghi one-dimensional consolidation equation conflict with the equation＇s initial condition. As such, the classical initial-boundary value problem for the Terzaghi one-dimensional consolidation equation is not well-posed. Moreover, the classical boundary conditions of the equation can only be applied to problems with either perfectly pervious or perfectly impervious boundaries. General boundary conditions are proposed to overcome these shortcomings and thus transfer the solution of the Terzaghi one-dimensional consolidation equation to a well-posed initial boundary value problem. The solution for proposed general boundary conditions is validated by comparing it to the classical solution. The actual field drainage conditions can be simulated by adjusting the values of parameters b and c given in the proposed general botmdary conditions. For relatively high coefficient of consolidation, just one term in series expansions is enough to obtain results with acceptable accuracy.

Numerical Simulation of the Flow around Two-dimensional Partially Cavitating Hydrofoils

In the present study, a new approach is applied to the cavity prediction for two-dimensional （2D） hydrofoils by the potential based boundary element method （BEM）. The boundary element method is treated with the source and doublet distributions on the panel surface and cavity surface by usethe of the Dirichlet type boundary conditions. An iterative solution approach is used to determine the cavity shape on partially cavitating hydrofoils. In the case of a specified cavitation number and cavity length, the iterative solution method proceeds by addition or subtraction of a displacement thickness on the cavity surface of the hydrofoil. The appropriate cavity shape is obtained by the dynamic boundary condition of the cavity surface and the kinematic boundary condition of the whole foil surface including the cavity. For a given cavitation number the cavity length of the 2D hydrofoil is determined according to the minimum error criterion among different cavity lengths, which satisfies the dynamic boundary condition on the cavity surface. The NACA 16006, NACA 16012 and NACA 16015 hydrofoil sections are investigated for two angles of attack. The results are compared with other potential based boundary element codes, the PCPAN and a commercial CFD code （FLUENT）. Consequently, it has been shown that the results obtained from the two dimensional approach are consistent with those obtained from the others.

Navier’s slip condition on time dependent Darcy-Forchheimer nanofluid using spectral relaxation method

In industrial applications involving metal and polymer sheets, the flow situation is strongly unsteady and the sheet temperature is a mixture of prescribed surface temperature and heat flux. Further, a proper choice of cooling liquid is also an important component of the analysis to achieve better outputs. In this paper, we numerically investigate Darcy-Forchheimer nanoliquid flows past an unsteady stretching surface by incorporating various effects, such as the Brownian and thermophoresis effects, Navier’s slip condition and convective thermal boundary conditions. To solve the governing equations, using suitable similarity transformations, the nonlinear ordinary differential equations are derived and the resulting coupled momentum and energy equations are numerically solved using the spectral relaxation method. Through the systematically numerical investigation, the important physical parameters of the present model are analyzed. We find that the presence of unsteadiness parameter has significant effects on velocity, temperature, concentration fields, the associated heat and mass transport rates. Also, an increase in inertia coefficient and porosity parameter causes an increase in the velocity at the boundary.

An Alternative Method to Study Wave Scattering by Semi-infinite Inertial Surfaces

A new method to solve the boundary value problem arising in the study of scattering of two-dimensional surface water waves by a discontinuity in the surface boundary conditions is presented in this paper. The discontinuity arises due to the floating of two semi-infinite inertial surfaces of different surface densities. Applying Green＇s second identity to the potential functions and appropriate Green＇s functions, this problem is reduced to solving two coupled Fredholm integral equations with regular kernels. The solutions to these integral equations are used to determine the reflection and the transmission coefficients. The results for the reflection coefficient are presented graphically and are compared to those obtained earlier using other research methods. It is observed from the graphs that the results computed from the present analysis match exactly with the previous results.

Laser-Manipulating Macroscopic Quantum States of a Bose-Einstein Condensate Held in a Kronig-Penny Potential

We investigate the boundary vaJue problem （BVP） of a quasi-one-dimensional Gross-Pitaevskii equation with the Kronig-Penney potential （KPP） of period d, which governs a repulsive Bose-Einstein condensate. Under the zero and periodic boundary conditions, we show how to determine n exact stationary eigenstates {Rn} corresponding to different chemical potentials {μn} from the known solutions of the system. The n-th eigenstate P~ is the Jacobian elliptic function with period 2din for n = 1,2,…, and with zero points containing the potential barrier positions. So Rn is differentiable at any spatial point and R2 describes n complete wave-packets in each period of the KPP. It is revealed that one can use a laser pulse modeled by a 5 potential at site xi to manipulate the transitions from the states of {Rn} with zero Point x≠xi to the states of {Rn＇} with zero Point x= Xi. The results suggest an experimental scheme for applying BEC to test the BVP and to observe the macroscopic quantum transitions.

Positive Solutions of Boundary Value Problem for a Coupled System of Nonlinear Third-order Differential Equations

By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee the boundary value problems for a coupled system of nonlinear third-order differential equation has at least one positive solution. Some examples are given to verify our results.

Second Reference State and Complete Eigenstates of Open XYZ Chain

The second reference state of the open XYZ spin chain with non-diagonal boundary terms is studied. The associated Bethe states exactly yield the second set of eigenvalues proposed recently by functional Bethe Ansatz.

Numerical Homogenization of the Elliptic Problem with Rapidly Varying Tensor or Boundary Condition

Homogenization is concerned with obtaining the average properties of a material. The problem on its own has no easy solution, except in cases like the periodic case, when it can be obtained in closed form. In this paper we consider a numerical solution of the elliptic homogenization problem for the case of rapidly varying tensor or boundary conditions. The method makes use of an adaptive finite element method to correctly capture the rapid change in the tensor or boundary condition. In the numerical experiments we vary the mesh size and do a posteriori error analysis on test problems.

Numerical Solution of Some Diffusion Problems in 3-Layered 3D Domain

In this paper we consider averaging and finite difference methods for solving the 3-D boundary-value problem in a multilayered domain. We consider the metal concentration in the 3 layered peat blocks. Using experimental data the mathematical model for calculating the concentration of metal at different points in peat layers is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for the partial differential equations （PDEs） of the elliptic type of second order with piece-wise diffusion coefficients in the three layer domain. We develop here a finite-difference method for solving a problem of the above type with the periodical boundary condition in x direction. This procedure allows reducing the 3-D problem to a system of 2-D problems by using a circulant matrix.