Particular solutions of the equation of transfer

Particular solutions corresponding to various forms for the inhomogeneous source term are established for the equation of transfer with Lth order anisotropic scattering.     

非齐次燃耗方程数值解法

非齐次燃耗方程常用于描述具有显著核素迁移效应的核能系统中核素含量随时间的变化规律.国内外许多燃耗计算程序无法求解方程非齐次项含时的情况.本文在方程非齐次项能够被有限阶关于时间的泰勒展开逼近这一前提下,研究了非齐次项含时情况下方程的两种解法.首先通过Laplace变换推导出了方程基于线性子链方法的解析解形式,然后使用Carathéodory-Fejér方法计算出了方程矩阵级数解的近最佳Chebyshev有理逼近式.将两种方法在燃耗计算程序JBURN中实现,并进行了数值计算,绝大部分计算结果符合很好,部分结果在较长有效数字内仍能保持一致,验证了方法的正确性和精度.同时为求解具有其他非齐次项形式的燃耗方程提供了一种思路.     

Sound propagation above an inhomogeneous plane: Boundary integral equation methods

A boundary integral equation method is used to compute the sound pressure emitted by a harmonic source above an inhomogeneous plane. First, the theoretical aspects of the problem (behaviour of the pressure around the discontinuities,…) are studied. Then, a comparison between theoretical levels and experimental levels obtained in an anechoic room is presented. It shows that the boundary integral equation (BIE) method is quite convenient for solving this kind of problem. Two interesting results are pointed out: (i) if only a prediction of maximum sound levels is needed, the attenuation is the same for a cylindrical source, a spherical source and N spherical sources, and so it is possible to transform some three-dimensional problems into two-dimensional ones; (ii) a numerical method of computation of the sound field above an inhomogeneous plane does not provide a correct prediction if each part of the plane is not accurately described by the boundary condition chosen.     

A perturbation technique for the prediction of the displacement of a line-driven plate with discontinuities

A novel technique is presented for obtaining approximate analytic expressions for an inhomogeneous line-driven plate. The equation of motion for the inhomogeneous plate is transformed, and the transform of the total displacement is written as a sum of the solution for a homogeneous line-driven plate plus a term due to the inhomogeneity. The result is an integral equation for the transform of the inhomogeneous contribution. This expression may in general be solved numerically. However, by introducing a small parameter into the problem, it may be solved approximately using perturbation techniques. This series may not be convergent, but its convergence may be improved using Pade approximation. Results are presented for the case of a single mass discontinuity, and a distribution of mass discontinuities.     

Massive strings in higher-dimensional inhomogeneous spacetime

An inhomogeneous cosmological model with massive strings as source term is developed in a Kaluza-Klein type of spacetime where the usual space exhibits spherical symmetry and the extra dimension is taken in the form of a circle. When an arbitrary constant in our solution vanishes we get an inhomogeneous cosmological model for pure dust, which further reduces to some particular forms of well known homogeneous models under suitable adjustment of arbitrary constants. The dynamical behaviour of the model is studied and it is found that the extra space admits of dimensional reduction via the successive appearances of one minimum and one maximum throughout its evolution.     

From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.     

Variationally optimized numerical orbitals for molecular calculations

The problem of variational optimization of the atomic orbitals used in molecular calculations is investigated. It is shown that the variational principle leads to an equation similar to the radial Schrodinger equation but containing an inhomogeneous term. As an example, the equations are solved for the minimum basis set orbitals for the methane molecule. The results show a substantial improvement over those of a previous calculation optimizing in a minimum basis of Slater orbitals.     

On Average Properties of Inhomogeneous Fluids in General Relativity: Perfect Fluid Cosmologies

For general relativistic spacetimes filled with an irrotational perfect fluid a generalized form of Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies is derived. The averaging problem for scalar quantities is condensed into the problem of finding an "effective equation of state" including kinematical as well as dynamical "backreaction" terms that measure the departure from a standard FLRW cosmology. Applications of the averaged models are outlined including radiation-dominated and scalar field cosmologies (inflationary and dilaton/string cosmologies). In particular, the averaged equations show that the averaged scalar curvature must generically change in the course of structure formation, that an averaged inhomogeneous radiation cosmos does not follow the evolution of the standard homogeneous-isotropic model, and that an averaged inhomogeneous perfect fluid features kinematical "backreaction" terms that, in some cases, act like a free scalar field source. The free scalar field (dilaton) itself, modelled by a "stiff" fluid, is singled out as a special inhomogeneous case where the averaged equations assume a simple form.     

Properties of homogeneous and inhomogeneous mass relations

Mass equations based on third-order partial difference equations have been investigated. A test which makes use of subsets of data has been developed to study long-range extrapolations. Inherent connections with the liquid-drop model and the shell model are established. Higher-order effects in isospin, presumably due to subshell mixing and core polarization, are recognized as the origin for an inhomogeneous source term which strongly affects long-range extrapolations.     

Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms

We compare four different approximate solvers for the generalized Riemann problem (GRP) for non-linear systems of hyperbolic equations with source terms. The GRP is a special Cauchy problem for a hyperbolic system with source terms whose initial condition is piecewise smooth. We briefly review the four solvers currently available and carry out a systematic assessment of these in terms of accuracy and computational efficiency. These solvers are the building block for constructing high-order numerical schemes of the ADER type for solving the general initial-boundary value problem for inhomogeneous systems in multiple space dimensions, in the frameworks of finite volume and discontinuous Galerkin finite element methods.     

A contrast source method for nonlinear acoustic wave fields in media with spatially inhomogeneous attenuation

Experimental data reveals that attenuation is an important phenomenon in medical ultrasound. Attenuation is particularly important for medical applications based on nonlinear acoustics, since higher harmonics experience higher attenuation than the fundamental. Here, a method is presented to accurately solve the wave equation for nonlinear acoustic media with spatially inhomogeneous attenuation. Losses are modeled by a spatially dependent compliance relaxation function, which is included in the Westervelt equation. Introduction of absorption in the form of a causal relaxation function automatically results in the appearance of dispersion. The appearance of inhomogeneities implies the presence of a spatially inhomogeneous contrast source in the presented full-wave method leading to inclusion of forward and backward scattering. The contrast source problem is solved iteratively using a Neumann scheme, similar to the iterative nonlinear contrast source (INCS) method. The presented method is directionally independent and capable of dealing with weakly to moderately nonlinear, large scale, three-dimensional wave fields occurring in diagnostic ultrasound. Convergence of the method has been investigated and results for homogeneous, lossy, linear media show full agreement with the exact results. Moreover, the performance of the method is demonstrated through simulations involving steered and unsteered beams in nonlinear media with spatially homogeneous and inhomogeneous attenuation.     

The Biermann battery effect at the interfaces in stratified plasmas

The Biermann battery arises because an inhomogeneous electron pressure acts as a source term for the magnetic field. In order to better understand its effects, we consider a simplified model formed by the boundary between two fluids with different mean molecular weight and look for magnetic fields generated by the battery and localized in a band around the interface. We show that a parallel field is generated, which tends to push the original flow away from the boundary creating a rarefaction band. The specific forms of magnetic field and velocity are detailed.     

The Intra-cavity Mode Competition and Laser Absorption Spectroscopy with CO2 Laser

A mathematical model is developed for analysis the dynamics of an intra-cavity laser spectroscopy and mode competition using the CO₂ laser as the coherent source. The governing equations of system are derived and effect of the absorbing material on the laser modes, as a time-dependent term is considered. Thus, the application of the CO₂ laser for intra-cavity laser spectroscopy with considering an inhomogeneous medium and effect of the absorbing material is investigated.     

Propagation of gravitational waves in Robertson-Walker backgrounds

The electric and magnetic parts of the linearized Weyl tensor, when the stress-energy tensor is that of a perfect fluid and the background is of Robertson-Walker type, are known to satisfy wave equations that differ by the presence of a source term for the electric part. It is shown here that all of the allowed solutions of the inhomogeneous equation can be obtained by applying a differential operator to the solutions of the homogeneous equation; consequently, electric-type and magnetic-type gravitational waves have the same propagation properties. The results of a complete integration of the appropriately linearized Newman-Penrose equations are given.     

Inhomogeneous similarity solutions of the Boltzmann equation with confining external forces

The Nikolskii transform makes it possible to construct inhomogeneous solutions of the Boltzmann equation from homogeneous ones. These solutions correspond to a gas in expansion, but if we introduce external forces, they can relax toward absolute Maxwellians. This property holds independently of the assumed intermolecular inverse power force. Consequently, for Maxwell molecules and from energy-dependent homogeneous distributions, we construct effectively a class of inhomogeneous similarity distributions with Maxwellian equilibrium relaxation. We review and investigate again the homogeneous distributions which can be written in closed form, for instance, we show that an elliptic exact solution proposed some years ago violates positivity. For Maxwell interaction with singular cross sections, we numerically construct inhomogeneous distributions having Maxwellian equilibrium states and study the Tjon overshoot effect. We show that both the sign and the time decrease of the external force as well as the microscopic model of the cross section contribute to the asymptotic behavior of the distribution. These inhomogeneous similarity solutions include a class of distributions that asymptotically oscillate between different Maxwellians. Two classes of external forces are considered: linear spatial-dependent forces or linear velocity-dependent forces plus source term.     

轴对称介质内辐射源项反演算法

结合Abel变换和离散坐标法,提出了一种基于CCD相机采集的单幅辐射图像重建轴对称发射-吸收介质内辐射源项分布的反演算法。通过在求解辐射正问题得到的准确值的基础上,添加随机噪声模拟试验测量数据,分析了网格数目、辐射源项分布形式、吸收系数和测量误差对算法反演精度的影响。测试结果表明:该算法对测量误差不敏感,在有测量误差的情况下也能够准确的重建介质内的辐射源项分布。     

On the scintillations of radiation from an extended source in a randomly inhomogeneous medium

Using the method of refractive scattering of radio waves (RSRW), we solve the problem of space correlation of scintillations of radiation from an extended source in a randomly inhomogeneous medium. General expressions are obtained for the index and radius of the space correlation of fluctuations of intensity of radiation from a source with finite angular dimensions as it propagates though a multilayered medium with refractive index fluctuations. The RSRW method is similar to the diffraction calculation of space correlation of scintillations of an extended source in a thick layer with inhomogeneities. We note that under certain conditions an increase in the number of inhomogeneous layers on the radio wave propagation path leads to a pronounced decrease in the scintillation index and an increase in the space correlation of fluctuations of intensity of received radiation from an extended source. The results obtained have a simple geometric-optical interpretation. We indicate a specific feature in determining the angular dimensions of extended radio sources using the known method of scintillations. Radiophysical Research Institute, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 40, No. 5, pp. 586–593, May, 1997.     

Radiation transfer by a finite-emitting inhomogeneous medium

A functional relation is obtained between radiative transfer in an inhomogeneous finite planar layer with an internal energy source and diffuse reflection. The intensity is derived for the emerging radiation of a polynomial energy source. We use Padé approximants to calculate the emitted intensity for a linear energy source when the single scattering albedo decreases exponentially with optical depth. Numerical results are given for both homogeneous and inhomogeneous media.     

Quantum effects in the Gowdy T3 cosmology

The Gowdy T³ Cosmology is an exact solution to the vacuum Einstein equations interpreted to be a single polarization of gravitational waves propagating in an anisotropic, spatially inhomogeneous background. The classical behavior is reviewed and related to standard cosmological parameters. Canonical quantization of the dynamical degrees of freedom is reviewed. An adiabatic vacuum state is constructed. Adiabatic regularization is used to obtain non-divergent stress-energy tensor vacuum expectation values. Casimir energy terms due to T³ imposed discrete modes are evaluated. The vacuum expectation values are analyzed in early and late time limits and evaluated numerically. The regularized expectation value is used as a source for the classical background spacetime in the spirit of semi-classical gravity. An entirely vacuum expectation value source term produces essentially the time reverse of the classical evolution. Classical stress-energy added to the source restores the classical behavior at late times only. The combined system collapses from infinite to small but non-zero volume and reexpands. The classical singularity is replaced by a symmetric bounce.