Solving the Difference Initial-Boundary Value Problems by the Operator Exponential Method

Abstract

We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions as the perturbation of the same operator for periodic ones. We analyze the error, stability and efficiency of the scheme for a model example of the one-dimensional operator of second difference.     

The exact response of a two-dimensional flat-layered medium to a probing pulse: An application to a layer stripping reconstruction procedure

Abstract

We consider a simple model problem that can be found in many fields of application such as, for example, reflection seismology. That is we consider an initial boundary value problem on a half-plane for a class of two-dimensional wave equations with a piecewise-constant coefficient. This coefficient describes the flat layered medium under consideration. An initial pulse located on the boundary of the half-plane is used to probe the medium. An integral representation of the solution of this problem is obtained by studying the spectral measures of some differential operators in one variable. This integral representation is exploited to obtain an ‘explicit’ formula for the solution of the problem considered evaluated at the location of the probing pulse. This ‘explicit’ formula is exploited to reconstruct the structure of the medium from its response to a probing pulse via a layer stripping procedure. Some numerical results obtained with this procedure on test problems are shown. The ‘explicit’ formula obtained can be used in several other contexts such as, for example, the study of perturbed flat layered media or the study of random flat layered media.     

Many-electron tunneling in atoms

A formula for describing the N-electron ionization of atoms by a dc field and laser radiation in the tunneling regime is derived theoretically, and numerical examples for noble-gas atoms are presented. Zh. éksp. Teor. Fiz. 116, 410–417 (August 1999)     

Nuclear size correction to the hyperfine splitting in low- hydrogen-like atoms

The finite nuclear size effect on the hyperfine splitting of low-Z hydrogen-like atoms is studied in the external field approximation. A simple non-relativistic formula is proposed which expresses the nuclear size correction to the hyperfine splitting in terms of moments of the nuclear charge and magnetization distribution. The numerical results obtained via this formula are compared with corresponding results derived by means of the Zemach formula. A relativistic formula for the nuclear size correction to the hyperfine splitting is also derived. Received 18 March 2002 / Received in final form 14 November 2002 Published online 28 January 2003 RID="a" ID="a"e-mail: plunien@physik.tu-dresden.de     

超导临界温度级数公式的应用

从超导临界温度级数公式(1)出发,得到了B类超导体同位素效应的表达。在μ^*=0及μ^*≠0时,我们得到了T_c上限的级数表达式,其结果与数值解符合甚好。最后,在μ^*=0时,讨论了T_c与各种谱参数的关系。由级数表达式可以清楚地看出,当固定不同参数时,T_c的行为完全不同,从而统一解释了不同作者关于声子谱对T_c影响的不同结论。我们着重指出,在大多数情况下,级数公式(1)仅取前两项,可作为数值解的很好近似,从而大大简化了级数公式。预计可以由此简化公式出发研究其它问题。 关键词：     

Zur Bestimmung des D2O-Gehaltes in schwerem Wasser nach der Schwimmer-Methode

For the evaluation of float measurements all individual steps of the calculations are comprised under the expression of “precise float formula”. From this simplified approximations can be deduced. Their errors are estimated under little restraining conditions. The rather simple “difference formula” in the range of high D₂O-concentration together with a density scale for pure D₂O gained by interpolation suffice for most practical purposes. This is proved by numerical comparing with the precise formula.     

Improvement of Unified Mobility Model with Analytic Current-Voltage Expression for Organic Diodes

The carrier mobility formula of Pasveer et al. [Phys. Rev. Lett. 94, 206601 (2005)] is modified through analyzing numerical results for mobility. The modified expression gives improved precision to numerical data, and is a function of temperature, carrier density, and electric field. The corresponding space-charge limited (SCL) equations can be strictly solved with carrier density being replaced by the average value evaluated based on the strict solution. An analytic current-voltage relationship is derived and is validated with experimental data collected from different materials in a wide range of operating conditions.     

Comment on “Scattering length for fermionic alkali atoms"

We suggest that Gautam and Angom [Eur. Phys. J. D 56, 173 (2010)] underestimated the influence of the van der Waals dispersion forces at long range on their calculated low energy scattering parameters and we provide numerical evidence in support. We also provide a simple proof of the relation between s-wave and p-wave scattering parameters and we comment briefly on the semiclassical formula derived by these authors.     

Dependence of transport characteristics of molecules on the rotational quantum number

We derive a simple analytic formula that describes the relative difference of transport collision rates, Δv/v, for collisions of molecules and atoms in the rovibrational excitation of the former by light, as a function of the rotational quantum numbers of the combining (i.e., affected by radiation) levels of the molecules. (The relative difference of transport collision rates can be measured in light-induced drift, or LID, experiments and is proportional to the LID effect.) The formula is valid in the energy sudden approximation and is based on the well-known factorization formula for cross sections of RT-transitions in linear molecules that collide with atoms. We show that in this approximation the factor Δv/v is the sum of two independent terms, the vibrational term (Δv/v)_vib and the rotational term (Δv/v)_rot. Each term can be measured individually in LID experiments. Zh. éksp. Teor. Fiz. 113, 1649–1660 (May 1998)     

Numerical solutions of matrix Riccati equations for radiative transfer in a plane-parallel geometry

Abstract

In this paper, we conduct numerical experiments with matrix Riccati equations (MREs) which describe the reflection (R) and transmission (T) matrices of the specific intensities in a layer containing randomly distributed scattering particles. The theoretical formulation of MREs is discussed in our previous paper where we show that R and T for a thick layer can be efficiently computed by successively doubling R and T matrices for a thin layer (with small optical thickness τ_Δ). We can compute R(τ_Δ) and T(τ_Δ) very accurately using either a fourth-order Runge–Kutta scheme or the fourth-order iterative solution. The differences between these results and those computed by the eigenmode expansion technique (EMET) are very small (<0.1%). Although the MRE formulation cannot be extended to handle the inhomogeneous term (source term) in the differential equation, we show that the force term can be reformulated as an equivalent boundary condition which is consistent with MRE methods. MRE methods offer an alternative way of solving plane-parallel radiative transport problems. For large problems that do not fit into computer memory, the MRE method provides a significant reduction in computer memory and computational time.     

Comments on the paper “Detection of the Absorption Spectra of Water Clusters under Atmospheric Conditions”

A numerical simulation has been performed, indicating that the statement on the detection of water dimer absorption by V.I. Serdyukov et al. in Pis’ma Zh. Eksp. Teor. Fiz. 89, 12 (2009) [JETP Lett. 89, 10 (2009)] had been premature. Most of the revealed induced absorption can be attributed to water monomers.     

Two-dimensional lattice models of the Peierls type

Two-dimensional discrete models are used to describe nonlinear interactions of 'atoms' within interface regions. An evolution algorithm is presented for a distribution of dislocations on a plane. Results of numerical simulations illustrate motion of a dislocation kink. A finite-thickness nonlinear interface is studied in two dimensions; it gives a further extension of the mathematical model to the analysis of contact problems with frictional interfaces.     

Numerical simulation methods for rough surface scattering

Abstract

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.     

On the R2⁎ relaxometry in complex multi-peak multi-Echo chemical shift-based water-fat quantification: Applications to the neuromuscular diseases

PurposeInvestigation of the feasibility of the R₂^⁎ mapping techniques by using latest theoretical models corrected for confounding factors and optimized for signal to noise ratio.Theory and methodsThe improvement of the performance of state of the art magnetic resonance imaging (MRI) relaxometry algorithms is challenging because of a non-negligible bias and still unresolved numerical instabilities. Here, R₂^⁎ mapping reconstructions, including complex fitting with multi-spectral fat-correction by using single-decay and double-decay formulation, are deeply studied in order to investigate and identify optimal configuration parameters and minimize the occurrence of numerical artifacts. The effects of echo number, echo spacing, and fat/water relaxation model type are evaluated through both simulated and in-vivo data. We also explore the stability and feasibility of the fat/water relaxation model by analyzing the impact of high percentage of fat infiltrations and local transverse relaxation differences among biological species.ResultsThe main limits of the MRI relaxometry are the presence of bias and the occurrence of artifacts, which significantly affect its accuracy. Chemical-shift complex R₂^⁎-correct single-decay reconstructions exhibit a large bias in presence of a significant difference in the relaxation rates of fat and water and with fat concentration larger than 30%. We find that for fat-dominated tissues or in patients affected by extensive iron deposition, MRI reconstructions accounting for multi-exponential relaxation time provide accurate R₂^⁎ measurements and are less prone to numerical artifacts.ConclusionsComplex fitting and fat-correction with multi-exponential decay formulation outperforms the conventional single-decay approximation in various diagnostic scenarios. Although it still lacks of numerical stability, which requires model enhancement and support from spectroscopy, it offers promising perspectives for the development of relaxometry as a reliable tool to improve tissue characterization and monitoring of neuromuscular disorders.     

A simple analytic method for estimating T2 in the knee from DESS

PurposeTo introduce a simple analytical formula for estimating T₂ from a single Double-Echo in Steady-State (DESS) scan.MethodsExtended Phase Graph (EPG) modeling was used to develop a straightforward linear approximation of the relationship between the two DESS signals, enabling accurate T₂ estimation from one DESS scan. Simulations were performed to demonstrate cancellation of different echo pathways to validate this simple model. The resulting analytic formula was compared to previous methods for T₂ estimation using DESS and fast spin-echo scans in agar phantoms and knee cartilage in three volunteers and three patients. The DESS approach allows 3D (256 × 256 × 44) T₂-mapping with fat suppression in scan times of 3–4 min.ResultsThe simulations demonstrated that the model approximates the true signal very well. If the T₁ is within 20% of the assumed T₁, the T₂ estimation error was shown to be less than 5% for typical scans. The inherent residual error in the model was demonstrated to be small both due to signal decay and opposing signal contributions. The estimated T₂ from the linear relationship agrees well with reference scans, both for the phantoms and in vivo. The method resulted in less underestimation of T₂ than previous single-scan approaches, with processing times 60 times faster than using a numerical fit.ConclusionA simplified relationship between the two DESS signals allows for rapid 3D T₂ quantification with DESS that is accurate, yet also simple. The simplicity of the method allows for immediate T₂ estimation in cartilage during the MRI examination.     

Zur Theorie der Coulombenergiedifferenzen isobarer Analogzustände. II

The conclusions of part I [Z. Physik211, 195 (1968)] are revised: the nuclear energy changes within an isomultiplet, so that the change in the charge radius cannot be derived from the energy difference. The three-parameter formula for the charge density does not remove the discrepancy found in heavy nuclei with the two-parameter formula.     

Modeling of certain nonuniform processes in semiconductor technology

In this paper two problems are formulated and solved: the problem of diffusion in a two-phase system with a moving boundary, and the problem of taking into account the finite reaction rate for formation of the new phase. A numerical solution is found by a variational method, which has a number of practical advantages. Zh. Tekh. Fiz. 68, 75–82 (April 1998)     

Magnetic phase diagram of perovskite Mn oxides at T = 0

Minimizing total free energy by numerical calculations, we obtain the magnetic phase diagram of perovskite Mn oxides, such as with , Ca, Sr, etc. in the whole doping region from x =0 to x =1 at temperature T =0. It is discovered that a spiral state is stable in a low concentration of X ions while a canted state is stable in a high concentration of X ions, and a ferromagnetic phase can exist in the intermediate concentrations when the antiferromagnetic interaction is weak. The energy difference between spiral and canted states is found to be small when the Hund coupling is large. Magnetic field induced spiral/canted phase transition is considered as a possible mechanism of the colossal magnetoresistance (CMR) in the Mn oxides. Received: 11 July 1996 / Revised: 7 December 1996 / Accepted: 24 July 1997     

Initial stochastic modulation and break up of soliton bound states in optical fibers

We analyzed the influence of the initial phase stochastic modulations of high power optical pulses on their propagation in optical fiber. Our numerical results are compared with those of Cavalcanti S.B., Agrawal G.P.: Phys. Rev. A 51, 4086–4092 (1995), Fattakhov A.M., Chirkin A.S.: Kvantovaya Elektronka 11, 1989 (1983), Kvantovaya Elektronka 11, 2349 (1984), Maimistov A.I., Manykin B.A., Skliyarov Y.M.: Kvantovaya Elektronka 13, 2243–2248 (1986). Break up of soliton bound states due to initial phase stochastic modulations is observed.     

New calculations of α-decay half-lives by the Viola-Seaborg formula

The Viola-Seaborg formula is a well-known formula for α-decay half-lives of heavy nuclei. In this work we obtain new parameters of this formula through a least-square fit to even-even nuclei between Z = 84 and Z = 110 with N greater than 126. On average, the formula can reproduce the half-lives of heavy even-even nuclei within a factor of 1.3. The formula with new parameters works well for the superheavy region which is a hot topic of nuclear physics. The numerical results from the formula are compared with those from the cluster model.