A Modified Kernel Method for Solving Cauchy Problem of Two-Dimensional Heat Conduction Equation

In this paper, a Cauchy problem of two-dimensional heat conduction equation is investigated. This is a severely ill-posed problem. Based on the solution of Cauchy problem of two-dimensional heat conduction equation, we propose to solve this problem by modifying the kernel, which generates a well-posed problem. Error estimates between the exact solution and the regularized solution are given. We provide a numerical experiment to illustrate the main results.     

线性瞬态涡流电磁场定解问题解的唯一性和稳定性

线性瞬态涡流电磁场定解问题的主要特点是边界条件使用磁感应强度的法向分量边界条件代替了电场强度的切向分量边界条件，约束方程中忽略了位移电流.这种具有特殊性的定解问题的解是否唯一和稳定对于求解瞬态涡流电磁场而言是一个基本问题.本文在非涡流区引入标量位函数，证明了在推导过程中起重要作用的辅助函数的存在性.通过推导线性瞬态涡流电磁场定解问题的能量估计式，证明了该定解问题的解是唯一的，并且关于初始条件和外源项是稳定的.本结果对于线性瞬态涡流电磁场的求解有一定的指导意义.作为应用，给出了通有单脉冲电流的单匝圆环线圈与球形导体共轴的涡流问题的解析解. 关键词： 瞬态涡流电磁场 能量估计式 唯一性 稳定性     

Three-dimensional MHD flow over a shrinking sheet: Analytical solution and stability analysis

The magnetohydrodynamic(MHD) steady and unsteady axisymmetric flows of a viscous fluid over a two-dimensional shrinking sheet are addressed. The mathematical analysis is carried out in the presence of a large magnetic field. The steady state problem results in a singular perturbation problem having an infinite domain singularity. The secular term appearing in the solution is removed and a two-term uniformly valid solution is derived using the Lindstedt–Poincaré technique. This asymptotic solution is validated by comparing it with the numerical solution. The solution for the unsteady problem is also presented analytically in the asymptotic limit of large magnetic field. The results of velocity profile and skin friction are shown graphically to explore the physical features of the flow field. The stability analysis of the unsteady flow is made to validate the asymptotic solution.     

Propagation of electromagnetic dipole waves through dielectric interfaces

The problem of divergent electromagnetic dipole waves propagating through parallel dielectric interfaces is solved. The solution is obtained in an analytic form that can be readily evaluated numerically. The result is obtained as a solution to a boundary-value problem. Applications of the solution are described.     

Knudsen Layer for Gas Mixtures

The Knudsen layer in rarefied gas dynamics is essentially described by a half-space boundary-value problem of the linearized Boltzmann equation, in which the incoming data are specified on the boundary and the solution is assumed to be bounded at infinity (Milne problem). This problem is considered for a binary mixture of hard-sphere gases, and the existence and uniqueness of the solution, as well as some asymptotic properties, are proved. The proof is an extension of that of the corresponding theorem for a single-component gas given by Bardos, Caflisch, and Nicolaenko [Comm. Pure Appl. Math. 39:323 (1986)]. Some estimates on the convergence of the solution in a finite slab to the solution of the Milne problem are also obtained.     

Model of electron beam transformation in a narrow tube

The problem of transformation of an electron beam in a long narrow conducting tube is considered. General wave solutions for the propagation of an arbitrary (including aperiodic) modulation of a homogeneous beam are obtained in the case of weak modulation. The solution corresponding to harmonic modulation is compared to the solutions obtained earlier and described in the literature. The problem of beam transformation corresponding to the bunching regime in high-power klystrons is also considered (including overtaking). A discrete analytic solution to the problem is obtained in the ??frozen beam approximation.?? This solution is tested for the problem of spilling of a monovelocity bunch and for the problem of the efficiency of a two-cavity klystron with infinitely narrow gaps. In both cases, physically correct results are obtained.     

Diffraction and radiation of flexural waves by a circular aperture loaded along the edge

An inhomogeneous wave exponentially decaying in all directions may exist near the edge of a circular aperture if the mass and the moment of inertia are appropriately chosen (i.e., their values coincide with the critical ones) [1]. In this case, the solution to the diffraction problem is not unique. This paper inquires into the feasibility of the solution to the wave diffraction and radiation problem in the case of critical loading. The solution is constructed according to the procedure suggested in [2]. It is shown that the diffraction problem always has a solution while the radiation problem may have no solution. The effects related to the critical mass and the critical moment of inertia do not manifest themselves in the far field.     

Scattering of plane waves by a wedge with different face impedances

The scattering process of plane waves by a wedge with different face impedances is examined in terms of the closed form series solution. A new boundary condition is derived using the solution of the reflection problem of plane waves by an impedance plane. The series solution is obtained for the wedge problem. The results are investigated numerically.     

On static,axially symmetric Einstein-Maxwell fields. Part I

The Weyl solution of the problem, obtained on the assumption thatg ₀₀ is a function of the electrostatic potential, is varied and the linearized field equations for the variation are discussed. The complete solution of the problem is determined for the special case of the Weyl solution that generalizes the Reissner-Nordstrom solution withm = ¦e¦.     

Total reflection and canalized states in a three-dimensional quantum model of thin films

A peculiar particular solution of the particle scattering problem in a three-dimensional quantum model of thin films is found. In the one-layer version, this solution is related to the total reflection phenomenon. So-called canalized states are found at positive energy in the two-layer version of the problem.     

Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem

The complex variable method for the plane elasticity theory of icosahedral quasi-crystals is developed. Based on the general solution obtained previously, complex representations of stress and displacement components of phonon and phason fields in the quasicrystals are given. With the help of conformal transformation, an analytic solution for the elliptic notch problem of the material is presented. The solution of the Griffith crack problem can be observed as a special case of the results. The stress intensity factor and energy release rate of the crack are also obtained.     

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     

Angular momentum projected Hartree-Fock-Theory

The Euler Lagrange equations corresponding to the Hartree-Fock variational problem after angular momentum projection (PHF) are derived. The exact equations are simplified employing the Gaussian overlap approximation. The solution of the PHF equations in this approximation is seen to be not much more complicated than the solution of the normal HF problem.     

A comparison of the stochastic theory of the crystal growth and the Stefan problem

The stochastic theory of the crystal growth is compared with the solution of the Stefan problem in the case of Sn solidification. It is shown that the stochastic theory gives the same results as the solution of the Stefan problem if kinetic processes at the solidification front are very rapid.     

ON THE DISCRETIZATION OF AN ELASTIC ROD WITH DISTRIBUTED SLIDING FRICTION

A one-dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction. A partial differential equation of motion is established and its exact solution is presented. Due to the friction the eigenvalue problem is non-self-adjoint. Mathematical methods for handling the non-self-adjoint system, such as the non-self-adjoint eigenvalue problem and the eigenvalue problem with a proper inner product, are reviewed and applied. The exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant. The assumed mode approximation and the lumped-parameter discretization method are evaluated and their solutions are compared with the exact solution. As a cautionary example the assumed modes approximation leads to false conclusions about stability. The lumped-parameter discretization algorithm generates reliable results.     

A Method for Solving Initial-value Problem and Finding Exact Solutions of Nonlinear Partial Differential Equations

Homogeneous balance method for solving nonlinear partial differential equation(s) is extended to solving initial-value problem and getting new solution(s) from a known solution of the equation(s) under consideration. The approximate equations for long water waves are chosen to illustrate the method, infinitely many simple-solitary-wave solutions and infinitely many rational function solutions, especially the closed form of the solution for initial-value problem, are obtained by using the extended homogeneous balance method given here.     

Optimized method for solving the inverse problem for a triangular plasmonic waveguide using a response of scanning heterodyne microscope

A method proposed previously to solve the inverse problem for triangular plasmonic waveguides using a TE-polarized response of scanning differential heterodyne microscope has been optimized. The use of the phase contrast of microscope response at two wavelengths as initial data made it possible to reduce significantly the solution error and eliminate the instability domains of initial data, in which the solution error dramatically increased. It is demonstrated that the accuracy of determining the waveguide parameters significantly improves in comparison with the case where the amplitude and phase contrast of response at one wavelength are used as initial data. It is shown that the solution of the inverse problem can be further optimized using contrasts of amplitude responses as additional initial data. The dependences of the error of inverse problem solution on the error in determining the initial data are calculated.     

Inverse problem of dyon and magnetic monopole dynamics

The problem of finding an electromagnetic field allowing the movement of dual-charged particles on the background of an arbitrary static space-time to have a priori given properties, is considered. A solution of this problem is given, and the degree of arbitrariness of the obtained solution is found. Two particular examples of application of the general results are considered.     

Capacitance of a tube

The electrostatic problem of a hollow, conducting tube of finite length held at a fixed potential is solved using two methods. A two-term Galerkin solution is constructed for the surface distribution of induced charge. The sum of a uniform component and a simple edge-condition term provides a variational solution to the dual integral equations that are the equations-of-motion for the mixed boundary value problem. Comparisons are made with the numerical results of an independent boundary element or moment method. The numerical solution uses collocation or point matching and a piecewise constant basis for the charge density.