Method of solving diffusion boundary-value problems for a region with a boundary moving in accordance with an arbitrary law

A general method is proposed for solving the boundary-value problem of the diffusion equation in a limited region with a boundary that moves in accordance with an arbitrary law. The method is used to solve the first linear diffusion problem. Other boundary-value problems can be solved in similar fashion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 97–101, December, 1970.     

The Theory of Block Structures in Problems on the Strength of Galleries and Constructions with Multiple Connections

Doklady Physics - It is shown that the boundary-value problem for a layered medium with parallel multiple cavities is reduced to the Riemann vector problem. To solve it, a factorization method is...     

Efficient method for solving the problems of wave diffraction by scatterers with broken boundaries

The pattern equations method is extended to solving the problems of wave scattering by bodies with piecewise smooth boundaries. The method is based on the reduction of the initial boundary-value problem to an integro-operator equation of the second kind in the scattering pattern of a body. With the use of the series expansion of the scattering pattern in angular spherical harmonics, the problem is ultimately reduced to solving an infinite algebraic system of equations in the expansion coefficients of the scattering pattern. The conditions at which this system can be solved by the method of reduction are formulated. Examples of solving the problems of wave scattering by bodies with impedance boundaries are considered. Essential advantages of the proposed method over other known methods are demonstrated.     

Analytical solutions to heat transfer equations by homotopy-perturbation method revisited

In this Letter, the problem of determining the temperature distribution of a straight rectangular fin with power-law temperature-dependent surface heat flux is revisited. The solution of the corresponding nonlinear boundary-value problem (BVP) based on the homotopy-perturbation method (HPM) presented earlier is corrected and extended.     

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.     

Electromagnetic-Wave Propagation in a Half-Space with a Curvilinear Impedance Boundary

We study propagation of electromagnetic waves in a vacuum half-space with a one-dimensional distortion of the impedance boundary whose profile is smooth on the wavelength scale. We suggest a method allowing one to obtain the solution of this problem on the basis of a more simple boundary-value problem with the same geometry of the boundary, but with a zero Dirichlet boundary condition.     

A new method for solving the problem of diffraction by a compact obstacle in a layered medium

We propose a new method for solving the two-dimensional problem of diffraction by a cylindrical body in a layered medium. Unlike the traditional methods, the boundary-value problem is reduced to solving the integral equation for the scattering pattern of the body. As an illustration of this method, we consider the problem of scattering by a round cylinder located in a dielectric layer between two homogeneous half-spaces. Technical University of Communication and Information Science, Moscow, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 7, pp. 874–888, July, 1998.     

Analytical approach for solving the radiative transfer equation in two-dimensional layered media

This study presents an analytical approach for obtaining Green's function of the two-dimensional radiative transfer equation to the boundary-value problem of a layered medium. A conventional Fourier transform and a modified Fourier series which is defined in a rotated reference frame are applied to derive an analytical solution of the radiance in the transformed space. The Monte Carlo method was used for a successful validation of the derived solutions.     

用边界元法分析非均匀介质中的传输线

 从静电场边值问题的积分解出发,推导出用边界元法求解分区均匀介质填充传输线问题的矩阵表达式,给出传输线电容参数的计算公式,介绍用边界元法求解分区均匀介质填充传输线问题的基本原理和求解过程。对两类传输线的计算结果表明：用边界元法求解分区均匀介质填充传输线问题,不仅具有较高的计算精度,而且可以很方便地应用于各类复杂截面分区均匀介质填充传输线问题的工程设计与计算,边界元法是求解分区均匀介质填充传输线问题的一种有效方法。     

Riemann-Hilbert problems for the Ernst equation and fibre bundles

Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The solution of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.     

An efficient high-order boundary element method for nonlinear wave–wave and wave-body interactions

A highly efficient high-order boundary element method is developed for the numerical simulation of nonlinear wave–wave and wave-body interactions in the context of potential flow. The method is based on the framework of the quadratic boundary element method (QBEM) for the boundary integral equation and uses the pre-corrected fast Fourier transform (PFFT) algorithm to accelerate the evaluation of far-field influences of source and/or normal dipole distributions on boundary elements. The resulting PFFT–QBEM reduces the computational effort of solving the associated boundary-value problem from O(N^2～3) (with the traditional QBEM) to O(N ln N) where N represents the total number of boundary unknowns. Significantly, it allows for reliable computations of nonlinear hydrodynamics useful in ship design and marine applications, which are forbidden with the traditional methods on the presently available computing platforms. The formulation and numerical issues in the development and implementation of the PFFT–QBEM are described in detail. The characteristics of accuracy and efficiency of the PFFT–QBEM for various boundary-value problems are studied and compared to those of the existing accelerated (lower- and higher-order) boundary element methods. To illustrate the usefulness of the PFFT–QBEM, it is applied to solve the initial boundary-value problem in the generation of three-dimensional nonlinear waves by a moving ship hull. The predicted wave profile and resistance on the ship are compared to available experimental measurements with satisfactory agreements.     

Temperature measurement of a surface exposed to a low-temperature plasma flux

A method of measuring the surface temperature in the region exposed to a low-temperature plasma flux is suggested. The boundary-value problem of heat conduction for the unsteady case is analytically solved by means of the Laplace transformation. Analysis of the analytical relationships shows that this method measures temperature with a high accuracy in real time. Its application in formation of micro-and nanostructures in a low-temperature plasma is considered.     

Bound State Solutions of the Klein-Gordon Equation for the Mathews-Lakshmanan Oscillator

We study a boundary-value problem for the Klein-Gordon equation that is inspired by the well-known Mathews-Lakshmanan oscillator model. By establishing a link to the spheroidal equation, we show that our problem admits an infinite number of discrete energies, together with associated solutions that form an orthogonal set in a weighted L ²-Hilbert space.     

Ground-force- or plate-displacement-based vibrator control?

The feedback system of seismic vibrators minimizes the difference between the theoretical sweep and the seismic signal radiated into the earth. This requires the determination of the correct source signature, that is, the characteristic of the baseplate motion that is a true representation of the downgoing wave. The prevalent systems use ground-force-based vibrator control, which assumes that the amplitudes of the seismic waves are proportional to the force applied to the ground. Mathematical solution of the pressure boundary-value problem of linear elasticity is the theoretical underpinning of this method. However, the ground force has a significant disadvantage of not being a directly measurable parameter. It has to be inferred, with significant uncertainty, as a “weighted sum” of the accelerations of the baseplate and the reaction mass based on a simplified linear model of oscillating masses, springs, and dashpots. On the other hand, the baseplate displacement (acceleration) can itself serve as an alternative, model-independent source signature. To provide a theoretical justification for the displacement-based vibrator control, one has to show that the amplitudes of the outgoing waves are proportional to the plate displacement. This has to be done by solving a “mixed” boundary-value problem, in which displacements are specified under the plate and pressures everywhere else on the surface. Such a problem has historically been considered unsolvable, unlike the mathematically easier pressure-source radiation problem. However, the analysis of approximate approaches to the solution of the mixed boundary-value problem, as well as its exact solution for the static-displacement case, reveal equivalence between the source displacement and source force in being the proportionality factors controlling the amplitudes of the downgoing waves. This provides mathematical justification for the displacement being the correct source signature. The displacement-based feedback, or feedback based on a combination of the weighted-sum and displacement, may be a more accurate method.     

Radiation Impedances of Disc-Shaped Antennas for an Electron Plasma Wave

The radiation impedances of disc and circular mesh antennas in the high-frequency region were measured. As a result, it is shown that measured impedances in the frequency region of the electron plasma-wave propagation agree with the theoretically predicted ones, which are obtained by solving a boundary-value problem.     

Simulation of deformation and fracture of coated material with account for propagation of a Lüders-Chernov band in the steel substrate

The paper studies deformation and fracture of boride-coated steel. The dynamic boundary-value problem is solved in the plane strain statement by the finite difference method. The geometry of the coating-substrate interface corresponds to those experimentally observed and is explicitly specified in the calculations. The mechanical response of the steel substrate is described by an elastoplastic model of isotropically hardening material with relations for slow flows. The peculiarities of plastic strain localization and fracture during the propagation of a Lüders-Chernov band in the steel substrate are investigated under tension.     

A diffusion problem in a two-medium system

A solution is obtained to the boundary-value diffusion-equation problem in a system of two anisotropic semiinfinite media, separated by a movable boundary region. The solution is obtained for arbitrary initial values and concentration on the boundary region. Thus, this class of boundary-value problems is completely solved in the given range.     

Interaction of an atom with a laser field of intraatomic strength

The specific features of the nonlinear optical response of an atom at a laser pulse field of intraatomic strength have been investigated. An effective mathematical method, based on the expansion of the wave function in the basis of eigenfunctions of an axially symmetric boundary-value problem, is proposed to calculate the atomic response.     

Mathematical modeling of waveguides with fractal insets

The paper is devoted to mathematical modeling of rectangular dielectric waveguides with local periodic and fractal insets. A three-dimensional scalar boundary-value problem for the Helmholtz equation is considered. The solution is obtained numerically by means of incomplete Galerkin method and finite-difference techniques. The transmission spectra for one-, two- and three-dimensional periodic and fractal insets are computed and compared with each other. Practical applications are suggested.