Analysis of Time–Domain Maxwell's Equations for 3-D Cavities

Time–domain Maxwell's equations are studied for the electromagnetic scattering of plane waves from an arbitrarily shaped cavity filled with nonhomogeneous medium. A transparent boundary condition is introduced to reduce the problem to the bounded cavity. Existence and uniqueness of the model problem are established by a variational approach and the Hodge decomposition. The analysis forms a basis for numerical solution of the model problem.     

On an exact solution of master equations for the model of reversible growth

The set of master equations for the monomer quasi-chemical reversible-growth model in a heterogeneous open medium material is studied. The exact solution to the master equations is obtained for the case where the velocity constants for the growth and decay reactions are linear in the particle number. It is shown that the model under consideration is canonically invariant w.r.t. the Polya distribution with a time-dependent mean size of the clusters.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 327–336, August, 1996.     

凝析油-气两相流偏微分方程组问题的求解及应用

根据凝析气藏开发的实际需要，引入凝析油－气两相拟压力、拟时间函数，建立了凝析气在双重渗透率介质中的不稳定渗流新模型，该模型为一类偏微分方程组的混合问题，本文研究了这类偏微分方程组问题的求解方法。     

A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.     

Two-dimensional isotropic scattering

An exact formulation for the source function, the radiative flux, and the intensity is presented for a two-dimensional, finite, planar, isotropically scattering medium. Exact expressions are obtained for both collimated and diffuse radiation. Cosine varying collimated and diffuse radiation is considered in detail. The solution for the cosine varying collimated radiation model is used to construct the solutions for the other boundary conditions. The two-dimensional integral equations are reduced to one-dimensional equations by the method of separation of variables.     

Mathematical modeling of drug delivery from cylindrical implantable devices

A mechanistic mathematical model applicable to the controlled dispersed‐drug release from cylindrical device such as implantable drug delivery system was derived. Analytical solutions based on the pseudosteady state approximation are derived taken account an exact external medium volume. The model prediction is accurate when the initial drug load is higher than the drug solubility in the polymer. The results obtained are compared with the analytical solutions available in the literature. The equations are corroborated by comparison with experimental profiles reported in the literature for sink conditions and non sink conditions. The evolution of concentration distribution profiles is compared for different volume of external medium. A reduction in the volume of the external solution leads to an increase in the concentration on the surface of the device, which determines decreases in the release of drug. One criterion for determining whether the volume of external solution should be considered for the prediction of drug release from cylindrical devices is established. This criterion is based on establishing a maximum percentage error allowed in the values of amount of drug released. The usefulness of the model is focused in the design of implant for controlled release of drug into a small volume of external medium of release. Copyright © 2014 John Wiley & Sons, Ltd.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

The breaking of a non-homogeneous fiber embedded in an infinite non-homogeneous medium

The torsion of an infinite non-homogeneous elastic cylindrical fiber, containing a penny-shaped crack embedded in an infinite non-homogeneous elastic material is considered. The cylinder and elastic medium have different shear moduli. Using integral transformation techniques the solution of the problem is reduced to the solution of dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind, which is solved numerically. Closed form expressions are obtained for the stress intensity factor and numerical values for the stress intensity factors are graphed to demonstrate the effect of non-homogeneity of the fiber and infinite medium. In the end the stress singularity is obtained when the crack touches the infinite non-homogeneous medium (matrix).     

The steady axisymmetric flow of ideally plastic materials in a conical channel

The problem of the asymmetric flow of an ideally plastic medium is formulated within the framework of the von Mises model and the total plasticity condition, using the invariant condition of compatibility for the deviator component of the stress tensor. Flow in a converging conical channel, on the boundary of which the shear stresses are specified, is considered. First-order differential equations are obtained, describing the shear-stress distribution in the moving medium, one of which corresponds to the von Mises model, and the other to the total plasticity condition. It is established from an analysis of the solution in the neighbourhood of singular points, that the minus sign in front of the radical in these equations corresponds to positive shear stresses and vice versa. The problem of the shear stresses reaching a maximum value on the specified boundary surface of the channel is investigated. The aperture angle of the channel, beginning from which this value is reached, is determined. It is established that the value of the angle, following from the total plasticity condition, somewhat exceeds its value obtained within the framework of the von Mises model.     

Solution of inverse problems in the class of quasi-one-dimensional functions

Solution of the inverse problem of magnetotelluric sounding in a two-dimensional medium is considered. The solution methods are based on the assumption that the distribution of electrical conductivity is representable as the sum of a slowly varying component and a small increment. This model makes it possible to introduce small parameters in the equations of the problem, which are used for the construction of a sequence of functions approximating the solution of the inverse problem.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 160–175, 1986.     

A Runge-Kutta/WENO method for solving equations for small-amplitude wave propagation in a saturated elastic porous medium

A high-accuracy Runge-Kutta/WENO method of up to fourth order with respect to time and fifth order with respect to space is developed for the numerical modeling of small-amplitude wave propagation in a steady fluid-saturated elastic porous medium. A system of governing equations is derived from a general thermodynamically consistent model of a compressible fluid flow through a saturated elastic porous medium, which is described by a hyperbolic system of conservation laws with allowance for finite deformations of the medium. The results of numerical solution of one- and two-dimensional wave fields demonstrate the efficiency of the method.     

横观各向同性弹性层点力解

本文根据弹性层状结构的传递矩阵法思想，由横观各向同性弹性力学基本方程，导出了含应力和位移两类变量的混合方程，利用Ｆｏｕｒｉｅｒ变换和文献［７］的位移函数通解，以及计算机代数软件，得到了横观各向同性层的点力解，这个点力解可直接退化到各同性情形的解．     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Traveling Wave Solutions for Combustion in a Porous Medium

Traveling wave solutions are sought for a model of combustion in a porous medium. The problem is formulated as a nonlinear eigenvalue problem for a system of ordinary differential equations of order four, defined over an infinite interval. A shooting method is used to prove existence, and a priori bounds for the solution and parameters are obtained.     

Two-dimensional radiative equilibrium

An exact formulation for the radiative flux and the emissive power is presented for a two-dimensional, finite, planar, absorbing-emitting, gray medium in radiative equilibrium. Exact expressions are obtained for a medium subjected to the following types of boundary conditions: (A) cosine varying collimated radiation, (B) a strip of collimated radiation, (C) cosine varying diffuse radiation, and (D) a uniform temperature strip. The solution for the cosine varying collimated radiation model is used to construct the solutions for the other boundary conditions. The two-dimensional equations are reduced to one-dimensional equations by the method of separation of variables.     

Algorithms and numerical analysis of dc fields in a piecewise-homogeneous medium by the boundary integral equation method

Boundary integral equation methods are considered for computing dc fields in three-dimensional regions filled with a piecewise-homogeneous medium. The problem is formulated and a system of Fredholm boundary integral equations of first kind is constructed, following directly from Green’s formula. The numerical solution stages are considered in detail, including construction and triangulation of the numerical surfaces, evaluation of surface integrals, and solution of a system of block-matrix equations. Translated from Prikladnaya Matematika i Informatika, No. 30, pp. 35–45, 2008.     

Wave Propagation in an Elastic Medium Intersected by Systems of Parallel Fractures

An elastic anisotropic medium intersected by systems of parallel fractures is investigated. Every fracture is considered as a plane boundary with jumps of displacements and stresses, and these jumps are linear functions of displacements and stresses averaged on the boundary. For this medium, an effective model is constructed by the method of matrix averaging. The equations of this model describe wave propagation in the given medium and are more complicated than the equations of elasticity theory. In particular cases, the equations obtained are converted to the equations of elastic media. On the basis of the equations of the effective model, expressions for the densities of the kinetic and potential energies are derived, and conditions of absoption in the medium are established. Bibliography: 15 titles.     

Solving the 2D Maxwell equations by a Laguerre spectral method

A spectral method for solving the 2D Maxwell equations with relaxation of electromagnetic parameters is presented. The method is based on an expansion of the solution in terms of Laguerre functions in time. The operation of convolution of functions, which is part of the formulas describing the relaxation processes, is reduced to a sum of products of the harmonics. The Maxwell equations transform to a system of linear algebraic equations for the solution harmonics. In the algorithm, an inner parameter of the Laguerre transformis used. With large values of this parameter, the solution is shifted to high harmonics. This is done to simplify the numerical algorithm and to increase the efficiency of the problem solution. Results of a comparison of the Laguerre method and a finite-difference method in accuracy both for a 2D medium structure and a layered medium are given. Results of a comparison of the spectral and finite-difference methods in efficiency for axial and plane geometries of the problem are presented.