Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of the MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of the MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.     

三维Helmholtz方程外问题的自然积分方程及其数值解

用文[2,3]提出的自然边界归化方法来处理三维Helmholtz方程的外边值问题。在简要介绍如何用球谐展开的方法得到Helmholtz问题在外球域上的自然积分方程后,给出求解该自然积分方程的一种数值方法及相应的数值算例。     

Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential

A numerical procedure to solve the linearized Boltzmann equation with an arbitrary intermolecular potential by the discrete velocity method is elaborated. The equation is written in terms of the kernel, which contains the differential cross section and represents a singularity. As an example, the Lennard-Jones potential is used and the corresponding differential cross section is calculated and tabulated. Then, the kernel is calculated so that to overcome its singularity. Once, the kernel is known and stored it can be used for many kinds of gas flows. In order to test the method, the transport coefficients, i.e. thermal conductivity and viscosity for all noble gases, are calculated and compared with those obtained by the variational method using the Sonine polynomials expansion. The fine agreement between the results obtained by the two different methods shows the feasibility of application of the proposed technique to calculate rarefied gas flows over the whole range of the Knudsen number.     

铅球最佳投掷角的精确方程及数值解

指出关于铅球最佳投掷角计算公式中存在的问题,提出了计算这一角度的精确方程和数值解。     

Poiseuille problem for an ellipsoidal-statistical equation and nearly specular boundary conditions

An analytical solution of the Poiseuille problem is obtained over a wide range of Knudsen numbers for the case when the tangential momentum accommodation coefficients of the channel walls are much less than unity. An expression for the mass flux is derived that is valid for Knudsen numbers much smaller than the reciprocal of the accommodation coefficients. A new intermediate flow regime, for which the mass flux is given by an expression that differs from the classical (macroscopic) form, is found to exist. Zh. Tekh. Fiz. 68, 27–31 (November 1998)     

Radiation pressure on an inclusion with a given scattering amplitude in an arbitrary external field

A method is proposed for calculating the radiation pressure forces that act on a body characterized by a given scattering amplitude and positioned in an ideal liquid in an arbitrary acoustic field. The efficiency of the method is tested by two examples: a fluid sphere in a spherical wave and a similar sphere in the field of a traveling wave with a Gaussian amplitude distribution. The agreement with the results obtained earlier by other methods is demonstrated.     

Blow-up of solution of an initial boundary value problem for a generalized Camassa-Holm equation

In this Letter, we study the following initial boundary value problem for a generalized Camassa-Holm equation

     

Solution of the Dirichlet problem for the Laplace equation for a multiply connected region with point symmetry

A method is proposed which uses an expansion of the potential in irreducible representations of the symmetry group of the field-defining elements of a system. A boundary-value problem is solved for multipole systems with planar plate electrodes for the C _nv symmetry group. A quadrature expression is obtained for the field potential of these systems. Constraints imposed on the electrode potentials, under which such a solution is possible, are determined. Results of calculations of the potential distribution are presented for various specific systems. Zh. Tekh. Fiz. 69, 1–9 (March 1999)     

一个追线问题的数值解法及推广

对一个追线问题进行了讨论,指出了用解析法求解的局限性和数值解法的优点;并将问题推广到了一般情况,得到了有意义的结果．     

On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering

The authors developed a numerical method of the boundary-value problem solution in the vectorial radiative transfer theory applicable to the turbid media with an arbitrary three-dimensional geometry. The method is based on the solution representation as the sum of an anisotropic part that contains all the singularities of the exact solution and a smooth regular part. The regular part of the solution could be found numerically by the finite element method that enables to extend the approach to the arbitrary medium geometry. The anisotropic part of the solution is determined analytically by the special form of the small-angle approximation. The method development is performed by the examples of the boundary-value problems for the plane unidirectional and point isotropic sources in a turbid medium slab.     

Analytical solution to the skin-effect problem for an arbitrary accommodation coefficient of electron tangential momentum

An analytical solution to the boundary-value problem of an electric field and electrons in a metal-filled half-space is obtained for arbitrary values of the tangential-momentum accommodation coefficient. The frequency of an external electromagnetic field directed tangentially to the surface is allowed to take on complex values. Both the normal and anomalous skin effects are considered. In the latter case, the low-and high-frequency limits are examined.     

Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation

We study the Cauchy problem of generalized Boussinesq equation u_tt−u_xx+(u_xx+f(u))_xx=0, where f(u)=±|u|^p or ±|u|^p−1u, p>1. By introducing a family of potential wells we obtain invariant sets, vacuum isolating and threshold result of global existence and nonexistence of solution.     

A comment on an exact solution with a stiff equation of state

Recently, an analytical stellar model with a stiff equation of state and density distribution= _c (1-r ²/r _o ² ) was presented. We show that such a solution cannot exist.     

Example of nonexistence of a positive generalized entropy solution of a Cauchy problem with unbounded positive initial data

A Cauchy problem for a first-order quasilinear equation with polynomial flux function and exponential initial data is studied. We prove that this problem has no positive solution in the class of locally bounded functions.     

Mode equation and its solution for dielectric waveguide with parabolic cross-section

The mode equation for the dielectric waveguide with parabolic cross-section is derived using both the effective index method and WKB theory, and the expression of the mode propagation constant or effective index is solved directly from this mode equation, so that the mode-propagation properties of this kind of waveguide can be analysed conveniently with comparative accuracy.     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

Using the Galerkin method to construct an analytical solution for the integral equation of a certain diffraction problem

Grodno State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 31, No. 11, pp. 1355–1361, November, 1988.