Kinetic equations for clean superconductors: Application to the flux flow hall effect

International Nuclear Information System (INIS)

Kopnin, N.B.

1994-01-01

The kinetic equations for clean superconductors (l>>ζ) are derived. expanding the equations for the time dependent Green functions in the quasiclassical parameter, the new contributions are found which contain the derivatives of the distribution functions with respect to the quasiparticle momentum. The transition from the ultra-clean case (no relaxation) to a relaxation-dominated behavior, for which the kinetic equations coincide with the usual quasiclassical approximation, occurs for the relaxation time of the order of ℎE F /Δ 2 . The kinetic equations can be used for various dynamic processes in superconductors including the flux-flow Hall effect. The derived equations, after necessary modifications for the p-wave pairing, are especially suitable for nonstationary problems in the theory of superfluidity of 3 He

Fractional neutron point kinetics equations for nuclear reactor dynamics

International Nuclear Information System (INIS)

Espinosa-Paredes, Gilberto; Polo-Labarrios, Marco-A.; Espinosa-Martinez, Erick-G.; Valle-Gallegos, Edmundo del

2011-01-01

The fractional point-neutron kinetics model for the dynamic behavior in a nuclear reactor is derived and analyzed in this paper. The fractional model retains the main dynamic characteristics of the neutron motion in which the relaxation time associated with a rapid variation in the neutron flux contains a fractional order, acting as exponent of the relaxation time, to obtain the best representation of a nuclear reactor dynamics. The physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. The numerical approximation to the solution of the fractional neutron point kinetics model, which can be represented as a multi-term high-order linear fractional differential equation, is calculated by reducing the problem to a system of ordinary and fractional differential equations. The numerical stability of the fractional scheme is investigated in this work. Results for neutron dynamic behavior for both positive and negative reactivity and for different values of fractional order are shown and compared with the classic neutron point kinetic equations. Additionally, a related review with the neutron point kinetics equations is presented, which encompasses papers written in English about this research topic (as well as some books and technical reports) published since 1940 up to 2010.

Solution of multigroup diffusion equations in cylindrical configuration by local polynomial approximation

International Nuclear Information System (INIS)

Jakab, J.

1979-05-01

Local approximations of neutron flux density by 2nd degree polynomials are used in calculating light water reactors. The calculations include spatial kinetics tasks for the models of two- and three-dimensional reactors in the Cartesian geometry. The resulting linear algebraic equations are considered to be formally identical to the results of the differential method of diffusion equation solution. (H.S.)

Some Aspects of Extended Kinetic Equation

Directory of Open Access Journals (Sweden)

Dilip Kumar

2015-09-01

Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.

Present status on numerical algorithms and benchmark tests for point kinetics and quasi-static approximate kinetics

International Nuclear Information System (INIS)

Ise, Takeharu

1976-12-01

Review studies have been made on algorithms of numerical analysis and benchmark tests on point kinetics and quasistatic approximate kinetics computer codes to perform efficiently benchmark tests on space-dependent neutron kinetics codes. Point kinetics methods have now been improved since they can be directly applied to the factorization procedures. Methods based on Pade rational function give numerically stable solutions and methods on matrix-splitting are interested in the fact that they are applicable to the direct integration methods. An improved quasistatic (IQ) approximation is the best and the most practical method; it is numerically shown that the IQ method has a high stability and precision and the computation time which is about one tenth of that of the direct method. IQ method is applicable to thermal reactors as well as fast reactors and especially fitted for fast reactors to which many time steps are necessary. Two-dimensional diffusion kinetics codes are most practicable though there exist also three-dimensional diffusion kinetics code as well as two-dimensional transport kinetics code. On developing a space-dependent kinetics code, in any case, it is desirable to improve the method so as to have a high computing speed for solving static diffusion and transport equations. (auth.)

A multi scale approximation solution for the time dependent Boltzmann-transport equation

International Nuclear Information System (INIS)

Merk, B.

2004-03-01

The basis of all transient simulations for nuclear reactor cores is the reliable calculation of the power production. The local power distribution is generally calculated by solving the space, time, energy and angle dependent neutron transport equation known as Boltzmann equation. The computation of exact solutions of the Boltzmann equation is very time consuming. For practical numerical simulations approximated solutions are usually unavoidable. The objective of this work is development of an effective multi scale approximation solution for the Boltzmann equation. Most of the existing methods are based on separation of space and time. The new suggested method is performed without space-time separation. This effective approximation solution is developed on the basis of an expansion for the time derivative of different approximations to the Boltzmann equation. The method of multiple scale expansion is used for the expansion of the time derivative, because the problem of the stiff time behaviour can't be expressed by standard expansion methods. This multiple scale expansion is used in this work to develop approximation solutions for different approximations of the Boltzmann equation, starting from the expansion of the point kinetics equations. The resulting analytic functions are used for testing the applicability and accuracy of the multiple scale expansion method for an approximation solution with 2 delayed neutron groups. The results are tested versus the exact analytical results for the point kinetics equations. Very good agreement between both solutions is obtained. The validity of the solution with 2 delayed neutron groups to approximate the behaviour of the system with 6 delayed neutron groups is demonstrated in an additional analysis. A strategy for a solution with 4 delayed neutron groups is described. A multiple scale expansion is performed for the space-time dependent diffusion equation for one homogenized cell with 2 delayed neutron groups. The result is

Kinetic equation solution by inverse kinetic method

International Nuclear Information System (INIS)

Salas, G.

1983-01-01

We propose a computer program (CAMU) which permits to solve the inverse kinetic equation. The CAMU code is written in HPL language for a HP 982 A microcomputer with a peripheral interface HP 9876 A ''thermal graphic printer''. The CAMU code solves the inverse kinetic equation by taking as data entry the output of the ionization chambers and integrating the equation with the help of the Simpson method. With this program we calculate the evolution of the reactivity in time for a given disturbance

Study of the stochastic point reactor kinetic equation

International Nuclear Information System (INIS)

Gotoh, Yorio

1980-01-01

Diagrammatic technique is used to solve the stochastic point reactor kinetic equation. The method gives exact results which are derived from Fokker-Plank theory. A Green's function dressed with the clouds of noise is defined, which is a transfer function of point reactor with fluctuating reactivity. An integral equation for the correlation function of neutron power is derived using the following assumptions: 1) Green's funntion should be dressed with noise, 2) The ladder type diagrams only contributes to the correlation function. For a white noise and the one delayed neutron group approximation, the norm of the integral equation and the variance to mean-squared ratio are analytically obtained. (author)

Hybrid dynamic modeling of Escherichia coli central metabolic network combining Michaelis–Menten and approximate kinetic equations

DEFF Research Database (Denmark)

Costa, Rafael S.; Machado, Daniel; Rocha, Isabel

2010-01-01

, represent nowadays the limiting factor in the construction of such models. In this study, we compare four alternative modeling approaches based on Michaelis–Menten kinetics for the bi-molecular reactions and different types of simplified rate equations for the remaining reactions (generalized mass action......The construction of dynamic metabolic models at reaction network level requires the use of mechanistic enzymatic rate equations that comprise a large number of parameters. The lack of knowledge on these equations and the difficulty in the experimental identification of their associated parameters...

The choice of optimal Discrete Interaction Approximation to the kinetic integral for ocean waves

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V. G. Polnikov

2003-01-01

Full Text Available A lot of discrete configurations for the four-wave nonlinear interaction processes have been calculated and tested by the method proposed earlier in the frame of the concept of Fast Discrete Interaction Approximation to the Hasselmann's kinetic integral (Polnikov and Farina, 2002. It was found that there are several simple configurations, which are more efficient than the one proposed originally in Hasselmann et al. (1985. Finally, the optimal multiple Discrete Interaction Approximation (DIA to the kinetic integral for deep-water waves was found. Wave spectrum features have been intercompared for a number of different configurations of DIA, applied to a long-time solution of kinetic equation. On the basis of this intercomparison the better efficiency of the configurations proposed was confirmed. Certain recommendations were given for implementation of new approximations to the wave forecast practice.

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

International Nuclear Information System (INIS)

Gelß, Patrick; Matera, Sebastian; Schütte, Christof

2016-01-01

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO 2 (110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

Science.gov (United States)

Gelß, Patrick; Matera, Sebastian; Schütte, Christof

2016-06-01

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

Solving the master equation without kinetic Monte Carlo: Tensor train approximations for a CO oxidation model

Energy Technology Data Exchange (ETDEWEB)

Gelß, Patrick, E-mail: p.gelss@fu-berlin.de; Matera, Sebastian, E-mail: matera@math.fu-berlin.de; Schütte, Christof, E-mail: schuette@mi.fu-berlin.de

2016-06-01

In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO{sub 2}(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.

Ultrafast dynamics of laser-pulse excited semiconductors: non-Markovian quantum kinetic equations with nonequilibrium correlations

Directory of Open Access Journals (Sweden)

V.V.Ignatyuk

2004-01-01

Full Text Available Non-Markovian kinetic equations in the second Born approximation are derived for a two-zone semiconductor excited by a short laser pulse. Both collision dynamics and running nonequilibrium correlations are taken into consideration. The energy balance and relaxation of the system to equilibrium are discussed. Results of numerical solution of the kinetic equations for carriers and phonons are presented.

Quasi-linear landau kinetic equations for magnetized plasmas: compact propagator formalism, rotation matrices and interaction

International Nuclear Information System (INIS)

Misguich, J.H.

2004-04-01

As a first step toward a nonlinear renormalized description of turbulence phenomena in magnetized plasmas, the lowest order quasi-linear description is presented here from a unified point of view for collisionless as well as for collisional plasmas in a constant magnetic field. The quasi-linear approximation is applied to a general kinetic equation obtained previously from the Klimontovich exact equation, by means of a generalised Dupree-Weinstock method. The so-obtained quasi-linear description of electromagnetic turbulence in a magnetoplasma is applied to three separate physical cases: -) weak electrostatic turbulence, -) purely magnetic field fluctuations (the classical quasi-linear results are obtained for cosmic ray diffusion in the 'slab model' of magnetostatic turbulence in the solar wind), and -) collisional kinetic equations of magnetized plasmas. This mathematical technique has allowed us to derive basic kinetic equations for turbulent plasmas and collisional plasmas, respectively in the quasi-linear and Landau approximation. In presence of a magnetic field we have shown that the systematic use of rotation matrices describing the helical particle motion allows for a much more compact derivation than usually performed. Moreover, from the formal analogy between turbulent and collisional plasmas, the results derived here in detail for the turbulent plasmas, can be immediately translated to obtain explicit results for the Landau kinetic equation

Quasi-linear landau kinetic equations for magnetized plasmas: compact propagator formalism, rotation matrices and interaction

Energy Technology Data Exchange (ETDEWEB)

Misguich, J.H

2004-04-01

As a first step toward a nonlinear renormalized description of turbulence phenomena in magnetized plasmas, the lowest order quasi-linear description is presented here from a unified point of view for collisionless as well as for collisional plasmas in a constant magnetic field. The quasi-linear approximation is applied to a general kinetic equation obtained previously from the Klimontovich exact equation, by means of a generalised Dupree-Weinstock method. The so-obtained quasi-linear description of electromagnetic turbulence in a magnetoplasma is applied to three separate physical cases: -) weak electrostatic turbulence, -) purely magnetic field fluctuations (the classical quasi-linear results are obtained for cosmic ray diffusion in the 'slab model' of magnetostatic turbulence in the solar wind), and -) collisional kinetic equations of magnetized plasmas. This mathematical technique has allowed us to derive basic kinetic equations for turbulent plasmas and collisional plasmas, respectively in the quasi-linear and Landau approximation. In presence of a magnetic field we have shown that the systematic use of rotation matrices describing the helical particle motion allows for a much more compact derivation than usually performed. Moreover, from the formal analogy between turbulent and collisional plasmas, the results derived here in detail for the turbulent plasmas, can be immediately translated to obtain explicit results for the Landau kinetic equation.

Legendre-tau approximations for functional differential equations

Science.gov (United States)

Ito, K.; Teglas, R.

1986-01-01

The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

Drift-free kinetic equations for turbulent dispersion

Science.gov (United States)

Bragg, A.; Swailes, D. C.; Skartlien, R.

2012-11-01

The dispersion of passive scalars and inertial particles in a turbulent flow can be described in terms of probability density functions (PDFs) defining the statistical distribution of relevant scalar or particle variables. The construction of transport equations governing the evolution of such PDFs has been the subject of numerous studies, and various authors have presented formulations for this type of equation, usually referred to as a kinetic equation. In the literature it is often stated, and widely assumed, that these PDF kinetic equation formulations are equivalent. In this paper it is shown that this is not the case, and the significance of differences among the various forms is considered. In particular, consideration is given to which form of equation is most appropriate for modeling dispersion in inhomogeneous turbulence and most consistent with the underlying particle equation of motion. In this regard the PDF equations for inertial particles are considered in the limit of zero particle Stokes number and assessed against the fully mixed (zero-drift) condition for fluid points. A long-standing question regarding the validity of kinetic equations in the fluid-point limit is answered; it is demonstrated formally that one version of the kinetic equation (derived using the Furutsu-Novikov method) provides a model that satisfies this zero-drift condition exactly in both homogeneous and inhomogeneous systems. In contrast, other forms of the kinetic equation do not satisfy this limit or apply only in a limited regime.

Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor

International Nuclear Information System (INIS)

Saha Ray, S.

2012-01-01

Highlights: ► In this paper stochastic neutron point kinetic equations have been analyzed. ► Euler–Maruyama method and Strong Taylor 1.5 order method have been discussed. ► These methods are applied for the solution of stochastic point kinetic equations. ► Comparison between the results of these methods and others are presented in tables. ► Graphs for neutron and precursor sample paths are also presented. -- Abstract: In the present paper, the numerical approximation methods, applied to efficiently calculate the solution for stochastic point kinetic equations () in nuclear reactor dynamics, are investigated. A system of Itô stochastic differential equations has been analyzed to model the neutron density and the delayed neutron precursors in a point nuclear reactor. The resulting system of Itô stochastic differential equations are solved over each time-step size. The methods are verified by considering different initial conditions, experimental data and over constant reactivities. The computational results indicate that the methods are simple and suitable for solving stochastic point kinetic equations. In this article, a numerical investigation is made in order to observe the random oscillations in neutron and precursor population dynamics in subcritical and critical reactors.

Derivation of a new kinetic equation. Application to the determination of viscosity coefficients

International Nuclear Information System (INIS)

Frey, Jean-Jacques

1970-01-01

By introducing a new hypothesis concerning the closure in the B.B.G.K.Y. equation system, an approximate expression for f 12 is obtained. By inserting this expression in the first B.B.G.K.Y. equation, a new kinetic equation results. It is verified that this equation does in fact give the fluid mechanics equations, and new expressions for the shear and expansion viscosity coefficients are obtained. The numerical calculations which have been carried out show that very satisfactory agreement exists with experimental results. (author) [fr

The analysis of the derivation principles of kinetic equations based on exactly solvable models of the bulk reaction A + B → Product

International Nuclear Information System (INIS)

Kipriyanov, A.A.; Doktorov, A.B.

2005-01-01

We have considered two many-particle models of the irreversible reaction A + B → Product for which closed kinetic equations for the mean concentration N A (t) of A species can be exactly obtained. These equations are identically recast into a unified form of integro-differential equation of general kinetic theory. It is shown that the memory functions for both models under consideration can be represented as a sum of the Markovian and non-Markovian parts. It is essential that the Markovian part of the Laplace transform of any kernel can be obtained using the Laplace transform of the kernel itself, and is the root of the non-Markovian part of the Laplace transform of the kernel. The properties established allowed us to perform correct approximation of the memory functions at small concentrations [B] of B species and derive the binary non-Markovian integro-differential equation. Within the binary theory accuracy this equation has been rewritten in a regular frame of a familiar rate equation satisfying general principles of binary kinetic equations. Thus using particular exactly solvable many-particle models, we have reproduced the most essential steps of the known general way for the derivation of the binary kinetic equation avoiding the sophisticated many-particle technique and the corresponding approximations. Besides, the results obtained can serve as an additional evidence of the approximations made in a general many-particle approach to the derivation of the binary kinetic equation

A new integral method for solving the point reactor neutron kinetics equations

International Nuclear Information System (INIS)

Li Haofeng; Chen Wenzhen; Luo Lei; Zhu Qian

2009-01-01

A numerical integral method that efficiently provides the solution of the point kinetics equations by using the better basis function (BBF) for the approximation of the neutron density in one time step integrations is described and investigated. The approach is based on an exact analytic integration of the neutron density equation, where the stiffness of the equations is overcome by the fully implicit formulation. The procedure is tested by using a variety of reactivity functions, including step reactivity insertion, ramp input and oscillatory reactivity changes. The solution of the better basis function method is compared to other analytical and numerical solutions of the point reactor kinetics equations. The results show that selecting a better basis function can improve the efficiency and accuracy of this integral method. The better basis function method can be used in real time forecasting for power reactors in order to prevent reactivity accidents.

Different seeds to solve the equations of stochastic point kinetics using the Euler-Maruyama method

International Nuclear Information System (INIS)

Suescun D, D.; Oviedo T, M.

2017-09-01

In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and

Comparative analysis among several methods used to solve the point kinetic equations

International Nuclear Information System (INIS)

Nunes, Anderson L.; Goncalves, Alessandro da C.; Martinez, Aquilino S.; Silva, Fernando Carvalho da

2007-01-01

The main objective of this work consists on the methodology development for comparison of several methods for the kinetics equations points solution. The evaluated methods are: the finite differences method, the stiffness confinement method, improved stiffness confinement method and the piecewise constant approximations method. These methods were implemented and compared through a systematic analysis that consists basically of confronting which one of the methods consume smaller computational time with higher precision. It was calculated the relative which function is to combine both criteria in order to reach the goal. Through the analyses of the performance factor it is possible to choose the best method for the solution of point kinetics equations. (author)

Comparative analysis among several methods used to solve the point kinetic equations

Energy Technology Data Exchange (ETDEWEB)

Nunes, Anderson L.; Goncalves, Alessandro da C.; Martinez, Aquilino S.; Silva, Fernando Carvalho da [Universidade Federal, Rio de Janeiro, RJ (Brazil). Coordenacao dos Programas de Pos-graduacao de Engenharia. Programa de Engenharia Nuclear; E-mails: alupo@if.ufrj.br; agoncalves@con.ufrj.br; aquilino@lmp.ufrj.br; fernando@con.ufrj.br

2007-07-01

The main objective of this work consists on the methodology development for comparison of several methods for the kinetics equations points solution. The evaluated methods are: the finite differences method, the stiffness confinement method, improved stiffness confinement method and the piecewise constant approximations method. These methods were implemented and compared through a systematic analysis that consists basically of confronting which one of the methods consume smaller computational time with higher precision. It was calculated the relative which function is to combine both criteria in order to reach the goal. Through the analyses of the performance factor it is possible to choose the best method for the solution of point kinetics equations. (author)

An efficient technique for the point reactor kinetics equations with Newtonian temperature feedback effects

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → An efficient technique for the nonlinear reactor kinetics equations is presented. → This method is based on Backward Euler or Crank Nicholson and fundamental matrix. → Stability of efficient technique is defined and discussed. → This method is applied to point kinetics equations of six-groups of delayed neutrons. → Step, ramp, sinusoidal and temperature feedback reactivities are discussed. - Abstract: The point reactor kinetics equations of multi-group of delayed neutrons in the presence Newtonian temperature feedback effects are a system of stiff nonlinear ordinary differential equations which have not any exact analytical solution. The efficient technique for this nonlinear system is based on changing this nonlinear system to a linear system by the predicted value of reactivity and solving this linear system using the fundamental matrix of the homogenous linear differential equations. The nonlinear point reactor kinetics equations are rewritten in the matrix form. The solution of this matrix form is introduced. This solution contains the exponential function of a variable coefficient matrix. This coefficient matrix contains the unknown variable, reactivity. The predicted values of reactivity in the explicit form are determined replacing the exponential function of the coefficient matrix by two kinds, Backward Euler and Crank Nicholson, of the rational approximations. The nonlinear point kinetics equations changed to a linear system of the homogenous differential equations. The fundamental matrix of this linear system is calculated using the eigenvalues and the corresponding eigenvectors of the coefficient matrix. Stability of the efficient technique is defined and discussed. The efficient technique is applied to the point kinetics equations of six-groups of delayed neutrons with step, ramp, sinusoidal and the temperature feedback reactivities. The results of these efficient techniques are compared with the traditional methods.

Nodal approximations in space and time for neutron kinetics

International Nuclear Information System (INIS)

Grossman, L.M.; Hennart, J.P.

2005-01-01

A general formalism is described of the nodal type in time and space for the neutron kinetics equations. In space, several nodal methods are given of the Raviart-Thomas type (RT0 and RT1), of the Brezzi-Douglas-Marini type (BDM0 and BDM1) and of the Brezzi-Douglas-Fortin-Marini type (BDFM 1). In time, polynomial and analytical approximations are derived. In the analytical case, they are based on the inclusion of an exponential term in the basis function. They can be continuous or discontinuous in time, leading in particular to the well-known Crank-Nicolson, Backward Euler and θ schemes

Transport and relaxation properties of superfluid 3He. I. Kinetic equation and Bogoliubov quasiparticle relaxation rate

International Nuclear Information System (INIS)

Einzel, D.; Woelfle, P.

1978-01-01

The kinetic equation for Bogoliubov quasiparticles for both the A and B phases of superfluid 3 He is derived from the general matrix kinetic equation. A condensed expression for the exact spin-symmetric collision integral is given. The quasiparticle relaxation rate is calculated for the BW state using the s--p approximation for the quasiparticle scattering amplitude. By using the results for the quasiparticle relaxation rate, the mean free path of Bogoliubov quasiparticles is calculated for all temperatures

Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks

Science.gov (United States)

Pogan, Alin; Zumbrun, Kevin

2018-06-01

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman-Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.

Evaluation of stochastic differential equation approximation of ion channel gating models.

Science.gov (United States)

Bruce, Ian C

2009-04-01

Fox and Lu derived an algorithm based on stochastic differential equations for approximating the kinetics of ion channel gating that is simpler and faster than "exact" algorithms for simulating Markov process models of channel gating. However, the approximation may not be sufficiently accurate to predict statistics of action potential generation in some cases. The objective of this study was to develop a framework for analyzing the inaccuracies and determining their origin. Simulations of a patch of membrane with voltage-gated sodium and potassium channels were performed using an exact algorithm for the kinetics of channel gating and the approximate algorithm of Fox & Lu. The Fox & Lu algorithm assumes that channel gating particle dynamics have a stochastic term that is uncorrelated, zero-mean Gaussian noise, whereas the results of this study demonstrate that in many cases the stochastic term in the Fox & Lu algorithm should be correlated and non-Gaussian noise with a non-zero mean. The results indicate that: (i) the source of the inaccuracy is that the Fox & Lu algorithm does not adequately describe the combined behavior of the multiple activation particles in each sodium and potassium channel, and (ii) the accuracy does not improve with increasing numbers of channels.

A Derivation of Source-based Kinetics Equation with Time Dependent Fission Kernel for Reactor Transient Analyses

International Nuclear Information System (INIS)

Kim, Song Hyun; Woo, Myeong Hyun; Shin, Chang Ho; Pyeon, Cheol Ho

2015-01-01

In this study, a new balance equation to overcome the problems generated by the previous methods is proposed using source-based balance equation. And then, a simple problem is analyzed with the proposed method. In this study, a source-based balance equation with the time dependent fission kernel was derived to simplify the kinetics equation. To analyze the partial variations of reactor characteristics, two representative methods were introduced in previous studies; (1) quasi-statics method and (2) multipoint technique. The main idea of quasistatics method is to use a low-order approximation for large integration times. To realize the quasi-statics method, first, time dependent flux is separated into the shape and amplitude functions, and shape function is calculated. It is noted that the method has a good accuracy; however, it can be expensive as a calculation cost aspect because the shape function should be fully recalculated to obtain accurate results. To improve the calculation efficiency, multipoint method was proposed. The multipoint method is based on the classic kinetics equation with using Green's function to analyze the flight probability from region r' to r. Those previous methods have been used to analyze the reactor kinetics analysis; however, the previous methods can have some limitations. First, three group variables (r g , E g , t g ) should be considered to solve the time dependent balance equation. This leads a big limitation to apply large system problem with good accuracy. Second, the energy group neutrons should be used to analyze reactor kinetics problems. In time dependent problem, neutron energy distribution can be changed at different time. It can affect the change of the group cross section; therefore, it can lead the accuracy problem. Third, the neutrons in a space-time region continually affect the other space-time regions; however, it is not properly considered in the previous method. Using birth history of the neutron sources

Numerical approximation of the Boltzmann equation : moment closure

NARCIS (Netherlands)

Abdel Malik, M.R.A.; Brummelen, van E.H.

2012-01-01

This work applies the moment method onto a generic form of kinetic equations to simplify kinetic models of particle systems. This leads to the moment closure problem which is addressed using entropy-based moment closure techniques utilizing entropy minimization. The resulting moment closure system

Using Equation-Free Computation to Accelerate Network-Free Stochastic Simulation of Chemical Kinetics.

Science.gov (United States)

Lin, Yen Ting; Chylek, Lily A; Lemons, Nathan W; Hlavacek, William S

2018-06-21

The chemical kinetics of many complex systems can be concisely represented by reaction rules, which can be used to generate reaction events via a kinetic Monte Carlo method that has been termed network-free simulation. Here, we demonstrate accelerated network-free simulation through a novel approach to equation-free computation. In this process, variables are introduced that approximately capture system state. Derivatives of these variables are estimated using short bursts of exact stochastic simulation and finite differencing. The variables are then projected forward in time via a numerical integration scheme, after which a new exact stochastic simulation is initialized and the whole process repeats. The projection step increases efficiency by bypassing the firing of numerous individual reaction events. As we show, the projected variables may be defined as populations of building blocks of chemical species. The maximal number of connected molecules included in these building blocks determines the degree of approximation. Equation-free acceleration of network-free simulation is found to be both accurate and efficient.

Receptor binding kinetics equations: Derivation using the Laplace transform method.

Science.gov (United States)

Hoare, Sam R J

Measuring unlabeled ligand receptor binding kinetics is valuable in optimizing and understanding drug action. Unfortunately, deriving equations for estimating kinetic parameters is challenging because it involves calculus; integration can be a frustrating barrier to the pharmacologist seeking to measure simple rate parameters. Here, a well-known tool for simplifying the derivation, the Laplace transform, is applied to models of receptor-ligand interaction. The method transforms differential equations to a form in which simple algebra can be applied to solve for the variable of interest, for example the concentration of ligand-bound receptor. The goal is to provide instruction using familiar examples, to enable investigators familiar with handling equilibrium binding equations to derive kinetic equations for receptor-ligand interaction. First, the Laplace transform is used to derive the equations for association and dissociation of labeled ligand binding. Next, its use for unlabeled ligand kinetic equations is exemplified by a full derivation of the kinetics of competitive binding equation. Finally, new unlabeled ligand equations are derived using the Laplace transform. These equations incorporate a pre-incubation step with unlabeled or labeled ligand. Four equations for measuring unlabeled ligand kinetics were compared and the two new equations verified by comparison with numerical solution. Importantly, the equations have not been verified with experimental data because no such experiments are evident in the literature. Equations were formatted for use in the curve-fitting program GraphPad Prism 6.0 and fitted to simulated data. This description of the Laplace transform method will enable pharmacologists to derive kinetic equations for their model or experimental paradigm under study. Application of the transform will expand the set of equations available for the pharmacologist to measure unlabeled ligand binding kinetics, and for other time

Development of kinetics equations from the Boltzmann equation; Etablissement des equations de la cinetique a partir de l'equation de Boltzmann

Energy Technology Data Exchange (ETDEWEB)

Plas, R.

1962-07-01

The author reports a study on kinetics equations for a reactor. He uses the conventional form of these equations but by using a dynamic multiplication factor. Thus, constants related to delayed neutrons are not modified by efficiency factors. The author first describes the theoretic kinetic operation of a reactor and develops the associated equations. He reports the development of equations for multiplication factors.

Hamiltonian formalism of two-dimensional Vlasov kinetic equation.

Science.gov (United States)

Pavlov, Maxim V

2014-12-08

In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo-Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.

Approximate solutions to Mathieu's equation

Science.gov (United States)

Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.

2018-06-01

Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.

Solution of the reactor point kinetics equations by MATLAB computing

Directory of Open Access Journals (Sweden)

Singh Sudhansu S.

2015-01-01

Full Text Available The numerical solution of the point kinetics equations in the presence of Newtonian temperature feedback has been a challenging issue for analyzing the reactor transients. Reactor point kinetics equations are a system of stiff ordinary differential equations which need special numerical treatments. Although a plethora of numerical intricacies have been introduced to solve the point kinetics equations over the years, some of the simple and straightforward methods still work very efficiently with extraordinary accuracy. As an example, it has been shown recently that the fundamental backward Euler finite difference algorithm with its simplicity has proven to be one of the most effective legacy methods. Complementing the back-ward Euler finite difference scheme, the present work demonstrates the application of ordinary differential equation suite available in the MATLAB software package to solve the stiff reactor point kinetics equations with Newtonian temperature feedback effects very effectively by analyzing various classic benchmark cases. Fair accuracy of the results implies the efficient application of MATLAB ordinary differential equation suite for solving the reactor point kinetics equations as an alternate method for future applications.

A Derivation of Source-based Kinetics Equation with Time Dependent Fission Kernel for Reactor Transient Analyses

Energy Technology Data Exchange (ETDEWEB)

Kim, Song Hyun; Woo, Myeong Hyun; Shin, Chang Ho [Hanyang University, Seoul (Korea, Republic of); Pyeon, Cheol Ho [Kyoto University, Osaka (Japan)

2015-10-15

In this study, a new balance equation to overcome the problems generated by the previous methods is proposed using source-based balance equation. And then, a simple problem is analyzed with the proposed method. In this study, a source-based balance equation with the time dependent fission kernel was derived to simplify the kinetics equation. To analyze the partial variations of reactor characteristics, two representative methods were introduced in previous studies; (1) quasi-statics method and (2) multipoint technique. The main idea of quasistatics method is to use a low-order approximation for large integration times. To realize the quasi-statics method, first, time dependent flux is separated into the shape and amplitude functions, and shape function is calculated. It is noted that the method has a good accuracy; however, it can be expensive as a calculation cost aspect because the shape function should be fully recalculated to obtain accurate results. To improve the calculation efficiency, multipoint method was proposed. The multipoint method is based on the classic kinetics equation with using Green's function to analyze the flight probability from region r' to r. Those previous methods have been used to analyze the reactor kinetics analysis; however, the previous methods can have some limitations. First, three group variables (r{sub g}, E{sub g}, t{sub g}) should be considered to solve the time dependent balance equation. This leads a big limitation to apply large system problem with good accuracy. Second, the energy group neutrons should be used to analyze reactor kinetics problems. In time dependent problem, neutron energy distribution can be changed at different time. It can affect the change of the group cross section; therefore, it can lead the accuracy problem. Third, the neutrons in a space-time region continually affect the other space-time regions; however, it is not properly considered in the previous method. Using birth history of the

Approximate radiative solutions of the Einstein equations

International Nuclear Information System (INIS)

Kuusk, P.; Unt, V.

1976-01-01

In this paper the external field of a bounded source emitting gravitational radiation is considered. A successive approximation method is used to integrate the Einstein equations in Bondi's coordinates (Bondi et al, Proc. R. Soc.; A269:21 (1962)). A method of separation of angular variables is worked out and the approximate Einstein equations are reduced to key equations. The losses of mass, momentum, and angular momentum due to gravitational multipole radiation are found. It is demonstrated that in the case of proper treatment a real mass occurs instead of a mass aspect in a solution of the Einstein equations. In an appendix Bondi's new function is given in terms of sources. (author)

Kinetic equations in dirty superconductors

International Nuclear Information System (INIS)

Kraehenbuehl, Y.

1981-01-01

Kinetic equations for superconductors in the dirty limit are derived using a method developed for superfluid systems, which allows a systematic expansion in small parameters; exact charge conservation is obeyed. (orig.)

Time discretization of the point kinetic equations using matrix exponential method and First-Order Hold

International Nuclear Information System (INIS)

Park, Yujin; Kazantzis, Nikolaos; Parlos, Alexander G.; Chong, Kil To

2013-01-01

Highlights: • Numerical solution for stiff differential equations using matrix exponential method. • The approximation is based on First Order Hold assumption. • Various input examples applied to the point kinetics equations. • The method shows superior useful and effective activity. - Abstract: A system of nonlinear differential equations is derived to model the dynamics of neutron density and the delayed neutron precursors within a point kinetics equation modeling framework for a nuclear reactor. The point kinetic equations are mathematically characterized as stiff, occasionally nonlinear, ordinary differential equations, posing significant challenges when numerical solutions are sought and traditionally resulting in the need for smaller time step intervals within various computational schemes. In light of the above realization, the present paper proposes a new discretization method inspired by system-theoretic notions and technically based on a combination of the matrix exponential method (MEM) and the First-Order Hold (FOH) assumption. Under the proposed time discretization structure, the sampled-data representation of the nonlinear point kinetic system of equations is derived. The performance of the proposed time discretization procedure is evaluated using several case studies with sinusoidal reactivity profiles and multiple input examples (reactivity and neutron source function). It is shown, that by applying the proposed method under a First-Order Hold for the neutron density and the precursor concentrations at each time step interval, the stiffness problem associated with the point kinetic equations can be adequately addressed and resolved. Finally, as evidenced by the aforementioned detailed simulation studies, the proposed method retains its validity and accuracy for a wide range of reactor operating conditions, including large sampling periods dictated by physical and/or technical limitations associated with the current state of sensor and

Numerical Calculation of Transport Based on the Drift Kinetic Equation for plasmas in General Toroidal Magnetic Geometry

International Nuclear Information System (INIS)

Reynolds, J. M.; Lopez-Bruna, D.

2009-01-01

This report is the first of a series dedicated to the numerical calculation of the evolution of fusion plasmas in general toroidal geometry, including TJ-II plasmas. A kinetic treatment has been chosen: the evolution equation of the distribution function of one or several plasma species is solved in guiding center coordinates. The distribution function is written as a Maxwellian one modulated by polynomial series in the kinetic coordinates with no other approximations than those of the guiding center itself and the computation capabilities. The code allows also for the inclusion of the three-dimensional electrostatic potential in a self-consistent manner, but the initial objective has been set to solving only the neoclassical transport. A high order conservative method (Spectral Difference Method) has been chosen in order to discretized the equation for its numerical solution. In this first report, in addition to justifying the work, the evolution equation and its approximations are described, as well as the baseline of the numerical procedures. (Author) 28 refs

Quantum-statistical kinetic equations

International Nuclear Information System (INIS)

Loss, D.; Schoeller, H.

1989-01-01

Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, the authors derive an exact quantum-statistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors, P q -rule, etc.) to nonequilibrium systems described by a density operator ρ(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantum-statistical versions (including exchange effects due to Fermi-Dirac or Bose-Einstein statistics) of the homogeneous Boltzmann (binary collisions) and Choh-Uhlenbeck (triple collisions) equations are derived

Kinetic Boltzmann, Vlasov and Related Equations

CERN Document Server

Sinitsyn, Alexander; Vedenyapin, Victor

2011-01-01

Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in

Reduced kinetic equations: An influence functional approach

International Nuclear Information System (INIS)

Wio, H.S.

1985-01-01

The author discusses a scheme for obtaining reduced descriptions of multivariate kinetic equations based on the 'influence functional' method of Feynmann. It is applied to the case of Fokker-Planck equations showing the form that results for the reduced equation. The possibility of Markovian or non-Markovian reduced description is discussed. As a particular example, the reduction of the Kramers equation to the Smoluchwski equation in the limit of high friction is also discussed

Kinetic equations for the collisional plasma model

International Nuclear Information System (INIS)

Rij, W.I. Van; Meier, H.K.; Beasley, C.O. Jr.; McCune, J.E.

1977-01-01

Using the Collisional Plasma Model (CPM) representation, expressions are derived for the Vlasov operator, both in its general form and in the drift-kinetic approximation following the recursive derivation by Hazeltine. The expressions for the operators give easily calculated couplings between neighbouring components of the CPM representation. Expressions for various macroscopic observables in the drift-kinetics approximation are also given. (author)

A new hybrid code (CHIEF) implementing the inertial electron fluid equation without approximation

Science.gov (United States)

Muñoz, P. A.; Jain, N.; Kilian, P.; Büchner, J.

2018-03-01

We present a new hybrid algorithm implemented in the code CHIEF (Code Hybrid with Inertial Electron Fluid) for simulations of electron-ion plasmas. The algorithm treats the ions kinetically, modeled by the Particle-in-Cell (PiC) method, and electrons as an inertial fluid, modeled by electron fluid equations without any of the approximations used in most of the other hybrid codes with an inertial electron fluid. This kind of code is appropriate to model a large variety of quasineutral plasma phenomena where the electron inertia and/or ion kinetic effects are relevant. We present here the governing equations of the model, how these are discretized and implemented numerically, as well as six test problems to validate our numerical approach. Our chosen test problems, where the electron inertia and ion kinetic effects play the essential role, are: 0) Excitation of parallel eigenmodes to check numerical convergence and stability, 1) parallel (to a background magnetic field) propagating electromagnetic waves, 2) perpendicular propagating electrostatic waves (ion Bernstein modes), 3) ion beam right-hand instability (resonant and non-resonant), 4) ion Landau damping, 5) ion firehose instability, and 6) 2D oblique ion firehose instability. Our results reproduce successfully the predictions of linear and non-linear theory for all these problems, validating our code. All properties of this hybrid code make it ideal to study multi-scale phenomena between electron and ion scales such as collisionless shocks, magnetic reconnection and kinetic plasma turbulence in the dissipation range above the electron scales.

Analytical solution of point kinetics equations for linear reactivity variation during the start-up of a nuclear reactor

International Nuclear Information System (INIS)

Palma, Daniel A.P.; Martinez, Aquilino S.; Goncalves, Alessandro C.

2009-01-01

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting the variation of neutron density during the start-up of a nuclear reactor. In the practical case of an increase of nuclear reactor power resulting from the linear insertion of reactivity, the exact analytical solution cannot be obtained. Approximate solutions have been obtained in previous articles, based on considerations that need to be verifiable in practice. In the present article, an alternative analytic solution is presented for point kinetics equations in which the only approximation consists of disregarding the term of the second derivative for neutron density in relation to time. The results proved satisfactory when applied to practical situations in the start-up of a nuclear reactor through the control rods withdraw.

Analytical solution of point kinetics equations for linear reactivity variation during the start-up of a nuclear reactor

Energy Technology Data Exchange (ETDEWEB)

Palma, Daniel A.P. [CEFET QUIMICA de Nilopolis/RJ, 21941-914 Rio de Janeiro (Brazil)], E-mail: agoncalves@con.ufrj.br; Martinez, Aquilino S.; Goncalves, Alessandro C. [COPPE/UFRJ - Programa de Engenharia Nuclear, Rio de Janeiro (Brazil)

2009-09-15

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting the variation of neutron density during the start-up of a nuclear reactor. In the practical case of an increase of nuclear reactor power resulting from the linear insertion of reactivity, the exact analytical solution cannot be obtained. Approximate solutions have been obtained in previous articles, based on considerations that need to be verifiable in practice. In the present article, an alternative analytic solution is presented for point kinetics equations in which the only approximation consists of disregarding the term of the second derivative for neutron density in relation to time. The results proved satisfactory when applied to practical situations in the start-up of a nuclear reactor through the control rods withdraw.

Metamaterial characterization using Boltzmann's kinetic equation for electrons

DEFF Research Database (Denmark)

Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.

2013-01-01

Statistical properties of electrons in metals are taken into consideration to describe the microscopic motion of electrons. Assuming degenerate electron gas in metal, we introduce the Boltzmann kinetic equation to supplement Maxwell's equations. The solution of these equations clearly shows...

Kinetic equation of heterogeneous catalytic isotope exchange

Energy Technology Data Exchange (ETDEWEB)

Trokhimets, A I [AN Belorusskoj SSR, Minsk. Inst. Fiziko-Organicheskoj Khimii

1979-12-01

A kinetic equation is derived for the bimolecular isotope exchange reaction between AXsub(n)sup(*) and BXsub(m)sup(o), all atoms of element X in each molecule being equivalent. The equation can be generalized for homogeneous and heterogeneous catalytic isotope exchange.

Rational approximations to solutions of linear differential equations.

Science.gov (United States)

Chudnovsky, D V; Chudnovsky, G V

1983-08-01

Rational approximations of Padé and Padé type to solutions of differential equations are considered. One of the main results is a theorem stating that a simultaneous approximation to arbitrary solutions of linear differential equations over C(x) cannot be "better" than trivial ones implied by the Dirichlet box principle. This constitutes, in particular, the solution in the linear case of Kolchin's problem that the "Roth's theorem" holds for arbitrary solutions of algebraic differential equations. Complete effective proofs for several valuations are presented based on the Wronskian methods and graded subrings of Picard-Vessiot extensions.

An accurate technique for the solution of the nonlinear point kinetics equations

International Nuclear Information System (INIS)

Picca, Paolo; Ganapol, Barry D.; Furfaro, Roberto

2011-01-01

A novel methodology for the solution of non-linear point kinetic (PK) equations is proposed. The technique is based on a piecewise constant approximation of PK system of ODEs and explicitly accounts for reactivity feedback effects, through an iterative cycle. High accuracy is reached by introducing a sub-mesh for the numerical evaluation of integrals involved and by correcting the source term to include the non-linear effect on a finer time scale. The use of extrapolation techniques for convergence acceleration is also explored. Results for adiabatic feedback model are reported and compared with other benchmarks in literature. The convergence trend makes the algorithm particularly attractive for applications, including in multi-point kinetics and quasi-static frameworks. (author)

A high-order method for the integration of the Galerkin semi-discretized nuclear reactor kinetics equations

International Nuclear Information System (INIS)

Vargas, L.

1988-01-01

The numerical approximate solution of the space-time nuclear reactor kinetics equation is investigated using a finite-element discretization of the space variable and a high order integration scheme for the resulting semi-discretized parabolic equation. The Galerkin method with spatial piecewise polynomial Lagrange basis functions are used to obtained a continuous time semi-discretized form of the space-time reactor kinetics equation. A temporal discretization is then carried out with a numerical scheme based on the Iterated Defect Correction (IDC) method using piecewise quadratic polynomials or exponential functions. The kinetics equations are thus solved with in a general finite element framework with respect to space as well as time variables in which the order of convergence of the spatial and temporal discretizations is consistently high. A computer code GALFEM/IDC is developed, to implement the numerical schemes described above. This issued to solve a one space dimensional benchmark problem. The results of the numerical experiments confirm the theoretical arguments and show that the convergence is very fast and the overall procedure is quite efficient. This is due to the good asymptotic properties of the numerical scheme which is of third order in the time interval

Hypocoercivity for linear kinetic equations conserving mass

KAUST Repository

Dolbeault, Jean; Mouhot, Clé ment; Schmeiser, Christian

2015-01-01

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

Hypocoercivity for linear kinetic equations conserving mass

KAUST Repository

Dolbeault, Jean

2015-02-03

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

Modeling Taylor series approximations for prompt neutron kinetics with lab view simulations

International Nuclear Information System (INIS)

Adzri, E. P.

2012-09-01

The reactor point kinetics equations have been subjected to intense research in an effort to find simple yet accurate numerical solutions methods. The equations are very stiff numerically, meaning that there is a wide variation in the decay constants, so that using a particular time step in the numerical solution may provide sufficient accuracy for the group, but not for another. Several solutions techniques have been presented on the point kinetics equations with varying degrees of complexity. These include Power Series Solutions, CORE, PCA, Genapol and Taylor series methods. In this research, algorithms were developed based on the first and second order Taylor series expansion and simulated in LabVIEW to solve the Reactor Point Kinetics equations using block diagram nodes implemented within stacked sequences. The algorithms developed were fast,accurate and simple to code. Several reactivity insertions were used to simulate the change in neutron population with time. The LabVIEW- Taylor series solutions were compared with other solution techniques such as Power Series Solutions, CORE, PCA, Genapol and McMahon and Pierson's Taylor series approximation. The results of LabVIEW-Taylor series technique used by McMahon and Pearson The LabVIEW-implemented techniques were found to agree very well with these other methods. At 1x10 -8 s the neutron population was 1.000220 neutrons / cm 3 , at 1 x 10 -2 s it was 2.007681 neutrons / cm 3 and at 1x10 -1 s it was 2.075317 neutrons / cm 3 ; same results reported by Genapol for a fast reactor, it produced good and accurate results and compared very favorably with other methods found in the literature. Using much smaller time steps to the order or 10 -8 s commensurate with fast reactor parameters also produced very satisfactory results, indicating that the LabVIEW-based Taylor series technique is suitable for simulating the kinetics of fast reactors as well as thermal reactors. Algorithms developed that included second order terms

A nondissipative simulation method for the drift kinetic equation

International Nuclear Information System (INIS)

Watanabe, Tomo-Hiko; Sugama, Hideo; Sato, Tetsuya

2001-07-01

With the aim to study the ion temperature gradient (ITG) driven turbulence, a nondissipative kinetic simulation scheme is developed and comprehensively benchmarked. The new simulation method preserving the time-reversibility of basic kinetic equations can successfully reproduce the analytical solutions of asymmetric three-mode ITG equations which are extended to provide a more general reference for benchmarking than the previous work [T.-H. Watanabe, H. Sugama, and T. Sato: Phys. Plasmas 7 (2000) 984]. It is also applied to a dissipative three-mode system, and shows a good agreement with the analytical solution. The nondissipative simulation result of the ITG turbulence accurately satisfies the entropy balance equation. Usefulness of the nondissipative method for the drift kinetic simulations is confirmed in comparisons with other dissipative schemes. (author)

A novel fractional technique for the modified point kinetics equations

Directory of Open Access Journals (Sweden)

Ahmed E. Aboanber

2016-10-01

Full Text Available A fractional model for the modified point kinetics equations is derived and analyzed. An analytical method is used to solve the fractional model for the modified point kinetics equations. This methodical technique is based on the representation of the neutron density as a power series of the relaxation time as a small parameter. The validity of the fractional model is tested for different cases of step, ramp and sinusoidal reactivity. The results show that the fractional model for the modified point kinetics equations is the best representation of neutron density for subcritical and supercritical reactors.

The solution of the neutron point kinetics equation with stochastic extension: an analysis of two moments

Energy Technology Data Exchange (ETDEWEB)

Silva, Milena Wollmann da; Vilhena, Marco Tullio M.B.; Bodmann, Bardo Ernst J.; Vasques, Richard, E-mail: milena.wollmann@ufrgs.br, E-mail: vilhena@mat.ufrgs.br, E-mail: bardobodmann@ufrgs.br, E-mail: richard.vasques@fulbrightmail.org [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

2015-07-01

The neutron point kinetics equation, which models the time-dependent behavior of nuclear reactors, is often used to understand the dynamics of nuclear reactor operations. It consists of a system of coupled differential equations that models the interaction between (i) the neutron population; and (II) the concentration of the delayed neutron precursors, which are radioactive isotopes formed in the fission process that decay through neutron emission. These equations are deterministic in nature, and therefore can provide only average values of the modeled populations. However, the actual dynamical process is stochastic: the neutron density and the delayed neutron precursor concentrations vary randomly with time. To address this stochastic behavior, Hayes and Allen have generalized the standard deterministic point kinetics equation. They derived a system of stochastic differential equations that can accurately model the random behavior of the neutron density and the precursor concentrations in a point reactor. Due to the stiffness of these equations, this system was numerically implemented using a stochastic piecewise constant approximation method (Stochastic PCA). Here, we present a study of the influence of stochastic fluctuations on the results of the neutron point kinetics equation. We reproduce the stochastic formulation introduced by Hayes and Allen and compute Monte Carlo numerical results for examples with constant and time-dependent reactivity, comparing these results with stochastic and deterministic methods found in the literature. Moreover, we introduce a modified version of the stochastic method to obtain a non-stiff solution, analogue to a previously derived deterministic approach. (author)

The solution of the neutron point kinetics equation with stochastic extension: an analysis of two moments

International Nuclear Information System (INIS)

Silva, Milena Wollmann da; Vilhena, Marco Tullio M.B.; Bodmann, Bardo Ernst J.; Vasques, Richard

2015-01-01

The neutron point kinetics equation, which models the time-dependent behavior of nuclear reactors, is often used to understand the dynamics of nuclear reactor operations. It consists of a system of coupled differential equations that models the interaction between (i) the neutron population; and (II) the concentration of the delayed neutron precursors, which are radioactive isotopes formed in the fission process that decay through neutron emission. These equations are deterministic in nature, and therefore can provide only average values of the modeled populations. However, the actual dynamical process is stochastic: the neutron density and the delayed neutron precursor concentrations vary randomly with time. To address this stochastic behavior, Hayes and Allen have generalized the standard deterministic point kinetics equation. They derived a system of stochastic differential equations that can accurately model the random behavior of the neutron density and the precursor concentrations in a point reactor. Due to the stiffness of these equations, this system was numerically implemented using a stochastic piecewise constant approximation method (Stochastic PCA). Here, we present a study of the influence of stochastic fluctuations on the results of the neutron point kinetics equation. We reproduce the stochastic formulation introduced by Hayes and Allen and compute Monte Carlo numerical results for examples with constant and time-dependent reactivity, comparing these results with stochastic and deterministic methods found in the literature. Moreover, we introduce a modified version of the stochastic method to obtain a non-stiff solution, analogue to a previously derived deterministic approach. (author)

Instabilities and chaos in a kinetic equation for active nematics

International Nuclear Information System (INIS)

Shi, Xia-qing; Ma, Yu-qiang; Chaté, Hugues

2014-01-01

We study dry active nematics at the kinetic equation level, stressing the differences with the well-known Doi theory for non-active rods near thermal equilibrium. By deriving hydrodynamic equations from the kinetic equation, we show analytically that these two description levels share the same qualitative phase diagram, as defined by the linear instability limits of spatially-homogeneous solutions. In particular, we show that the ordered, homogeneous state is unstable in a region bordering the linear onset of nematic order, and is only linearly stable deeper in the ordered phase. Direct simulations of the kinetic equation reveal that its solutions are chaotic in the region of linear instability of the ordered homogeneous state. The local mechanisms for this large-scale chaos are discussed. (paper)

GENERAL EQUATIONS OF CARBONIZATION OF EUCALYPTUS SPP KINETIC MECHANISMS

Directory of Open Access Journals (Sweden)

Túlio Jardim Raad

2006-06-01

Full Text Available In the present work, a set of general equations related to kinetic mechanism of wood compound carbonization: hemicelluloses, cellulose and lignin was obtained by Avrami-Eroffev and Arrhenius equations and Thermogravimetry of Eucalyptus cloeziana, Eucalyptus camaldulensis, Corymbia citriodora, Eucalyptus urophylla and Eucalyptus grandis samples, TG-Isothermal and TG-Dynamic. The different thermal stabilities and decomposition temperature bands of those species compounds were applied as strategy to obtain the kinetic parameters: activation energy, exponential factor and reaction order. The kinetic model developed was validated by thermogravimetric curves from carbonization of others biomass such as coconut. The kinetic parameters found were - Hemicelluloses: E=98,6 kJmol, A=3,5x106s-1 n=1,0; - Cellulose: E=182,2 kJmol, A=1,2x1013s-1 n=1,5; - Lignin: E=46,6 kJmol, A=2,01s-1 n=0,41. The set of equations can be implemented in a mathematical model of wood carbonization simulation (with heat and mass transfer equations with the aim of optimizing the control and charcoal process used to produce pig iron.

Approximate variational solutions of the Grad-Shafranov equation

International Nuclear Information System (INIS)

Ludwig, G.O.

2001-01-01

Approximate solutions of the Grad-Schlueter-Shafranov equation based on variational methods are developed. The power series solutions of the Euler-Lagrange equations for equilibrium are compared with direct variational results for a low aspect ratio tokamak equilibrium. (author)

approximate controllability of a non-autonomous differential equation

Indian Academy of Sciences (India)

53

for a non-autonomous functional differential equation using the theory of linear ... approximate controllability of various functional differential equations in abstract ...... the operator A(t) and into the requirement that x(t) ∈ D(A) for all t ≥ 0.

An effective rate equation approach to reaction kinetics in small volumes: theory and application to biochemical reactions in nonequilibrium steady-state conditions.

Science.gov (United States)

Grima, R

2010-07-21

Chemical master equations provide a mathematical description of stochastic reaction kinetics in well-mixed conditions. They are a valid description over length scales that are larger than the reactive mean free path and thus describe kinetics in compartments of mesoscopic and macroscopic dimensions. The trajectories of the stochastic chemical processes described by the master equation can be ensemble-averaged to obtain the average number density of chemical species, i.e., the true concentration, at any spatial scale of interest. For macroscopic volumes, the true concentration is very well approximated by the solution of the corresponding deterministic and macroscopic rate equations, i.e., the macroscopic concentration. However, this equivalence breaks down for mesoscopic volumes. These deviations are particularly significant for open systems and cannot be calculated via the Fokker-Planck or linear-noise approximations of the master equation. We utilize the system-size expansion including terms of the order of Omega(-1/2) to derive a set of differential equations whose solution approximates the true concentration as given by the master equation. These equations are valid in any open or closed chemical reaction network and at both the mesoscopic and macroscopic scales. In the limit of large volumes, the effective mesoscopic rate equations become precisely equal to the conventional macroscopic rate equations. We compare the three formalisms of effective mesoscopic rate equations, conventional rate equations, and chemical master equations by applying them to several biochemical reaction systems (homodimeric and heterodimeric protein-protein interactions, series of sequential enzyme reactions, and positive feedback loops) in nonequilibrium steady-state conditions. In all cases, we find that the effective mesoscopic rate equations can predict very well the true concentration of a chemical species. This provides a useful method by which one can quickly determine the

Finite element approximation to the even-parity transport equation

International Nuclear Information System (INIS)

Lewis, E.E.

1981-01-01

This paper studies the finite element method, a procedure for reducing partial differential equations to sets of algebraic equations suitable for solution on a digital computer. The differential equation is cast into the form of a variational principle, the resulting domain then subdivided into finite elements. The dependent variable is then approximated by a simple polynomial, and these are linked across inter-element boundaries by continuity conditions. The finite element method is tailored to a variety of transport problems. Angular approximations are formulated, and the extent of ray effect mitigation is examined. Complex trial functions are introduced to enable the inclusion of buckling approximations. The ubiquitous curved interfaces of cell calculations, and coarse mesh methods are also treated. A concluding section discusses limitations of the work to date and suggests possible future directions

A kinetic equation for irreversible aggregation

International Nuclear Information System (INIS)

Zanette, D.H.

1990-09-01

We introduce a kinetic equation for describing irreversible aggregation in the ballistic regime, including velocity distributions. The associated evolution for the macroscopic quantities is studied, and the general solution for Maxwell interaction models is obtained in the Fourier representation. (author). 23 refs

Fractional Bhatnagar-Gross-Krook kinetic equation

Science.gov (United States)

Goychuk, Igor

2017-11-01

The linear Boltzmann equation (LBE) approach is generalized to describe fractional superdiffusive transport of the Lévy walk type in external force fields. The time distribution between scattering events is assumed to have a finite mean value and infinite variance. It is completely characterized by the two scattering rates, one fractional and a normal one, which defines also the mean scattering rate. We formulate a general fractional LBE approach and exemplify it with a particularly simple case of the Bohm and Gross scattering integral leading to a fractional generalization of the Bhatnagar, Gross and Krook (BGK) kinetic equation. Here, at each scattering event the particle velocity is completely randomized and takes a value from equilibrium Maxwell distribution at a given fixed temperature. We show that the retardation effects are indispensable even in the limit of infinite mean scattering rate and argue that this novel fractional kinetic equation provides a viable alternative to the fractional Kramers-Fokker-Planck (KFP) equation by Barkai and Silbey and its generalization by Friedrich et al. based on the picture of divergent mean time between scattering events. The case of divergent mean time is also discussed at length and compared with the earlier results obtained within the fractional KFP. Also a phenomenological fractional BGK equation without retardation effects is proposed in the limit of infinite scattering rates. It cannot be, however, rigorously derived from a scattering model, being rather clever postulated. It this respect, this retardationless equation is similar to the fractional KFP by Barkai and Silbey. However, it corresponds to the opposite, much more physical limit and, therefore, also presents a viable alternative.

Derivation of a reduced kinetic equation using Lie-transform techniques

International Nuclear Information System (INIS)

Brizard, A.

1991-01-01

The asymptotic elimination of fast time scales from a general kinetic equation, of the form: ∂ t f+z·∂ x f = C[f], facilitates the study of the long time behavior of its solution f(z,t). Here z describe the single-particle Hamiltonian dynamics and the operator C, which may possess nonlinear functional dependence on f, describes processes (such as discrete-particle effects, resonant wave-particle effects, or effects due to external sources) which cause changes in f as it is convectively transported along a Hamiltonian phase-space trajectory. When a fast time scale is associated with z through the dependence on a fast angle θ (whose frequency θ = Ω satisfies ε ≡ 1/Ωτ much-lt 1, where τ is a slow time scale of interest), a near-identity phase-space transformation T ε :z→Z (carried out with Lie-transform techniques) yields reduced Hamiltonian dynamical equations Z ε which are θ-independent. The corresponding transformed kinetic equation is derived. Averaging this equation over the fast angle θ yields a kinetic equation for left-angle F right-angle, the θ-averaged part of F. In general, the θ-dependence of C ε couples the kinetic equations for left-angle F right-angle and F, the θ-dependent part of F. One solves for the Fourier coefficient F l (associated with e ilθ ) as a functional of left-angle F right-angle. One obtains a reduced kinetic equation for left-angle F right-angle: d R left-angle F right-angle/dt = C R [left-angle F right-angle]. General expressions for C R are given, as well as expressions for the guiding-center and oscillation-center phase-space transformations of a linear Fokker-Planck operator. A discussion of the relationship with Mynick's work is presented

On Approximate Solutions of Functional Equations in Vector Lattices

Directory of Open Access Journals (Sweden)

Bogdan Batko

2014-01-01

Full Text Available We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra. The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x+y+F(x+F(y≠0⇒F(x+y=F(x+F(y in Riesz spaces, the Cauchy equation with squares F(x+y2=(F(x+F(y2 in f-algebras, and the quadratic functional equation F(x+y+F(x-y=2F(x+2F(y in Riesz spaces.

Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations.

Science.gov (United States)

Vorobev, Anatoliy

2010-11-01

We use the Cahn-Hilliard approach to model the slow dissolution dynamics of binary mixtures. An important peculiarity of the Cahn-Hilliard-Navier-Stokes equations is the necessity to use the full continuity equation even for a binary mixture of two incompressible liquids due to dependence of mixture density on concentration. The quasicompressibility of the governing equations brings a short time-scale (quasiacoustic) process that may not affect the slow dynamics but may significantly complicate the numerical treatment. Using the multiple-scale method we separate the physical processes occurring on different time scales and, ultimately, derive the equations with the filtered-out quasiacoustics. The derived equations represent the Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. This approximation can be further employed as a universal theoretical model for an analysis of slow thermodynamic and hydrodynamic evolution of the multiphase systems with strongly evolving and diffusing interfacial boundaries, i.e., for the processes involving dissolution/nucleation, evaporation/condensation, solidification/melting, polymerization, etc.

Approximate analytical solution of two-dimensional multigroup P-3 equations

International Nuclear Information System (INIS)

Matausek, M.V.; Milosevic, M.

1981-01-01

Iterative solution of multigroup spherical harmonics equations reduces, in the P-3 approximation and in two-dimensional geometry, to a problem of solving an inhomogeneous system of eight ordinary first order differential equations. With appropriate boundary conditions, these equations have to be solved for each energy group and in each iteration step. The general solution of the corresponding homogeneous system of equations is known in analytical form. The present paper shows how the right-hand side of the system can be approximated in order to derive a particular solution and thus an approximate analytical expression for the general solution of the inhomogeneous system. This combined analytical-numerical approach was shown to have certain advantages compared to the finite-difference method or the Lie-series expansion method, which have been used to solve similar problems. (author)

The Balescu kinetic equation with exchange interaction

International Nuclear Information System (INIS)

Belyi, V V; Kukharenko, Yu A

2009-01-01

Starting with the quantum BBGKY hierarchy for the distribution functions, we have obtained the quantum kinetic equation including the dynamical screening of the interaction potential, which exactly takes into account the exchange scattering in the plasma. The collision integral is expressed in terms of the Green function of the linearized Hartree–Fock equation. The potential energy takes into account the polarization and exchange interaction too

Approximate equations at breaking for nearshore wave transformation coefficients

Digital Repository Service at National Institute of Oceanography (India)

Chandramohan, P.; Nayak, B.U.; SanilKumar, V.

Based on small amplitude wave theory approximate equations are evaluated for determining the coefficients of shoaling, refraction, bottom friction, bottom percolation and viscous dissipation at breaking. The results obtainEd. by these equations...

Evaluation of rate law approximations in bottom-up kinetic models of metabolism

DEFF Research Database (Denmark)

Du, Bin; Zielinski, Daniel C.; Kavvas, Erol S.

2016-01-01

mass action rate law that removes the role of the enzyme from the reaction kinetics. We utilized in vivo data for the human red blood cell to compare the effect of rate law choices against the backdrop of physiological flux and concentration differences. We found that the Michaelis-Menten rate law......Background: The mechanistic description of enzyme kinetics in a dynamic model of metabolism requires specifying the numerical values of a large number of kinetic parameters. The parameterization challenge is often addressed through the use of simplifying approximations to form reaction rate laws....... These approximate rate laws were: 1) a Michaelis-Menten rate law with measured enzyme parameters, 2) a Michaelis-Menten rate law with approximated parameters, using the convenience kinetics convention, 3) a thermodynamic rate law resulting from a metabolite saturation assumption, and 4) a pure chemical reaction...

Approximate solution fuzzy pantograph equation by using homotopy perturbation method

Science.gov (United States)

Jameel, A. F.; Saaban, A.; Ahadkulov, H.; Alipiah, F. M.

2017-09-01

In this paper, Homotopy Perturbation Method (HPM) is modified and formulated to find the approximate solution for its employment to solve (FDDEs) involving a fuzzy pantograph equation. The solution that can be obtained by using HPM is in the form of infinite series that converge to the actual solution of the FDDE and this is one of the benefits of this method In addition, it can be used for solving high order fuzzy delay differential equations directly without reduction to a first order system. Moreover, the accuracy of HPM can be detected without needing the exact solution. The HPM is studied for fuzzy initial value problems involving pantograph equation. Using the properties of fuzzy set theory, we reformulate the standard approximate method of HPM and obtain the approximate solutions. The effectiveness of the proposed method is demonstrated for third order fuzzy pantograph equation.

Approximate analytical solution of two-dimensional multigroup P-3 equations

International Nuclear Information System (INIS)

Matausek, M.V.; Milosevic, M.

1981-01-01

Iterative solution of multigroup spherical harmonics equations reduces, in the P-3 approximation and in two-dimensional geometry, to a problem of solving an inhomogeneous system of eight ordinary first order differential equations. With appropriate boundary conditions, these equations have to be solved for each energy group and in each iteration step. The general solution of the corresponding homogeneous system of equations is known in analytical form. The present paper shows how the right-hand side of the system can be approximated in order to derive a particular solution and thus an approximate analytical expression for the general solution of the inhomogeneous system. This combined analytical-numerical approach was shown to have certain advantages compared to the finite-difference method or the Lie-series expansion method, which have been used to solve similar problems. (orig./RW) [de

Steepest descent approximations for accretive operator equations

International Nuclear Information System (INIS)

Chidume, C.E.

1993-03-01

A necessary and sufficient condition is established for the strong convergence of the steepest descent approximation to a solution of equations involving quasi-accretive operators defined on a uniformly smooth Banach space. (author). 49 refs

Improved stochastic approximation methods for discretized parabolic partial differential equations

Science.gov (United States)

Guiaş, Flavius

2016-12-01

We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).

Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

International Nuclear Information System (INIS)

Goreac, D.

2009-01-01

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (Stochastic Partial Differential Equations and Applications, Series of Lecture Notes in Pure and Appl. Math., vol. 245, pp. 253-260, Chapman and Hall, London, 2006) and Goreac (Applied Analysis and Differential Equations, pp. 153-164, World Scientific, Singapore, 2007) from the finite dimensional to the infinite dimensional case

On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity

Directory of Open Access Journals (Sweden)

Nikolai N. Bogoliubov (Jr.

2007-01-01

Full Text Available A two-component particle model of Boltzmann-Vlasov type kinetic equations in the form of special nonlinear integro-differential hydrodynamic systems on an infinite-dimensional functional manifold is discussed. We show that such systems are naturally connected with the nonlinear kinetic Boltzmann-Vlasov equations for some one-dimensional particle flows with pointwise interaction potential between particles. A new type of hydrodynamic two-component Benney equations is constructed and their Hamiltonian structure is analyzed.

The soliton solution of BBGKY quantum kinetic equations chain for different type particles system

International Nuclear Information System (INIS)

Rasulova, M.Yu.; Avazov, U.; Hassan, T.

2006-12-01

In the present paper on the basis of BBGKY chain of quantum kinetic equations the chain of equations for correlation matrices is derived, describing the evolution of a system of different types particles, which interact by pair potential. The series, which is the solution of this chain of equations for correlation matrices, is suggested. Using this series the solution of the last chain of equations is reduced to a solution of a set of homogeneous and nonhomogeneous von-Neumann's kinetic equations (analogue of Vlasov equations for quantum case). The first and second equations of this set of equations coincide with the first and second kinetic equations of the set, which is used in plasma physics. For an potential in the form of Dirac delta function, the solution of von-Neumann equation is defined through soliton solution of nonlinear Schrodinger equations. Based on von-Neumann equation one can define all terms of series, which is a solution of a chain of equations for correlation matrices. On the basis of these correlation matrices for a system of different types of particles we can define exact solution of BBGKY chain of quantum kinetic equations

Parameter Estimation for Partial Differential Equations by Collage-Based Numerical Approximation

Directory of Open Access Journals (Sweden)

Xiaoyan Deng

2009-01-01

into a minimization problem of a function of several variables after the partial differential equation is approximated by a differential dynamical system. Then numerical schemes for solving this minimization problem are proposed, including grid approximation and ant colony optimization. The proposed schemes are applied to a parameter estimation problem for the Belousov-Zhabotinskii equation, and the results show that the proposed approximation method is efficient for both linear and nonlinear partial differential equations with respect to unknown parameters. At worst, the presented method provides an excellent starting point for traditional inversion methods that must first select a good starting point.

Taylor's series method for solving the nonlinear point kinetics equations

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → Taylor's series method for nonlinear point kinetics equations is applied. → The general order of derivatives are derived for this system. → Stability of Taylor's series method is studied. → Taylor's series method is A-stable for negative reactivity. → Taylor's series method is an accurate computational technique. - Abstract: Taylor's series method for solving the point reactor kinetics equations with multi-group of delayed neutrons in the presence of Newtonian temperature feedback reactivity is applied and programmed by FORTRAN. This system is the couples of the stiff nonlinear ordinary differential equations. This numerical method is based on the different order derivatives of the neutron density, the precursor concentrations of i-group of delayed neutrons and the reactivity. The r th order of derivatives are derived. The stability of Taylor's series method is discussed. Three sets of applications: step, ramp and temperature feedback reactivities are computed. Taylor's series method is an accurate computational technique and stable for negative step, negative ramp and temperature feedback reactivities. This method is useful than the traditional methods for solving the nonlinear point kinetics equations.

Approximate solution of the transport equation by methods of Galerkin type

International Nuclear Information System (INIS)

Pitkaranta, J.

1977-01-01

Questions of the existence, uniqueness, and convergence of approximate solutions of transport equations by methods of the Galerkin type (where trial and weighting functions are the same) are discussed. The results presented do not exclude the infinite-dimensional case. Two strategies can be followed in the variational approximation of the transport operator: one proceeds from the original form of the transport equation, while the other is based on the partially symmetrized equation. Both principles are discussed in this paper. The transport equation is assumed in a discretized multigroup form

Dislocation kinetics and the acoustic-wave approximation for liquids

International Nuclear Information System (INIS)

Stout, R.B.

1983-03-01

A dislocation-dependent model for liquids describes the lattice deformation and the fluidity deformation as additive deformations. The lattice deformation represents distortions of an atom's potential energy structure and is a recoverable deformation response. The fluidity deformation represents discontinuous repositioning of atoms by dislocation kinetics in the lattice structure and is a nonrecoverable deformation response. From this model, one concludes that in liquids the acoustic-wave approximation is a description of a recoverable oscillation deformation that has dissipation because of dislocation kinetics. Other more-complex waves may exist, but such waves would rapidly disappear because of the small thermodynamic potential for dislocation kinetics in liquids

Symmetries of th-Order Approximate Stochastic Ordinary Differential Equations

OpenAIRE

Fredericks, E.; Mahomed, F. M.

2012-01-01

Symmetries of $n$ th-order approximate stochastic ordinary differential equations (SODEs) are studied. The determining equations of these SODEs are derived in an Itô calculus context. These determining equations are not stochastic in nature. SODEs are normally used to model nature (e.g., earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.

The C{sub n} method for approximation of the Boltzmann equation; La methode C{sub n} d'approximation de l'equation de Boltzmann

Energy Technology Data Exchange (ETDEWEB)

Benoist, P; Kavenoky, A [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires

1968-01-15

In a new method of approximation of the Boltzmann equation, one starts from a particular form of the equation which involves only the angular flux at the boundary of the considered medium and where the space variable does not appear explicitly. Expanding in orthogonal polynomials the angular flux of neutrons leaking from the medium and making no assumption about the angular flux within the medium, very good approximations to several classical plane geometry problems, i.e. the albedo of slabs and the transmission by slabs, the extrapolation length of the Milne problem, the spectrum of neutrons reflected by a semi-infinite slowing down medium. The method can be extended to other geometries. (authors) [French] On etablit une nouvelle methode d'approximation pour l'equation de Boltzmann en partant d'une forme particuliere de cette equation qui n'implique que le flux angulaire a la frontiere du milieu et ou les variables d'espace n'apparaissent pas explicitement. Par un developpement en polynomes orthogonaux du flux angulaire sortant du milieu et sans faire d'hypothese sur le flux angulaire a l'interieur du milieu, on obtient de tres bonnes approximations pour plusieurs problemes classiques en geometrie plane: l'albedo et le facteur de transmission des plaques, la longueur d'extrapolation du probleme de Milne, le spectre des neutrons reflechis par un milieu semi-infini ralentisseur. La methode se generalise a d'autres geometries. (auteurs)

Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

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S. Narayanamoorthy

2015-01-01

Full Text Available We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

On a closed form approach to the fractional neutron point kinetics equation with temperature feedback

International Nuclear Information System (INIS)

Schramm, Marcelo; Bodmann, Bardo E.J.; Vilhena, Marco T.M.B.; Petersen, Claudio Z.; Alvim, Antonio C.M.

2013-01-01

Following the quest to find analytical solutions, we extend the methodology applied successfully to timely fractional neutron point kinetics (FNPK) equations by adding the effects of temperature. The FNPK equations with temperature feedback correspond to a nonlinear system and “stiff” type for the neutron density and the concentration of delayed neutron precursors. These variables determine the behavior of a nuclear reactor power with time and are influenced by the position of control rods, for example. The solutions of kinetics equations provide time information about the dynamics in a nuclear reactor in operation and are useful, for example, to understand the power fluctuations with time that occur during startup or shutdown of the reactor, due to adjustments of the control rods. The inclusion of temperature feedback in the model introduces an estimate of the transient behavior of the power and other variables, which are strongly coupled. Normally, a single value of reactivity is used across the energy spectrum. Especially in case of power change, the neutron energy spectrum changes as well as physical parameters such as the average cross sections. However, even knowing the importance of temperature effects on the control of the reactor power, the character of the set of nonlinear equations governing this system makes it difficult to obtain a purely analytical solution. Studies have been published in this sense, using numerical approaches. Here the idea is to consider temperature effects to make the model more realistic and thus solve it in a semi-analytical way. Therefore, the main objective of this paper is to obtain an analytical representation of fractional neutron point kinetics equations with temperature feedback, without having to resort to approximations inherent in numerical methods. To this end, we will use the decomposition method, which has been successfully used by the authors to solve neutron point kinetics problems. The results obtained will

On a closed form approach to the fractional neutron point kinetics equation with temperature feedback

Energy Technology Data Exchange (ETDEWEB)

Schramm, Marcelo; Bodmann, Bardo E.J.; Vilhena, Marco T.M.B., E-mail: marceloschramm@hotmail.com, E-mail: bardo.bodmann@ufrgs.br, E-mail: mtmbvilhena@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Engenharia Mecanica; Petersen, Claudio Z., E-mail: claudiopetersen@yahoo.com.br [Universidade Federal de Pelotas (UFPel), RS (Brazil). Departamento de Matematica; Alvim, Antonio C.M., E-mail: alvim@nuclear.ufrj.br [Universidade Federal do Rio de Janeiro (UFRJ), RJ (Brazil). Instituto Alberto Luiz Coimbra de Pos-Graduacao e Pesquisa em Engenharia

2013-07-01

Following the quest to find analytical solutions, we extend the methodology applied successfully to timely fractional neutron point kinetics (FNPK) equations by adding the effects of temperature. The FNPK equations with temperature feedback correspond to a nonlinear system and “stiff” type for the neutron density and the concentration of delayed neutron precursors. These variables determine the behavior of a nuclear reactor power with time and are influenced by the position of control rods, for example. The solutions of kinetics equations provide time information about the dynamics in a nuclear reactor in operation and are useful, for example, to understand the power fluctuations with time that occur during startup or shutdown of the reactor, due to adjustments of the control rods. The inclusion of temperature feedback in the model introduces an estimate of the transient behavior of the power and other variables, which are strongly coupled. Normally, a single value of reactivity is used across the energy spectrum. Especially in case of power change, the neutron energy spectrum changes as well as physical parameters such as the average cross sections. However, even knowing the importance of temperature effects on the control of the reactor power, the character of the set of nonlinear equations governing this system makes it difficult to obtain a purely analytical solution. Studies have been published in this sense, using numerical approaches. Here the idea is to consider temperature effects to make the model more realistic and thus solve it in a semi-analytical way. Therefore, the main objective of this paper is to obtain an analytical representation of fractional neutron point kinetics equations with temperature feedback, without having to resort to approximations inherent in numerical methods. To this end, we will use the decomposition method, which has been successfully used by the authors to solve neutron point kinetics problems. The results obtained will

An analytical solution of the one-dimensional neutron diffusion kinetic equation in cartesian geometry

International Nuclear Information System (INIS)

Ceolin, Celina; Vilhena, Marco T.; Petersen, Claudio Z.

2009-01-01

In this work we report an analytical solution for the monoenergetic neutron diffusion kinetic equation in cartesian geometry. Bearing in mind that the equation for the delayed neutron precursor concentration is a first order linear differential equation in the time variable, to make possible the application of the GITT approach to the kinetic equation, we introduce a fictitious diffusion term multiplied by a positive small value ε. By this procedure, we are able to solve this set of equations. Indeed, applying the GITT technique to the modified diffusion kinetic equation, we come out with a matrix differential equation which has a well known analytical solution when ε goes to zero. We report numerical simulations as well study of numerical convergence of the results attained. (author)

Numerical approximations of difference functional equations and applications

Directory of Open Access Journals (Sweden)

Zdzisław Kamont

2005-01-01

Full Text Available We give a theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type. We apply this general result in the investigation of the stability of difference schemes generated by nonlinear first order partial differential functional equations and by parabolic problems. We show that all known results on difference methods for initial or initial boundary value problems can be obtained as particular cases of this general and simple result. We assume that the right hand sides of equations satisfy nonlinear estimates of the Perron type with respect to functional variables.

Modelling opinion formation by means of kinetic equations

OpenAIRE

Boudin , Laurent; Salvarani , Francesco

2010-01-01

In this chapter, we review some mechanisms of opinion dynamics that can be modelled by kinetic equations. Beside the sociological phenomenon of compromise, naturally linked to collisional operators of Boltzmann kind, many other aspects, already mentioned in the sociophysical literature or no, can enter in this framework. While describing some contributions appeared in the literature, we enlighten some mathematical tools of kinetic theory that can be useful in the context of sociophysics.

Analytic solutions of the multigroup space-time reactor kinetics equations

International Nuclear Information System (INIS)

Lee, C.E.; Rottler, S.

1986-01-01

The development of analytical and numerical solutions to the reactor kinetics equations is reviewed. Analytic solutions of the multigroup space-time reactor kinetics equations are developed for bare and reflected slabs and spherical reactors for zero flux, zero current and extrapolated endpoint boundary conditions. The material properties of the reactors are assumed constant in space and time, but spatially-dependent source terms and initial conditions are investigated. The system of partial differential equations is reduced to a set of linear ordinary differential equations by the Laplace transform method. These equations are solved by matrix Green's functions yielding a general matrix solution for the neutron flux and precursor concentration in the Laplace transform space. The detailed pole structure of the Laplace transform matrix solutions is investigated. The temporally- and spatially-dependent solutions are determined from the inverse Laplace transform using the Cauchy residue theorem, the theorem of Frobenius, a knowledge of the detailed pole structure and matrix operators. (author)

An Explicit Finite Difference scheme for numerical solution of fractional neutron point kinetic equation

International Nuclear Information System (INIS)

Saha Ray, S.; Patra, A.

2012-01-01

Highlights: ► In this paper fractional neutron point kinetic equation has been analyzed. ► The numerical solution for fractional neutron point kinetic equation is obtained. ► Explicit Finite Difference Method has been applied. ► Supercritical reactivity, critical reactivity and subcritical reactivity analyzed. ► Comparison between fractional and classical neutron density is presented. - Abstract: In the present article, a numerical procedure to efficiently calculate the solution for fractional point kinetics equation in nuclear reactor dynamics is investigated. The Explicit Finite Difference Method is applied to solve the fractional neutron point kinetic equation with the Grunwald–Letnikov (GL) definition (). Fractional Neutron Point Kinetic Model has been analyzed for the dynamic behavior of the neutron motion in which the relaxation time associated with a variation in the neutron flux involves a fractional order acting as exponent of the relaxation time, to obtain the best operation of a nuclear reactor dynamics. Results for neutron dynamic behavior for subcritical reactivity, supercritical reactivity and critical reactivity and also for different values of fractional order have been presented and compared with the classical neutron point kinetic (NPK) equation as well as the results obtained by the learned researchers .

Nonlinear Schroedinger Approximations for Partial Differential Equations with Quadratic and Quasilinear Terms

Science.gov (United States)

Cummings, Patrick

We consider the approximation of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use approximation results of this kind to model these physical systems. The approximations are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the approximation captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be approximated by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the approximation holds for the full interval of existence of the approximate NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave approximation result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the order 1 over epsilon squared time scale.

Approximate Solution of LR Fuzzy Sylvester Matrix Equations

Directory of Open Access Journals (Sweden)

Xiaobin Guo

2013-01-01

Full Text Available The fuzzy Sylvester matrix equation AX~+X~B=C~ in which A,B are m×m and n×n crisp matrices, respectively, and C~ is an m×n LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

Solution of the chemical master equation by radial basis functions approximation with interface tracking

NARCIS (Netherlands)

Kryven, I.; Röblitz, S; Schütte, C.

2015-01-01

Background: The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents

On a closed form solution of the point kinetics equations with reactivity feedback of temperature

International Nuclear Information System (INIS)

Silva, Jeronimo J.A.; Vilhena, Marco T.M.B.; Petersen, Claudio Z.; Bodmann, Bardo E.J.; Alvim, Antonio C.M.

2011-01-01

An analytical solution of the point kinetics equations to calculate reactivity as a function of time by the Decomposition method has recently appeared in the literature. In this paper, we go one step forward, by considering the neutron point kinetics equations together with temperature feedback effects. To accomplish that, we extended the point kinetics by a temperature perturbation, obtaining a second order nonlinear ordinary differential equation. This equation is then solved by the Decomposition Method, that is, by expanding the neutron density in a series and the nonlinear terms into Adomian Polynomials. Substituting these expansions into the nonlinear ordinary equation, we construct a recursive set of linear problems that can be solved by the methodology previously mentioned for the point kinetics equation. We also report on numerical simulations and comparisons against literature results. (author)

Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

International Nuclear Information System (INIS)

Barth, Andrea; Lang, Annika

2012-01-01

In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.

Multi-scale method for the resolution of the neutronic kinetics equations

International Nuclear Information System (INIS)

Chauvet, St.

2008-10-01

In this PhD thesis and in order to improve the time/precision ratio of the numerical simulation calculations, we investigate multi-scale techniques for the resolution of the reactor kinetics equations. We choose to focus on the mixed dual diffusion approximation and the quasi-static methods. We introduce a space dependency for the amplitude function which only depends on the time variable in the standard quasi-static context. With this new factorization, we develop two mixed dual problems which can be solved with Cea's solver MINOS. An algorithm is implemented, performing the resolution of these problems defined on different scales (for time and space). We name this approach: the Local Quasi-Static method. We present here this new multi-scale approach and its implementation. The inherent details of amplitude and shape treatments are discussed and justified. Results and performances, compared to MINOS, are studied. They illustrate the improvement on the time/precision ratio for kinetics calculations. Furthermore, we open some new possibilities to parallelize computations with MINOS. For the future, we also introduce some improvement tracks with adaptive scales. (author)

Is the kinetic equation for turbulent gas-particle flows ill posed?

Science.gov (United States)

Reeks, M; Swailes, D C; Bragg, A D

2018-02-01

This paper is about the kinetic equation for gas-particle flows, in particular its well-posedness and realizability and its relationship to the generalized Langevin model (GLM) probability density function (PDF) equation. Previous analyses, e.g. [J.-P. Minier and C. Profeta, Phys. Rev. E 92, 053020 (2015)PLEEE81539-375510.1103/PhysRevE.92.053020], have concluded that this kinetic equation is ill posed, that in particular it has the properties of a backward heat equation, and as a consequence, its solution will in the course of time exhibit finite-time singularities. We show that this conclusion is fundamentally flawed because it ignores the coupling between the phase space variables in the kinetic equation and the time and particle inertia dependence of the phase space diffusion tensor. This contributes an extra positive diffusion that always outweighs the negative diffusion associated with the dispersion along one of the principal axes of the phase space diffusion tensor. This is confirmed by a numerical evaluation of analytic solutions of these positive and negative contributions to the particle diffusion coefficient along this principal axis. We also examine other erroneous claims and assumptions made in previous studies that demonstrate the apparent superiority of the GLM PDF approach over the kinetic approach. In so doing, we have drawn attention to the limitations of the GLM approach, which these studies have ignored or not properly considered, to give a more balanced appraisal of the benefits of both PDF approaches.

Bethe-Salpeter equation for fermion-antifermion system in the ladder approximation

International Nuclear Information System (INIS)

Fukui, Ichio; Seto, Noriaki; Yoshida, Toshihiro.

1977-01-01

The Bethe-Salpeter (B-S) equation is important for studying hadron physics. Especially intensive investigation on the fermion-antifermion B-S equation is indispensable for the phenomenological studies of hardrons. However, many components of the B-S amplitude and the Wick-rotated integral kernel of non-Fredholm type have prevented from knowing details the solutions even in the ladder approximation. Some particular solutions are known in case of the vanishing four-momenta of bound states. The B-S equation for the bound state of fermion-anti-fermion system interacting through vector (axial-vector) particle exchange was studied in the ladder approximation with Feynman gauge. The reduced equations were obtained for suitably decomposed amplitude, and it is shown that, in the S-wave case, the coupled equations separate into two parts. In the nonrelativistic limit, large components of the amplitude satisfy the Wick-Cutkosky equation, and small components are expressed in terms of the large ones. Equations are derived for the equal-time amplitudes. (Kobatake, H.)

An Approximate Method for the Acoustic Attenuating VTI Eikonal Equation

KAUST Repository

Hao, Q.

2017-05-26

We present an approximate method to solve the acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis (VTI). A perturbation method is used to derive the perturbation formula for complex-valued traveltimes. The application of Shanks transform further enhances the accuracy of approximation. We derive both analytical and numerical solutions to the acoustic eikonal equation. The analytic solution is valid for homogeneous VTI media with moderate anellipticity and strong attenuation and attenuation-anisotropy. The numerical solution is applicable for inhomogeneous attenuating VTI media.

An Approximate Method for the Acoustic Attenuating VTI Eikonal Equation

KAUST Repository

Hao, Q.; Alkhalifah, Tariq Ali

2017-01-01

We present an approximate method to solve the acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis (VTI). A perturbation method is used to derive the perturbation formula for complex-valued traveltimes. The application of Shanks transform further enhances the accuracy of approximation. We derive both analytical and numerical solutions to the acoustic eikonal equation. The analytic solution is valid for homogeneous VTI media with moderate anellipticity and strong attenuation and attenuation-anisotropy. The numerical solution is applicable for inhomogeneous attenuating VTI media.

A nonlinear bounce kinetic equation for trapped electrons

International Nuclear Information System (INIS)

Gang, F.Y.

1990-03-01

A nonlinear bounce averaged drift kinetic equation for trapped electrons is derived. This equation enables one to compute the nonlinear response of the trapped electron distribution function in terms of the field-line projection of a potential fluctuation left-angle e -inqθ φ n right-angle b . It is useful for both analytical and computational studies of the nonlinear evolution of short wavelength (n much-gt 1) trapped electron mode-driven turbulence. 7 refs

Symmetric approximations of the Navier-Stokes equations

International Nuclear Information System (INIS)

Kobel'kov, G M

2002-01-01

A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as ε→0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established

Initial state dependence of nonlinear kinetic equations: The classical electron gas

International Nuclear Information System (INIS)

Marchetti, M.C.; Cohen, E.G.D.; Dorfman, J.R.; Kirkpatrick, T.R.

1985-01-01

The method of nonequilibrium cluster expansion is used to study the decay to equilibrium of a weakly coupled inhomogeneous electron gas prepared in a local equilibrium state at the initial time, t=0. A nonlinear kinetic equation describing the long time behavior of the one-particle distribution function is obtained. For consistency, initial correlations have to be taken into account. The resulting kinetic equation-differs from that obtained when the initial state of the system is assumed to be factorized in a product of one-particle functions. The question of to what extent correlations in the initial state play an essential role in determining the form of the kinetic equation at long times is discussed. To that end, the present calculations are compared wih results obtained before for hard sphere gases and in general with strong short-range forces. A partial answer is proposed and some open questions are indicated

Analytical approximate solutions for a general class of nonlinear delay differential equations.

Science.gov (United States)

Căruntu, Bogdan; Bota, Constantin

2014-01-01

We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

Approximation of entropy solutions to degenerate nonlinear parabolic equations

Science.gov (United States)

Abreu, Eduardo; Colombeau, Mathilde; Panov, Evgeny Yu

2017-12-01

We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space L^∞, whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and L^1-stability. We prove that the sequence of approximate solutions is strongly L^1-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.

Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains

Science.gov (United States)

Wang, Xiaohu; Lu, Kening; Wang, Bixiang

2018-01-01

In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction-diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of stochastic reaction-diffusion equation. Then, we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic reaction-diffusion equation for both additive and multiplicative noise.

Direct application of Padé approximant for solving nonlinear differential equations.

Science.gov (United States)

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

2014-01-01

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

Validity of various approximations for the Bethe-Salpeter equation and their WKB quantization

International Nuclear Information System (INIS)

Silvestre-Brac, B.; Bilal, A.; Gignoux, C.; Schuck, P.

1984-01-01

The validity of the instantaneous approximation for the Bethe-Salpeter equation is questioned within the framework of the simple scalar-scalar model of Cutkosky. Detailed numerous results for various approximations are compared to the exact ones. WKB quantization is applied to these relativistic approximations. An unexpected question arises: is the currently used Bethe-Salpeter equation (i.e., the ladder approximation) well suited to describe two interacting relativistic particles

Approximate analytical methods for solving ordinary differential equations

CERN Document Server

Radhika, TSL; Rani, T Raja

2015-01-01

Approximate Analytical Methods for Solving Ordinary Differential Equations (ODEs) is the first book to present all of the available approximate methods for solving ODEs, eliminating the need to wade through multiple books and articles. It covers both well-established techniques and recently developed procedures, including the classical series solution method, diverse perturbation methods, pioneering asymptotic methods, and the latest homotopy methods.The book is suitable not only for mathematicians and engineers but also for biologists, physicists, and economists. It gives a complete descripti

Accelerated procedure to solve kinetic equation for neutral atoms in a hot plasma

Science.gov (United States)

Tokar, Mikhail Z.

2017-12-01

The recombination of plasma charged components, electrons and ions of hydrogen isotopes, on the wall of a fusion reactor is a source of neutral molecules and atoms, recycling back into the plasma volume. Here neutral species participate, in particular, in charge-exchange (c-x) collisions with the plasma ions and, as a result, atoms of high energies with chaotically directed velocities are generated. Some fraction of these hot atoms hit the wall. Statistical Monte Carlo methods normally used to model c-x atoms are too time consuming for reasonably small level of accident errors and extensive parameter studies are problematic. By applying pass method to evaluate integrals from functions, including the ion velocity distribution, an iteration approach to solve one-dimensional kinetic equation [1], being alternative to Monte Carlo procedure, has been tremendously accelerated, at least by a factor of 30-50 [2]. Here this approach is developed further to solve the 2-D kinetic equation, applied to model the transport of c-x atoms in the vicinity of an opening in the wall, e.g., the entrance of the duct guiding to a diagnostic installation. This is necessary to determine firmly the energy spectrum of c-x atoms penetrating into the duct and to assess the erosion of the installation there. The results of kinetic modeling are compared with those obtained with the diffusion description for c-x atoms, being strictly relevant under plasma conditions of low temperature and high density, where the mean free path length between c-x collisions is much smaller than that till the atom ionization by electrons. It is demonstrated that the previous calculations [3], done with the diffusion approximation for c-x atoms, overestimate the erosion rate of Mo mirrors in a reactor by a factor of 3 compared to the result of the present kinetic study.

Elements of plasma kinetic theory

International Nuclear Information System (INIS)

Guasp, J.

1976-01-01

The physical foundations of plasma kinetic equations are exposed inside a series of seminars on plasma and fusion physics. The Vlasov and collisional equations with its application range have been discussed. The momenta equations for the macroscopic magnitudes and the more usual approximations have been obtained: two fluid equations for cold and warm plasmas, magnetohydrodynamic equations and the double-adiabatic theory. (author)

Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

Directory of Open Access Journals (Sweden)

Berenguer MI

2010-01-01

Full Text Available This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and .

Approximate method in estimation sensitivity responses to variations in delayed neutron energy spectra

Energy Technology Data Exchange (ETDEWEB)

Yoo, J; Shin, H S; Song, T Y; Park, W S [Korea Atomic Energy Research Institute, Taejon (Korea, Republic of)

1998-12-31

Previous our numerical results in computing point kinetics equations show a possibility in developing approximations to estimate sensitivity responses of nuclear reactor. We recalculate sensitivity responses by maintaining the corrections with first order of sensitivity parameter. We present a method for computing sensitivity responses of nuclear reactor based on an approximation derived from point kinetics equations. Exploiting this approximation, we found that the first order approximation works to estimate variations in the time to reach peak power because of their linear dependence on a sensitivity parameter, and that there are errors in estimating the peak power in the first order approximation for larger sensitivity parameters. To confirm legitimacy of out approximation, these approximate results are compared with exact results obtained from out previous numerical study. 4 refs., 2 figs., 3 tabs. (Author)

Approximate method in estimation sensitivity responses to variations in delayed neutron energy spectra

Energy Technology Data Exchange (ETDEWEB)

Yoo, J.; Shin, H. S.; Song, T. Y.; Park, W. S. [Korea Atomic Energy Research Institute, Taejon (Korea, Republic of)

1997-12-31

Previous our numerical results in computing point kinetics equations show a possibility in developing approximations to estimate sensitivity responses of nuclear reactor. We recalculate sensitivity responses by maintaining the corrections with first order of sensitivity parameter. We present a method for computing sensitivity responses of nuclear reactor based on an approximation derived from point kinetics equations. Exploiting this approximation, we found that the first order approximation works to estimate variations in the time to reach peak power because of their linear dependence on a sensitivity parameter, and that there are errors in estimating the peak power in the first order approximation for larger sensitivity parameters. To confirm legitimacy of out approximation, these approximate results are compared with exact results obtained from out previous numerical study. 4 refs., 2 figs., 3 tabs. (Author)

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

Directory of Open Access Journals (Sweden)

M. Bishehniasar

2017-01-01

Full Text Available The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs. The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE. Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD method and standard finite difference (SFD technique, which are popular in the literature for solving engineering problems.

Effective computation of stochastic protein kinetic equation by reducing stiffness via variable transformation

Energy Technology Data Exchange (ETDEWEB)

Wang, Lijin, E-mail: ljwang@ucas.ac.cn [School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049 (China)

2016-06-08

The stochastic protein kinetic equations can be stiff for certain parameters, which makes their numerical simulation rely on very small time step sizes, resulting in large computational cost and accumulated round-off errors. For such situation, we provide a method of reducing stiffness of the stochastic protein kinetic equation by means of a kind of variable transformation. Theoretical and numerical analysis show effectiveness of this method. Its generalization to a more general class of stochastic differential equation models is also discussed.

The accuracy of time dependent transport equation ergodic approximation

International Nuclear Information System (INIS)

Stancic, V.

1995-01-01

In order to predict the accuracy of the ergodic approximation for solving the time dependent transport equation, a comparison with respect to multiple collision and time finite difference methods, has been considered. (author)

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

OpenAIRE

Leibov Roman

2017-01-01

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems ...

Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary

Science.gov (United States)

Tang, Yuehao; Jiang, Qinghui; Zhou, Chuangbing

2016-04-01

An approximate solution is presented to the 1-D Boussinesq equation (BEQ) characterizing transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. The flow equation is decomposed to a linearized BEQ and a head correction equation. The linearized BEQ is solved using a Laplace transform. By means of the frozen-coefficient technique and Gauss function method, the approximate solution for the head correction equation can be obtained, which is further simplified to a closed-form expression under the condition of local energy equilibrium. The solutions of the linearized and head correction equations are discussed from physical concepts. Especially for the head correction equation, the well posedness of the approximate solution obtained by the frozen-coefficient method is verified to demonstrate its boundedness, which can be further embodied as the upper and lower error bounds to the exact solution of the head correction by statistical analysis. The advantage of this approximate solution is in its simplicity while preserving the inherent nonlinearity of the physical phenomenon. Comparisons between the analytical and numerical solutions of the BEQ validate that the approximation method can achieve desirable precisions, even in the cases with strong nonlinearity. The proposed approximate solution is applied to various hydrological problems, in which the algebraic expressions that quantify the water flow processes are derived from its basic solutions. The results are useful for the quantification of stream-aquifer exchange flow rates, aquifer response due to the sudden reservoir release, bank storage and depletion, and front position and propagation speed.

Symbolic computation of analytic approximate solutions for nonlinear fractional differential equations

Science.gov (United States)

Lin, Yezhi; Liu, Yinping; Li, Zhibin

2013-01-01

The Adomian decomposition method (ADM) is one of the most effective methods to construct analytic approximate solutions for nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, Rach (2008) [22], the Adomian decomposition method and the Padé approximants technique, a new algorithm is proposed to construct analytic approximate solutions for nonlinear fractional differential equations with initial or boundary conditions. Furthermore, a MAPLE software package is developed to implement this new algorithm, which is user-friendly and efficient. One only needs to input the system equation, initial or boundary conditions and several necessary parameters, then our package will automatically deliver the analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for non-smooth initial value problems. Our package provides a helpful and easy-to-use tool in science and engineering simulations. Program summaryProgram title: ADMP Catalogue identifier: AENE_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENE_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 12011 No. of bytes in distributed program, including test data, etc.: 575551 Distribution format: tar.gz Programming language: MAPLE R15. Computer: PCs. Operating system: Windows XP/7. RAM: 2 Gbytes Classification: 4.3. Nature of problem: Constructing analytic approximate solutions of nonlinear fractional differential equations with initial or boundary conditions. Non-smooth initial value problems can be solved by this program. Solution method: Based on the new definition of the Adomian polynomials [1], the Adomian decomposition method and the Pad

Solving Simple Kinetics without Integrals

Science.gov (United States)

de la Pen~a, Lisandro Herna´ndez

2016-01-01

The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…

An accurate solution of point reactor neutron kinetics equations of multi-group of delayed neutrons

International Nuclear Information System (INIS)

Yamoah, S.; Akaho, E.H.K.; Nyarko, B.J.B.

2013-01-01

Highlights: ► Analytical solution is proposed to solve the point reactor kinetics equations (PRKE). ► The method is based on formulating a coefficient matrix of the PRKE. ► The method was applied to solve the PRKE for six groups of delayed neutrons. ► Results shows good agreement with other traditional methods in literature. ► The method is accurate and efficient for solving the point reactor kinetics equations. - Abstract: The understanding of the time-dependent behaviour of the neutron population in a nuclear reactor in response to either a planned or unplanned change in the reactor conditions is of great importance to the safe and reliable operation of the reactor. In this study, an accurate analytical solution of point reactor kinetics equations with multi-group of delayed neutrons for specified reactivity changes is proposed to calculate the change in neutron density. The method is based on formulating a coefficient matrix of the homogenous differential equations of the point reactor kinetics equations and calculating the eigenvalues and the corresponding eigenvectors of the coefficient matrix. A small time interval is chosen within which reactivity relatively stays constant. The analytical method was applied to solve the point reactor kinetics equations with six-groups delayed neutrons for a representative thermal reactor. The problems of step, ramp and temperature feedback reactivities are computed and the results compared with other traditional methods. The comparison shows that the method presented in this study is accurate and efficient for solving the point reactor kinetics equations of multi-group of delayed neutrons

Approximate solution to neutron transport equation with linear anisotropic scattering

International Nuclear Information System (INIS)

Coppa, G.; Ravetto, P.; Sumini, M.

1983-01-01

A method to obtain an approximate solution to the transport equation, when both sources and collisions show a linearly anisotropic behavior, is outlined and the possible implications for numerical calculations in applied neutronics as well as shielding evaluations are investigated. The form of the differential system of equations taken by the method is quite handy and looks simpler and more manageable than any other today available technique. To go deeper into the efficiency of the method, some typical calculations concerning critical dimension of multiplying systems are then performed and the results are compared with the ones coming from the classical Ssub(N) approximations. The outcome of such calculations leads us to think of interesting developments of the method which could be quite useful in alternative to other today widespread approximate procedures, for any geometry, but especially for curved ones. (author)

Approximate damped oscillatory solutions and error estimates for the perturbed Klein–Gordon equation

International Nuclear Information System (INIS)

Ye, Caier; Zhang, Weiguo

2015-01-01

Highlights: • Analyze the dynamical behavior of the planar dynamical system corresponding to the perturbed Klein–Gordon equation. • Present the relations between the properties of traveling wave solutions and the perturbation coefficient. • Obtain all explicit expressions of approximate damped oscillatory solutions. • Investigate error estimates between exact damped oscillatory solutions and the approximate solutions and give some numerical simulations. - Abstract: The influence of perturbation on traveling wave solutions of the perturbed Klein–Gordon equation is studied by applying the bifurcation method and qualitative theory of dynamical systems. All possible approximate damped oscillatory solutions for this equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. The results of numerical simulations also establish our analysis

On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation

International Nuclear Information System (INIS)

Kawashima, S.; Matsumara, A.; Nishida, T.

1979-01-01

The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay rate tsup(-5/4)) as t → + infinitely to that of the compressible Navier-Stokes equation for the corresponding initial data. (orig.) 891 HJ/orig. 892 MKO

Choosing of optimal start approximation for laplace equation ...

African Journals Online (AJOL)

We investigate Dirichlet problem for a case of two-dimensional area with lime border, numerical scheme for solving this equation is widely knowns it finite difference method. One of the major stages in the algorithm for that numerical solution is choosing of start approximation, usually as the initial values of the unknown ...

Comparison of two forms of Vlasov-type relativistic kinetic equations in hadrodynamics

International Nuclear Information System (INIS)

Mashnik, S.G.; Maino, G.

1996-01-01

A comparison of two methods in the relativistic kinetic theory of the Fermi systems is carried out assuming, as an example, the simplest σω-version of quantum hadrodynamics with allowance for strong mean meson fields. It is shown that the Vlasov-type relativistic kinetic equation (VRKE) obtained by means of the procedure of squaring at an intermediate step is responsible for unphysical features. A direct method of derivation of kinetic equations is proposed. This method does not contain such drawback and gives rise to VRKE in hydrodynamics of a non-contradictory form in which both spin degrees of freedom and states with positive and negative energies are taken into account. 17 refs

Nodal approximations of varying order by energy group for solving the diffusion equation

International Nuclear Information System (INIS)

Broda, J.T.

1992-02-01

The neutron flux across the nuclear reactor core is of interest to reactor designers and others. The diffusion equation, an integro-differential equation in space and energy, is commonly used to determine the flux level. However, the solution of a simplified version of this equation when automated is very time consuming. Since the flux level changes with time, in general, this calculation must be made repeatedly. Therefore solution techniques that speed the calculation while maintaining accuracy are desirable. One factor that contributes to the solution time is the spatial flux shape approximation used. It is common practice to use the same order flux shape approximation in each energy group even though this method may not be the most efficient. The one-dimensional, two-energy group diffusion equation was solved, for the node average flux and core k-effective, using two sets of spatial shape approximations for each of three reactor types. A fourth-order approximation in both energy groups forms the first set of approximations used. The second set used combines a second-order approximation with a fourth-order approximation in energy group two. Comparison of the results from the two approximation sets show that the use of a different order spatial flux shape approximation results in considerable loss in accuracy for the pressurized water reactor modeled. However, the loss in accuracy is small for the heavy water and graphite reactors modeled. The use of different order approximations in each energy group produces mixed results. Further investigation into the accuracy and computing time is required before any quantitative advantage of the use of the second-order approximation in energy group one and the fourth-order approximation in energy group two can be determined

Balance of liquid-phase turbulence kinetic energy equation for bubble-train flow

International Nuclear Information System (INIS)

Ilic, Milica; Woerner, Martin; Cacuci, Dan Gabriel

2004-01-01

In this paper the investigation of bubble-induced turbulence using direct numerical simulation (DNS) of bubbly two-phase flow is reported. DNS computations are performed for a bubble-driven liquid motion induced by a regular train of ellipsoidal bubbles rising through an initially stagnant liquid within a plane vertical channel. DNS data are used to evaluate balance terms in the balance equation for the liquid phase turbulence kinetic energy. The evaluation comprises single-phase-like terms (diffusion, dissipation and production) as well as the interfacial term. Special emphasis is placed on the procedure for evaluation of interfacial quantities. Quantitative analysis of the balance equation for the liquid phase turbulence kinetic energy shows the importance of the interfacial term which is the only source term. The DNS results are further used to validate closure assumptions employed in modelling of the liquid phase turbulence kinetic energy transport in gas-liquid bubbly flows. In this context, the performance of respective closure relations in the transport equation for liquid turbulence kinetic energy within the two-phase k-ε and the two-phase k-l model is evaluated. (author)

Inverse periodic problem for the discrete approximation of the Schroedinger nonlinear equation

International Nuclear Information System (INIS)

Bogolyubov, N.N.; Prikarpatskij, A.K.; AN Ukrainskoj SSR, Lvov. Inst. Prikladnykh Problem Mekhaniki i Matematiki)

1982-01-01

The problem of numerical solution of the Schroedinger nonlinear equation (1) iPSIsub(t) = PSIsub(xx)+-2(PSI)sup(2)PSI. The numerical solution of nonlinear differential equation supposes its discrete approximation is required for the realization of the computer calculation process. Tor the equation (1) there exists the following discrete approximation by variable x(2) iPSIsub(n, t) = (PSIsub(n+1)-2PSIsub(n)+PSIsub(n-1))/(Δx)sup(2)+-(PSIsub(n))sup(2)(PSIsub(n+1)+PSIsub(n-1)), n=0, +-1, +-2... where PSIsub(n)(+) is the corresponding value of PSI(x, t) function in the node and divisions with the equilibrium step Δx. The main problem is obtaining analytically exact solutions of the equations (2). The analysis of the equation system (2) is performed on the base of the discrete analogue of the periodic variant of the inverse scattering problem method developed with the aid of nonlinear equations of the Korteweg-de Vries type. Obtained in explicit form are analytical solutions of the equations system (2). The solutions are expressed through the Riemann THETA-function [ru

Beyond mean-field approximations for accurate and computationally efficient models of on-lattice chemical kinetics

Science.gov (United States)

Pineda, M.; Stamatakis, M.

2017-07-01

Modeling the kinetics of surface catalyzed reactions is essential for the design of reactors and chemical processes. The majority of microkinetic models employ mean-field approximations, which lead to an approximate description of catalytic kinetics by assuming spatially uncorrelated adsorbates. On the other hand, kinetic Monte Carlo (KMC) methods provide a discrete-space continuous-time stochastic formulation that enables an accurate treatment of spatial correlations in the adlayer, but at a significant computation cost. In this work, we use the so-called cluster mean-field approach to develop higher order approximations that systematically increase the accuracy of kinetic models by treating spatial correlations at a progressively higher level of detail. We further demonstrate our approach on a reduced model for NO oxidation incorporating first nearest-neighbor lateral interactions and construct a sequence of approximations of increasingly higher accuracy, which we compare with KMC and mean-field. The latter is found to perform rather poorly, overestimating the turnover frequency by several orders of magnitude for this system. On the other hand, our approximations, while more computationally intense than the traditional mean-field treatment, still achieve tremendous computational savings compared to KMC simulations, thereby opening the way for employing them in multiscale modeling frameworks.

Approximate analytical solutions of Klein-Gordon equation with Hulthen potentials for nonzero angular momentum

International Nuclear Information System (INIS)

Chen Changyuan; Sun Dongsheng; Lu Falin

2007-01-01

Using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Klein-Gordon equation with the vector and scalar Hulthen potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of bound states are attained for different l. The analytical energy equation and the unnormalized radial wave functions expressed in terms of hypergeometric polynomials are given

New finite volume methods for approximating partial differential equations on arbitrary meshes

International Nuclear Information System (INIS)

Hermeline, F.

2008-12-01

This dissertation presents some new methods of finite volume type for approximating partial differential equations on arbitrary meshes. The main idea lies in solving twice the problem to be dealt with. One addresses the elliptic equations with variable (anisotropic, antisymmetric, discontinuous) coefficients, the parabolic linear or non linear equations (heat equation, radiative diffusion, magnetic diffusion with Hall effect), the wave type equations (Maxwell, acoustics), the elasticity and Stokes'equations. Numerous numerical experiments show the good behaviour of this type of method. (author)

Stability of generalized Runge-Kutta methods for stiff kinetics coupled differential equations

International Nuclear Information System (INIS)

Aboanber, A E

2006-01-01

A stability and efficiency improved class of generalized Runge-Kutta methods of order 4 are developed for the numerical solution of stiff system kinetics equations for linear and/or nonlinear coupled differential equations. The determination of the coefficients required by the method is precisely obtained from the so-called equations of condition which in turn are derived by an approach based on Butcher series. Since the equations of condition are fewer in number, free parameters can be chosen for optimizing any desired feature of the process. A further related coefficient set with different values of these parameters and the region of absolute stability of the method have been introduced. In addition, the A(α) stability properties of the method are investigated. Implementing the method in a personal computer estimated the accuracy and speed of calculations and verified the good performances of the proposed new schemes for several sample problems of the stiff system point kinetics equations with reactivity feedback

On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind

Directory of Open Access Journals (Sweden)

Dinesh Kumar

2015-01-01

Full Text Available We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably new results.

Laser driven electron-positron pair creation-kinetic theory versus analytical approximations

International Nuclear Information System (INIS)

Smolyansky, S.A.; Prozorkevich, A.V.; Bonitz, M.

2013-01-01

The dynamical Schwinger effect of vacuum pair creation driven by an intense external laser pulse is studied on the basis of quantum kinetic theory. The numerical solutions of these kinetic equations exhibit a complex time dependence which makes an analysis of the physical processes difficult. In particular, the question of secondary effects, such as creation of secondary annihilation photons from the focus spot of the colliding laser beams, remains an important open problem. In the present work we, therefore, develop a perturbation theory which is able to capture the dominant time dependence of the produced electron-positron pair density. The theory shows excellent agreement with the exact kinetic results during the laser pulse, but fails to reproduce the residual pair density remaining in the system after termination of the pulse. (copyright 2013 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) (orig.)

ON ASYMTOTIC APPROXIMATIONS OF FIRST INTEGRALS FOR DIFFERENTIAL AND DIFFERENCE EQUATIONS

Directory of Open Access Journals (Sweden)

W.T. van Horssen

2007-04-01

Full Text Available In this paper the concept of integrating factors for differential equations and the concept of invariance factors for difference equations to obtain first integrals or invariants will be presented. It will be shown that all integrating factors have to satisfya system of partial differential equations, and that all invariance factors have to satisfy a functional equation. In the period 1997-2001 a perturbation method based on integrating vectors was developed to approximate first integrals for systems of ordinary differential equations. This perturbation method will be reviewed shortly. Also in the paper the first results in the development of a perturbation method for difference equations based on invariance factors will be presented.

Finite-dimensional approximation for operator equations of Hammerstein type

International Nuclear Information System (INIS)

Buong, N.

1992-11-01

The purpose of this paper is to establish convergence rate for a method of finite-dimensional approximation to solve operator equation of Hammerstein type in real reflexive Banach space. In order to consider a numerical example an iteration method is proposed in Hilbert space. (author). 25 refs

Simulating Chemical Kinetics Without Differential Equations: A Quantitative Theory Based on Chemical Pathways.

Science.gov (United States)

Bai, Shirong; Skodje, Rex T

2017-08-17

A new approach is presented for simulating the time-evolution of chemically reactive systems. This method provides an alternative to conventional modeling of mass-action kinetics that involves solving differential equations for the species concentrations. The method presented here avoids the need to solve the rate equations by switching to a representation based on chemical pathways. In the Sum Over Histories Representation (or SOHR) method, any time-dependent kinetic observable, such as concentration, is written as a linear combination of probabilities for chemical pathways leading to a desired outcome. In this work, an iterative method is introduced that allows the time-dependent pathway probabilities to be generated from a knowledge of the elementary rate coefficients, thus avoiding the pitfalls involved in solving the differential equations of kinetics. The method is successfully applied to the model Lotka-Volterra system and to a realistic H 2 combustion model.

Relaxation and kinetics in scalar field theories

International Nuclear Information System (INIS)

Boyanovsky, D.; Lawrie, I.D.; Lee, D.

1996-01-01

A new approach to the dynamics of relaxation and kinetics of thermalization in a scalar field theory is presented that incorporates the relevant time scales through the resummation of hard thermal loops. An alternative derivation of the kinetic equations for the open-quote open-quote quasiparticle close-quote close-quote distribution functions is obtained that allows a clear understanding of the different open-quote open-quote coarse-graining close-quote close-quote approximations usually involved in a kinetic description. This method leads to a systematic perturbative expansion to obtain the kinetic equations including hard thermal loop resummation and to an improvement including renormalization, off-shell effects, and contributions that change chemical equilibrium on short time scales. As a by-product of these methods we establish the equivalence between the relaxation time scale in the linearized equation of motion of the quasiparticles and the thermalization time scale of the quasiparticle distribution function in the open-quote open-quote relaxation time approximation close-quote close-quote including hard thermal loop effects. Hard thermal loop resummation dramatically modifies the scattering rate for long wavelength modes as compared to the usual (semi)classical estimate. Relaxation and kinetics are studied both in the unbroken and broken symmetry phases of the theory. The broken symmetry phase also provides the setting to obtain the contribution to the kinetic equations from processes that involve decay of a heavy scalar into light scalar particles in the medium. copyright 1996 The American Physical Society

Non-markovian boltzmann equation

International Nuclear Information System (INIS)

Kremp, D.; Bonitz, M.; Kraeft, W.D.; Schlanges, M.

1997-01-01

A quantum kinetic equation for strongly interacting particles (generalized binary collision approximation, ladder or T-matrix approximation) is derived in the framework of the density operator technique. In contrast to conventional kinetic theory, which is valid on large time scales as compared to the collision (correlation) time only, our approach retains the full time dependencies, especially also on short time scales. This means retardation and memory effects resulting from the dynamics of binary correlations and initial correlations are included. Furthermore, the resulting kinetic equation conserves total energy (the sum of kinetic and potential energy). The second aspect of generalization is the inclusion of many-body effects, such as self-energy, i.e., renormalization of single-particle energies and damping. To this end we introduce an improved closure relation to the Bogolyubov endash Born endash Green endash Kirkwood endash Yvon hierarchy. Furthermore, in order to express the collision integrals in terms of familiar scattering quantities (Mo/ller operator, T-matrix), we generalize the methods of quantum scattering theory by the inclusion of medium effects. To illustrate the effects of memory and damping, the results of numerical simulations are presented. copyright 1997 Academic Press, Inc

Analytical solution of point kinetic equations for sub-critical systems

International Nuclear Information System (INIS)

Henrice Junior, Edson; Goncalves, Alessandro C.

2013-01-01

This article presents an analytical solution for the set of point kinetic equations for sub-critical reactors. This solution stems from the ordinary, non-homogeneous differential equation that rules the neutron density and that presents the incomplete Gamma function in its functional form. The method used proved advantageous and allowed practical applications such as the linear insertion of reactivity, considering an external constant source or with both varying linearly. (author)

Comparison of approximate gravitational lens equations and a proposal for an improved new one

International Nuclear Information System (INIS)

Bozza, V.

2008-01-01

Keeping the exact general relativistic treatment of light bending as a reference, we compare the accuracy of commonly used approximate lens equations. We conclude that the best approximate lens equation is the Ohanian lens equation, for which we present a new expression in terms of distances between observer, lens, and source planes. We also examine a realistic gravitational lensing case, showing that the precision of the Ohanian lens equation might be required for a reliable treatment of gravitational lensing and a correct extraction of the full information about gravitational physics.

Diffusion approximations to the chemical master equation only have a consistent stochastic thermodynamics at chemical equilibrium.

Science.gov (United States)

Horowitz, Jordan M

2015-07-28

The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.

Kinetic equation for spin-polarized plasmas

International Nuclear Information System (INIS)

Cowley, S.C.; Kulsrud, R.M.; Valeo, E.

1984-07-01

The usual kinetic description of a plasma is extended to include variables to describe the spin. The distribution function, over phase-space and the new spin variables, provides a sufficient description of a spin-polarized plasma. The evolution equation for the distribution function is given. The equations derived are used to calculate depolarization due to four processes, inhomogeneous fields, collisions, collisions in inhomogeneous fields, and waves. It is found that depolarization by field inhomogeneity on scales large compared with the gyroradius is totally negligible. The same is true for collisional depolarization. Collisions in inhomogeneous fields yield a depolarization rate of order 10 -4 S -1 for deuterons and a negligible rate for tritons in a typical fusion reactor design. This is still sufficiently small on reactor time scales. However, small amplitude magnetic fluctuations (of order one gauss) resonant with the spin precession frequency can lead to significant depolarization (depolarises triton in ten seconds and deuteron in a hundred seconds.)

Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach

KAUST Repository

Collier, Nathan; Radwan, Hany; Dalcin, Lisandro; Calo, Victor M.

2011-01-01

We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity

Beyond the Cahn-Hilliard equation: a vacancy-based kinetic theory

International Nuclear Information System (INIS)

Nastar, M.

2011-01-01

A Self-Consistent Mean Field (SCMF) kinetic theory including an explicit description of the vacancy diffusion mechanism is developed. The present theory goes beyond the usual local equilibrium hypothesis. It is applied to the study of the early time spinodal decomposition in alloys. The resulting analytical expression of the structure function highlights the contribution of the vacancy diffusion mechanism. Instead of the single amplification rate of the Cahn-Hillard linear theory, the linearized SCMF kinetic equations involve three constant rates, first one describing the vacancy relaxation kinetics, second one related to the kinetic coupling between local concentrations and pair correlations and the third one representing the spinodal amplification rate. Starting from the same vacancy diffusion model, we perform kinetic Monte Carlo simulations of a Body Centered Cubic (BCC) demixting alloy. The resulting spherically averaged structure function is compared to the SCMF predictions. Both qualitative and quantitative agreements are satisfying. (authors)

Efficient solution of parabolic equations by Krylov approximation methods

Science.gov (United States)

Gallopoulos, E.; Saad, Y.

1990-01-01

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.

Analysis of atmospheric flow over a surface protrusion using the turbulence kinetic energy equation with reference to aeronautical operating systems

Science.gov (United States)

Frost, W.; Harper, W. L.

1975-01-01

Flow over surface obstructions can produce significantly large wind shears such that adverse flying conditions can occur for aeronautical systems (helicopters, STOL vehicles, etc.). Atmospheric flow fields resulting from a semi-elliptical surface obstruction in an otherwise horizontally homogeneous statistically stationary flow are modelled with the boundary-layer/Boussinesq-approximation of the governing equation of fluid mechanics. The turbulence kinetic energy equation is used to determine the dissipative effects of turbulent shear on the mean flow. Iso-lines of turbulence kinetic energy and turbulence intensity are plotted in the plane of the flow and highlight regions of high turbulence intensity in the stagnation zone and sharp gradients in intensity along the transition from adverse to favourable pressure gradient. Discussion of the effects of the disturbed wind field in CTOL and STOL aircraft flight path and obstruction clearance standards is given. The results indicate that closer inspection of these presently recommended standards as influenced by wind over irregular terrains is required.

Numerical instability of time-discretized one-point kinetic equations

International Nuclear Information System (INIS)

Hashimoto, Kengo; Ikeda, Hideaki; Takeda, Toshikazu

2000-01-01

The one-point kinetic equations with numerical errors induced by the explicit, implicit and Crank-Nicolson integration methods are derived. The zero-power transfer functions based on the present equations are demonstrated to investigate the numerical stability of the discretized systems. These demonstrations indicate unconditional stability for the implicit and Crank-Nicolson methods but present the possibility of numerical instability for the explicit method. An upper limit of time mesh spacing for the stability is formulated and several numerical calculations are made to confirm the validity of this formula

Different seeds to solve the equations of stochastic point kinetics using the Euler-Maruyama method; Diferentes semillas para solucionar las ecuaciones de la cinetica puntual estocastica empleando el metodo de Euler-Maruyama

Energy Technology Data Exchange (ETDEWEB)

Suescun D, D.; Oviedo T, M., E-mail: daniel.suescun@usco.edu.co [Universidad Surcolombiana, Av. Pastrana Borrero - Carrera 1, Neiva, Huila (Colombia)

2017-09-15

In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and

Calculation of statistic estimates of kinetic parameters from substrate uncompetitive inhibition equation using the median method

Directory of Open Access Journals (Sweden)

Pedro L. Valencia

2017-04-01

Full Text Available We provide initial rate data from enzymatic reaction experiments and tis processing to estimate the kinetic parameters from the substrate uncompetitive inhibition equation using the median method published by Eisenthal and Cornish-Bowden (Cornish-Bowden and Eisenthal, 1974; Eisenthal and Cornish-Bowden, 1974. The method was denominated the direct linear plot and consists in the calculation of the median from a dataset of kinetic parameters Vmax and Km from the Michaelis–Menten equation. In this opportunity we present the procedure to applicate the direct linear plot to the substrate uncompetitive inhibition equation; a three-parameter equation. The median method is characterized for its robustness and its insensibility to outlier. The calculations are presented in an Excel datasheet and a computational algorithm was developed in the free software Python. The kinetic parameters of the substrate uncompetitive inhibition equation Vmax, Km and Ks were calculated using three experimental points from the dataset formed by 13 experimental points. All the 286 combinations were calculated. The dataset of kinetic parameters resulting from this combinatorial was used to calculate the median which corresponds to the statistic estimator of the real kinetic parameters. A comparative statistical analyses between the median method and the least squares was published in Valencia et al. [3].

Error Estimates for Approximate Solutions of the Riccati Equation with Real or Complex Potentials

Science.gov (United States)

Finster, Felix; Smoller, Joel

2010-09-01

A method is presented for obtaining rigorous error estimates for approximate solutions of the Riccati equation, with real or complex potentials. Our main tool is to derive invariant region estimates for complex solutions of the Riccati equation. We explain the general strategy for applying these estimates and illustrate the method in typical examples, where the approximate solutions are obtained by gluing together WKB and Airy solutions of corresponding one-dimensional Schrödinger equations. Our method is motivated by, and has applications to, the analysis of linear wave equations in the geometry of a rotating black hole.

The Pathwise Numerical Approximation of Stationary Solutions of Semilinear Stochastic Evolution Equations

International Nuclear Information System (INIS)

Caraballo, T.; Kloeden, P.E.

2006-01-01

Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions

Approximate solution to the Kolmogorov equation for a fission chain-reacting system

International Nuclear Information System (INIS)

Ruby, L.; McSwine, T.L.

1986-01-01

An approximate solution has been obtained for the Kolmogorov equation describing a fission chain-reacting system. The method considers the population of neutrons, delayed-neutron precursors, and detector counts. The effect of the detector is separated from the statistics of the chain reaction by a weak coupling assumption that predicts that the detector responds to the average rather than to the instantaneous neutron population. An approximate solution to the remaining equation, involving the populations of neutrons and precursors, predicts a negative-binomial behaviour for the neutron probability distribution

Numerical solution of the point reactor kinetics equations with fuel burn-up and temperature feedback

International Nuclear Information System (INIS)

Tashakor, S.; Jahanfarnia, G.; Hashemi-Tilehnoee, M.

2010-01-01

Point reactor kinetics equations are solved numerically using one group of delayed neutrons and with fuel burn-up and temperature feedback included. To calculate the fraction of one-group delayed neutrons, a group of differential equations are solved by an implicit time method. Using point reactor kinetics equations, changes in mean neutrons density, temperature, and reactivity are calculated in different times during the reactor operation. The variation of reactivity, temperature, and maximum power with time are compared with the predictions by other methods.

A direct method for numerical solution of a class of nonlinear Volterra integro-differential equations and its application to the nonlinear fission and fusion reactor kinetics

International Nuclear Information System (INIS)

Nakahara, Yasuaki; Ise, Takeharu; Kobayashi, Kensuke; Itoh, Yasuyuki

1975-12-01

A new method has been developed for numerical solution of a class of nonlinear Volterra integro-differential equations with quadratic nonlinearity. After dividing the domain of the variable into subintervals, piecewise approximations are applied in the subintervals. The equation is first integrated over a subinterval to obtain the piecewise equation, to which six approximate treatments are applied, i.e. fully explicit, fully implicit, Crank-Nicolson, linear interpolation, quadratic and cubic spline. The numerical solution at each time step is obtained directly as a positive root of the resulting algebraic quadratic equation. The point reactor kinetics with a ramp reactivity insertion, linear temperature feedback and delayed neutrons can be described by one of this type of nonlinear Volterra integro-differential equations. The algorithm is applied to the Argonne benchmark problem and a model problem for a fast reactor without delayed neutrons. The fully implicit method has been found to be unconditionally stable in the sense that it always gives the positive real roots. The cubic spline method is divergent, and the other four methods are intermediate in between. From the estimation of the stability, convergency, accuracy and CPU time, it is concluded that the Crank-Nicolson method is best, then the linear interpolation method comes closely next to it. Discussions are also made on the possibility of applying the algorithm to the fusion reactor kinetics in the form of a nonlinear partial differential equation. (auth.)

On the use of the Lie group technique for differential equations with a small parameter: Approximate solutions and integrable equations

International Nuclear Information System (INIS)

Burde, G.I.

2002-01-01

A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the approach allows constructing integrable equations that have solutions with (partially) prescribed features. Examples of application of the approach to partial differential equations are given

Solution of the neutron point kinetics equations with temperature feedback effects applying the polynomial approach method

International Nuclear Information System (INIS)

Tumelero, Fernanda; Petersen, Claudio Z.; Goncalves, Glenio A.; Lazzari, Luana

2015-01-01

In this work, we present a solution of the Neutron Point Kinetics Equations with temperature feedback effects applying the Polynomial Approach Method. For the solution, we consider one and six groups of delayed neutrons precursors with temperature feedback effects and constant reactivity. The main idea is to expand the neutron density, delayed neutron precursors and temperature as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions of the problem and the analytical continuation is used to determine the solutions of the next intervals. With the application of the Polynomial Approximation Method it is possible to overcome the stiffness problem of the equations. In such a way, one varies the time step size of the Polynomial Approach Method and performs an analysis about the precision and computational time. Moreover, we compare the method with different types of approaches (linear, quadratic and cubic) of the power series. The answer of neutron density and temperature obtained by numerical simulations with linear approximation are compared with results in the literature. (author)

Solution of the neutron point kinetics equations with temperature feedback effects applying the polynomial approach method

Energy Technology Data Exchange (ETDEWEB)

Tumelero, Fernanda, E-mail: fernanda.tumelero@yahoo.com.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Petersen, Claudio Z.; Goncalves, Glenio A.; Lazzari, Luana, E-mail: claudiopeteren@yahoo.com.br, E-mail: gleniogoncalves@yahoo.com.br, E-mail: luana-lazzari@hotmail.com [Universidade Federal de Pelotas (DME/UFPEL), Capao do Leao, RS (Brazil). Instituto de Fisica e Matematica

2015-07-01

In this work, we present a solution of the Neutron Point Kinetics Equations with temperature feedback effects applying the Polynomial Approach Method. For the solution, we consider one and six groups of delayed neutrons precursors with temperature feedback effects and constant reactivity. The main idea is to expand the neutron density, delayed neutron precursors and temperature as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions of the problem and the analytical continuation is used to determine the solutions of the next intervals. With the application of the Polynomial Approximation Method it is possible to overcome the stiffness problem of the equations. In such a way, one varies the time step size of the Polynomial Approach Method and performs an analysis about the precision and computational time. Moreover, we compare the method with different types of approaches (linear, quadratic and cubic) of the power series. The answer of neutron density and temperature obtained by numerical simulations with linear approximation are compared with results in the literature. (author)

Approximations to the Probability of Failure in Random Vibration by Integral Equation Methods

DEFF Research Database (Denmark)

Nielsen, Søren R.K.; Sørensen, John Dalsgaard

Close approximations to the first passage probability of failure in random vibration can be obtained by integral equation methods. A simple relation exists between the first passage probability density function and the distribution function for the time interval spent below a barrier before...... passage probability density. The results of the theory agree well with simulation results for narrow banded processes dominated by a single frequency, as well as for bimodal processes with 2 dominating frequencies in the structural response....... outcrossing. An integral equation for the probability density function of the time interval is formulated, and adequate approximations for the kernel are suggested. The kernel approximation results in approximate solutions for the probability density function of the time interval, and hence for the first...

Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations

KAUST Repository

Giraldi, Loic; Nouy, Anthony

2017-01-01

This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.

Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations

KAUST Repository

Giraldi, Loic

2017-06-30

This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.

Parametrized post-Newtonian approximation and Rastall's gravitational field equations

International Nuclear Information System (INIS)

Smalley, L.L.

1978-01-01

The parametrized post-Newtonian (PPN) approximation is generalized to accomodate Rastall's modification of Einstein's theory of gravity, which allows nonzero divergence of the energy-momentum tensor. Rastall's theory is then shown to have consistent field equations, gauge conditions, and the correct Newtonian limit of the equations of motion. The PPN parameters are obtained and shown to agree experimentally with those for the Einstein theory. In light of the nonzero divergence condition, integral conservation laws are investigated and shown to yield conserved energy-momentum and angular-momentum. We conclude that the above generalization of metric theories, within the PPN framework, is a natural extension of the concept of metric theories

Kinetic theory of flocking: derivation of hydrodynamic equations.

Science.gov (United States)

Ihle, Thomas

2011-03-01

It is shown how to explicitly coarse-grain the microscopic dynamics of the rule-based Vicsek model for self-propelled agents. The hydrodynamic equations are derived by means of an Enskog-type kinetic theory. Expressions for all transport coefficients are given. The transition from a disordered to a flocking state, which at large particle speeds appears to be a fluctuation-induced first-order phase transition, is studied numerically and analytically.

Comparative analysis of solution methods of the punctual kinetic equations

International Nuclear Information System (INIS)

Hernandez S, A.

2003-01-01

The following one written it presents a comparative analysis among different analytical solutions for the punctual kinetics equation, which present two variables of interest: a) the temporary behavior of the neutronic population, and b) The temporary behavior of the different groups of precursors of delayed neutrons. The first solution is based on a method that solves the transfer function of the differential equation for the neutronic population, in which intends to obtain the different poles that give the stability of this transfer function. In this section it is demonstrated that the temporary variation of the reactivity of the system can be managed as it is required, since the integration time for this method doesn't affect the result. However, the second solution is based on an iterative method like that of Runge-Kutta or the Euler method where the algorithm was only used to solve first order differential equations giving this way solution to each differential equation that conforms the equations of punctual kinetics. In this section it is demonstrated that only it can obtain a correct temporary behavior of the neutronic population when it is integrated on an interval of very short time, forcing to the temporary variation of the reactivity to change very quick way without one has some control about the time. In both methods the same change is used so much in the reactivity of the system like in the integration times, giving validity to the results graph the one the temporary behavior of the neutronic population vs. time. (Author)

Kinetics of subdiffusion-assisted reactions: non-Markovian stochastic Liouville equation approach

International Nuclear Information System (INIS)

Shushin, A I

2005-01-01

Anomalous specific features of the kinetics of subdiffusion-assisted bimolecular reactions (time-dependence, dependence on parameters of systems, etc) are analysed in detail with the use of the non-Markovian stochastic Liouville equation (SLE), which has been recently derived within the continuous-time random-walk (CTRW) approach. In the CTRW approach, subdiffusive motion of particles is modelled by jumps whose onset probability distribution function is of a long-tailed form. The non-Markovian SLE allows for rigorous describing of some peculiarities of these reactions; for example, very slow long-time behaviour of the kinetics, non-analytical dependence of the reaction rate on the reactivity of particles, strong manifestation of fluctuation kinetics showing itself in very slowly decreasing behaviour of the kinetics at very long times, etc

Differential equation methods for simulation of GFP kinetics in non-steady state experiments.

Science.gov (United States)

Phair, Robert D

2018-03-15

Genetically encoded fluorescent proteins, combined with fluorescence microscopy, are widely used in cell biology to collect kinetic data on intracellular trafficking. Methods for extraction of quantitative information from these data are based on the mathematics of diffusion and tracer kinetics. Current methods, although useful and powerful, depend on the assumption that the cellular system being studied is in a steady state, that is, the assumption that all the molecular concentrations and fluxes are constant for the duration of the experiment. Here, we derive new tracer kinetic analytical methods for non-steady state biological systems by constructing mechanistic nonlinear differential equation models of the underlying cell biological processes and linking them to a separate set of differential equations governing the kinetics of the fluorescent tracer. Linking the two sets of equations is based on a new application of the fundamental tracer principle of indistinguishability and, unlike current methods, supports correct dependence of tracer kinetics on cellular dynamics. This approach thus provides a general mathematical framework for applications of GFP fluorescence microscopy (including photobleaching [FRAP, FLIP] and photoactivation to frequently encountered experimental protocols involving physiological or pharmacological perturbations (e.g., growth factors, neurotransmitters, acute knockouts, inhibitors, hormones, cytokines, and metabolites) that initiate mechanistically informative intracellular transients. When a new steady state is achieved, these methods automatically reduce to classical steady state tracer kinetic analysis. © 2018 Phair. This article is distributed by The American Society for Cell Biology under license from the author(s). Two months after publication it is available to the public under an Attribution–Noncommercial–Share Alike 3.0 Unported Creative Commons License (http://creativecommons.org/licenses/by-nc-sa/3.0).

Modified Gompertz equation for electrotherapy murine tumor growth kinetics: predictions and new hypotheses

International Nuclear Information System (INIS)

Cabrales, Luis E Bergues; Mateus, Miguel A O'Farril; Brooks, Soraida C Acosta; Palencia, Fabiola Suárez; Zamora, Lisset Ortiz; Quevedo, María C Céspedes; Seringe, Sarah Edward; Cuitié, Vladimir Crombet; Cabrales, Idelisa Bergues; González, Gustavo Sierra; Nava, Juan J Godina; Aguilera, Andrés Ramírez; Joa, Javier A González; Ciria, Héctor M Camué; González, Maraelys Morales; Salas, Miriam Fariñas; Jarque, Manuel Verdecia; González, Tamara Rubio

2010-01-01

Electrotherapy effectiveness at different doses has been demonstrated in preclinical and clinical studies; however, several aspects that occur in the tumor growth kinetics before and after treatment have not yet been revealed. Mathematical modeling is a useful instrument that can reveal some of these aspects. The aim of this paper is to describe the complete growth kinetics of unperturbed and perturbed tumors through use of the modified Gompertz equation in order to generate useful insight into the mechanisms that underpin this devastating disease. The complete tumor growth kinetics for control and treated groups are obtained by interpolation and extrapolation methods with different time steps, using experimental data of fibrosarcoma Sa-37. In the modified Gompertz equation, a delay time is introduced to describe the tumor's natural history before treatment. Different graphical strategies are used in order to reveal new information in the complete kinetics of this tumor type. The first stage of complete tumor growth kinetics is highly non linear. The model, at this stage, shows different aspects that agree with those reported theoretically and experimentally. Tumor reversibility and the proportionality between regions before and after electrotherapy are demonstrated. In tumors that reach partial remission, two antagonistic post-treatment processes are induced, whereas in complete remission, two unknown antitumor mechanisms are induced. The modified Gompertz equation is likely to lead to insights within cancer research. Such insights hold promise for increasing our understanding of tumors as self-organizing systems and, the possible existence of phase transitions in tumor growth kinetics, which, in turn, may have significant impacts both on cancer research and on clinical practice

Determination of kinetic coefficients for proton-nucleus collisions at high energy

International Nuclear Information System (INIS)

Rizzato, C.M.

1987-01-01

From the effective proton dynamics, the approximations in the context of high energy collisions which lead to the Boltzmann equation, are established. From this equation, general expressions for the kinetic coefficients are deduced. Using a simple model, analytical expressions for kinetic coefficients are obtained. The importance of the effect of Pauli blocking is also shown. (author) [pt

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

Science.gov (United States)

Bronstein, Leo; Koeppl, Heinz

2018-01-01

Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.

Computational modeling of fully-ionized, magnetized plasmas using the fluid approximation

Science.gov (United States)

Schnack, Dalton

2005-10-01

Strongly magnetized plasmas are rich in spatial and temporal scales, making a computational approach useful for studying these systems. The most accurate model of a magnetized plasma is based on a kinetic equation that describes the evolution of the distribution function for each species in six-dimensional phase space. However, the high dimensionality renders this approach impractical for computations for long time scales in relevant geometry. Fluid models, derived by taking velocity moments of the kinetic equation [1] and truncating (closing) the hierarchy at some level, are an approximation to the kinetic model. The reduced dimensionality allows a wider range of spatial and/or temporal scales to be explored. Several approximations have been used [2-5]. Successful computational modeling requires understanding the ordering and closure approximations, the fundamental waves supported by the equations, and the numerical properties of the discretization scheme. We review and discuss several ordering schemes, their normal modes, and several algorithms that can be applied to obtain a numerical solution. The implementation of kinetic parallel closures is also discussed [6].[1] S. Chapman and T.G. Cowling, ``The Mathematical Theory of Non-Uniform Gases'', Cambridge University Press, Cambridge, UK (1939).[2] R.D. Hazeltine and J.D. Meiss, ``Plasma Confinement'', Addison-Wesley Publishing Company, Redwood City, CA (1992).[3] L.E. Sugiyama and W. Park, Physics of Plasmas 7, 4644 (2000).[4] J.J. Ramos, Physics of Plasmas, 10, 3601 (2003).[5] P.J. Catto and A.N. Simakov, Physics of Plasmas, 11, 90 (2004).[6] E.D. Held et al., Phys. Plasmas 11, 2419 (2004)

Critique of the Brownian approximation to the generalized Langevin equation in lattice dynamics

International Nuclear Information System (INIS)

Diestler, D.J.; Riley, M.E.

1985-01-01

We consider the classical motion of a harmonic lattice in which only those atoms in a certain subset of the lattice (primary zone) may interact with an external force. The formally exact generalized Langevin equation (GLE) for the primary zone is an appropriate description of the dynamics. We examine a previously proposed Brownian, or frictional damping, approximation that reduces the GLE to a set of coupled ordinary Langevin equations for the primary atoms. It is shown that the solution of these equations can contain undamped motion if there is more than one atom in the primary zone. Such motion is explicitly demonstrated for a model that has been used to describe energy transfer in atom--surface collisions. The inability of the standard Brownian approximation to yield an acceptable, physically meaningful result for primary zones comprising more than one atom suggests that the Brownian approximation may introduce other spurious dynamical effects. Further work on damping of correlated motion in lattices is needed

Non-equilibrium reaction rates in chemical kinetic equations

Science.gov (United States)

Gorbachev, Yuriy

2018-05-01

Within the recently proposed asymptotic method for solving the Boltzmann equation for chemically reacting gas mixture, the chemical kinetic equations has been derived. Corresponding one-temperature non-equilibrium reaction rates are expressed in terms of specific heat capacities of the species participate in the chemical reactions, bracket integrals connected with the internal energy transfer in inelastic non-reactive collisions and energy transfer coefficients. Reactions of dissociation/recombination of homonuclear and heteronuclear diatomic molecules are considered. It is shown that all reaction rates are the complex functions of the species densities, similarly to the unimolecular reaction rates. For determining the rate coefficients it is recommended to tabulate corresponding bracket integrals, additionally to the equilibrium rate constants. Correlation of the obtained results with the irreversible thermodynamics is established.

Homotopy analysis solutions of point kinetics equations with one delayed precursor group

International Nuclear Information System (INIS)

Zhu Qian; Luo Lei; Chen Zhiyun; Li Haofeng

2010-01-01

Homotopy analysis method is proposed to obtain series solutions of nonlinear differential equations. Homotopy analysis method was applied for the point kinetics equations with one delayed precursor group. Analytic solutions were obtained using homotopy analysis method, and the algorithm was analysed. The results show that the algorithm computation time and precision agree with the engineering requirements. (authors)

Numerical nodal simulation of the axial power distribution within nuclear reactors using a kinetics diffusion model. I

International Nuclear Information System (INIS)

Barros, R.C. de.

1992-05-01

Presented here is a new numerical nodal method for the simulation of the axial power distribution within nuclear reactors using the one-dimensional one speed kinetics diffusion model with one group of delayed neutron precursors. Our method is based on a spectral analysis of the nodal kinetics equations. These equations are obtained by integrating the original kinetics equations separately over a time step and over a spatial node, and then considering flat approximations for the forward difference terms. These flat approximations are the only approximations that are considered in the method. As a result, the spectral nodal method for space - time reactor kinetics generates numerical solutions for space independent problems or for time independent problems that are completely free from truncation errors. We show numerical results to illustrate the method's accuracy for coarse mesh calculations. (author)

Gluon transport equation in the small angle approximation and the onset of Bose–Einstein condensation

Energy Technology Data Exchange (ETDEWEB)

Blaizot, Jean-Paul [Institut de Physique Théorique, CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette (France); Liao, Jinfeng [Physics Dept. and CEEM, Indiana University, 2401 N Milo B. Sampson Lane, Bloomington, IN 47408 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); McLerran, Larry [Physics Dept., Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); RIKEN BNL Research Center, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973 (United States); Physics Department, China Central Normal University, Wuhan (China)

2014-11-15

To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study.

Gluon transport equation in the small angle approximation and the onset of Bose–Einstein condensation

International Nuclear Information System (INIS)

Blaizot, Jean-Paul; Liao, Jinfeng; McLerran, Larry

2014-01-01

To understand the evolution of a dense system of gluons, such as those produced in the early stages of ultra-relativistic heavy ion collisions, is an important and challenging problem. We describe the approach to thermal equilibrium using the small angle approximation for gluon scattering in a Boltzmann equation that includes the effects of Bose statistics. The role of Bose statistical factors in amplifying the rapid growth of the population of the soft modes is essential. With these factors properly taken into account, one finds that elastic scattering alone provides an efficient mechanism for populating soft modes, and in fact leads to rapid infrared local thermalization. Furthermore, recent developments suggest that high initial overpopulation plays a key role and may lead to dynamical Bose–Einstein condensation. The kinetics of condensation is an interesting problem in itself. By solving the transport equation for initial conditions with a large enough initial phase-space density the equilibrium state contains a Bose condensate, and we present numerical evidence that such over-occupied systems reach the onset of Bose–Einstein condensation in a finite time. It is also found that the approach to condensation is characterized by a scaling behavior. Finally we discuss a number of extensions of the present study

Nonlinear ordinary differential equations analytical approximation and numerical methods

CERN Document Server

Hermann, Martin

2016-01-01

The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...

Gaussian approximations for stochastic systems with delay: Chemical Langevin equation and application to a Brusselator system

International Nuclear Information System (INIS)

Brett, Tobias; Galla, Tobias

2014-01-01

We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period

Gaussian approximations for stochastic systems with delay: chemical Langevin equation and application to a Brusselator system.

Science.gov (United States)

Brett, Tobias; Galla, Tobias

2014-03-28

We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.

An alternative technique for the implementation of an analytical approximation for transients with temperature feedback

Energy Technology Data Exchange (ETDEWEB)

Palma, Daniel A.P. [Instituto Federal do Rio de Janeiro, Nilopolis, RJ (Brazil)], e-mail: dpalmaster@gmail.com; Silva, Adilson C. da; Goncalves, Alessandro C.; Martinez, Aquilino S. [Coordenacao dos Programas de Pos-graduacao de Engenharia (COPPE/UFRJ), Rio de Janeiro, RJ (Brazil). Programa de Engenharia Nuclear], e-mail: asilva@con.ufrj.br, e-mail: agoncalves@con.ufrj.br, e-mail: aquilino@lmp.ufrj.br

2009-07-01

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting neutron density variation during the operation of a nuclear reactor. Although different approximate solutions for the system of point kinetics equations with temperature feedback may be found in literature, some of them do not present an explicit dependence in time, which makes the computing implementation difficult and, as a result, its applicability in practical cases. The present paper uses the polynomial adjustment technique to overcome this problem in the analytical approximation as proposed by Nahla. In a systematic comparison with other existing approximations it is concluded that the method is adequate, presenting small deviations in relation to the reference values obtained from the reference numerical method. (author)

An alternative technique for the implementation of an analytical approximation for transients with temperature feedback

International Nuclear Information System (INIS)

Palma, Daniel A.P.; Silva, Adilson C. da; Goncalves, Alessandro C.; Martinez, Aquilino S.

2009-01-01

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting neutron density variation during the operation of a nuclear reactor. Although different approximate solutions for the system of point kinetics equations with temperature feedback may be found in literature, some of them do not present an explicit dependence in time, which makes the computing implementation difficult and, as a result, its applicability in practical cases. The present paper uses the polynomial adjustment technique to overcome this problem in the analytical approximation as proposed by Nahla. In a systematic comparison with other existing approximations it is concluded that the method is adequate, presenting small deviations in relation to the reference values obtained from the reference numerical method. (author)

Local density approximation for a perturbative equation of state

International Nuclear Information System (INIS)

Astrakharchik, G. E.

2005-01-01

Knowledge of a series expansion of the equation of state provides a deep insight into the physical nature of a quantum system. Starting from a generic 'perturbative' equation of state of a homogeneous ultracold gas we make predictions for the properties of the gas in the presence of harmonic confinement. The local density approximation is used to obtain the chemical potential, total and release energies, Thomas-Fermi size, and density profile of a trapped system in three-, two-, and one-dimensional geometries. The frequencies of the lowest breathing modes are calculated using scaling and sum-rule approaches and could be used in an experiment as a high-precision tool for obtaining the expansion terms of the equation of state. The derived formalism is applied to dilute Bose and Fermi gases in different dimensions and to integrable one-dimensional models. The physical meaning of the expansion terms in a number of systems is discussed

An approximation method for nonlinear integral equations of Hammerstein type

International Nuclear Information System (INIS)

Chidume, C.E.; Moore, C.

1989-05-01

The solution of a nonlinear integral equation of Hammerstein type in Hilbert spaces is approximated by means of a fixed point iteration method. Explicit error estimates are given and, in some cases, convergence is shown to be at least as fast as a geometric progression. (author). 25 refs

Equations for the kinetic modeling of supersonically flowing electrically excited lasers

International Nuclear Information System (INIS)

Lind, R.C.

1973-01-01

The equations for the kinetic modeling of a supersonically flowing electrically excited laser system are presented. The work focuses on the use of diatomic gases, in particular carbon monoxide mixtures. The equations presented include the vibrational rate equation which describes the vibrational population distribution, the electron, ion and electronic level rate equations, the gasdynamic equations for an ionized gas in the presence of an applied electric field, and the free electron Boltzmann equation including flow and gradient coupling terms. The model developed accounts for vibration--vibration collisions, vibration-translation collisions, electron-molecule inelastic excitation and superelastic de-excitation collisions, charge particle collisions, ionization and three body recombination collisions, elastic collisions, and radiative decay, all of which take place in such a system. A simplified form of the free electron Boltzmann equation is developed and discussed with emphasis placed on its coupling with the supersonic flow. A brief description of a possible solution procedure for the set of coupled equations is discussed

Analytic solution of boundary-value problems for nonstationary model kinetic equations

International Nuclear Information System (INIS)

Latyshev, A.V.; Yushkanov, A.A.

1993-01-01

A theory for constructing the solutions of boundary-value problems for non-stationary model kinetic equations is constructed. This theory was incorrectly presented equation, separation of the variables is used, this leading to a characteristic equation. Eigenfunctions are found in the space of generalized functions, and the eigenvalue spectrum is investigated. An existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved. The proof is constructive and gives explicit expressions for the expansion coefficients. An application to the Rayleigh problem is obtained, and the corresponding result of Cercignani is corrected

A new numerical approximation of the fractal ordinary differential equation

Science.gov (United States)

Atangana, Abdon; Jain, Sonal

2018-02-01

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.

Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations

International Nuclear Information System (INIS)

FAN, WESLEY C.; DRUMM, CLIFTON R.; POWELL, JENNIFER L. email wcfan@sandia.gov

2002-01-01

The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations

Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations

CERN Document Server

Fan, W C; Powell, J L

2002-01-01

The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations.

Nodal collocation approximation for the multidimensional PL equations applied to transport source problems

Energy Technology Data Exchange (ETDEWEB)

Verdu, G. [Departamento de Ingenieria Quimica Y Nuclear, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain); Capilla, M.; Talavera, C. F.; Ginestar, D. [Dept. of Nuclear Engineering, Departamento de Matematica Aplicada, Universitat Politecnica de Valencia, Cami de Vera, 14, 46022. Valencia (Spain)

2012-07-01

PL equations are classical high order approximations to the transport equations which are based on the expansion of the angular dependence of the angular neutron flux and the nuclear cross sections in terms of spherical harmonics. A nodal collocation method is used to discretize the PL equations associated with a neutron source transport problem. The performance of the method is tested solving two 1D problems with analytical solution for the transport equation and a classical 2D problem. (authors)

Nodal collocation approximation for the multidimensional PL equations applied to transport source problems

International Nuclear Information System (INIS)

Verdu, G.; Capilla, M.; Talavera, C. F.; Ginestar, D.

2012-01-01

PL equations are classical high order approximations to the transport equations which are based on the expansion of the angular dependence of the angular neutron flux and the nuclear cross sections in terms of spherical harmonics. A nodal collocation method is used to discretize the PL equations associated with a neutron source transport problem. The performance of the method is tested solving two 1D problems with analytical solution for the transport equation and a classical 2D problem. (authors)

An analytical solution for the two-group kinetic neutron diffusion equation in cylindrical geometry

International Nuclear Information System (INIS)

Fernandes, Julio Cesar L.; Vilhena, Marco Tullio; Bodmann, Bardo Ernst

2011-01-01

Recently the two-group Kinetic Neutron Diffusion Equation with six groups of delay neutron precursor in a rectangle was solved by the Laplace Transform Technique. In this work, we report on an analytical solution for this sort of problem but in cylindrical geometry, assuming a homogeneous and infinite height cylinder. The solution is obtained applying the Hankel Transform to the Kinetic Diffusion equation and solving the transformed problem by the same procedure used in the rectangle. We also present numerical simulations and comparisons against results available in literature. (author)

Approximate, analytic solutions of the Bethe equation for charged particle range

OpenAIRE

Swift, Damian C.; McNaney, James M.

2009-01-01

By either performing a Taylor expansion or making a polynomial approximation, the Bethe equation for charged particle stopping power in matter can be integrated analytically to obtain the range of charged particles in the continuous deceleration approximation. Ranges match reference data to the expected accuracy of the Bethe model. In the non-relativistic limit, the energy deposition rate was also found analytically. The analytic relations can be used to complement and validate numerical solu...

Review of Kaganove's solution for the reactor point kinetics equations

International Nuclear Information System (INIS)

Couto, R.T.; Santo, A.C.F. de.

1993-09-01

A review of Kaganove's method for the reactor point kinetics equations solution is performed. This was method chosen to calculate the power in ATR, a computer program for the analysis of reactivity transients. The reasons for this choice and the adaptation of the method to the purposes of ATR are presented. (author)

Hamilton-Jacobi equation and the breaking of the WKB approximation

Energy Technology Data Exchange (ETDEWEB)

Canfora, F. [Istituto Nazionale di Fisica Nucleare, GC di Salerno (Italy) and Dipartimento di Fisica E.R. Caianiello, Universita di Salerno, Via S. Allende, 84081 Baronissi (Salerno) (Italy)]. E-mail: canfora@sa.infn.it

2005-03-17

A simple method to deal with four-dimensional Hamilton-Jacobi equation for null hypersurfaces is introduced. This method allows to find simple geometrical conditions which give rise to the failure of the WKB approximation on curved spacetimes. The relation between such failure, extreme blackholes and the Cosmic Censor hypothesis is briefly discussed.

Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics

Science.gov (United States)

Pratt, D. T.

1984-01-01

Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.

Numerical Calculation of Transport Based on the Drift Kinetic Equation for plasmas in General Toroidal Magnetic Geometry; Calculo Numerico del Transporte mediante la Ecuacion Cinetica de Deriva para Plasmas en Geometria Magnetica Toroidal: Preliminares

Energy Technology Data Exchange (ETDEWEB)

Reynolds, J. M.; Lopez-Bruna, D.

2009-12-11

This report is the first of a series dedicated to the numerical calculation of the evolution of fusion plasmas in general toroidal geometry, including TJ-II plasmas. A kinetic treatment has been chosen: the evolution equation of the distribution function of one or several plasma species is solved in guiding center coordinates. The distribution function is written as a Maxwellian one modulated by polynomial series in the kinetic coordinates with no other approximations than those of the guiding center itself and the computation capabilities. The code allows also for the inclusion of the three-dimensional electrostatic potential in a self-consistent manner, but the initial objective has been set to solving only the neoclassical transport. A high order conservative method (Spectral Difference Method) has been chosen in order to discretized the equation for its numerical solution. In this first report, in addition to justifying the work, the evolution equation and its approximations are described, as well as the baseline of the numerical procedures. (Author) 28 refs.

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Science.gov (United States)

Gorban, Alexander N.; Karlin, Iliya V.

2004-05-01

Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given anzatz manifold which is not tangent to the Lyapunov function levels. Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. ( The theorem about entropy overproduction.) In example, we apply the thermodynamic projector to derive the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2 m, and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of “molecular individualism”. This is the third result.

Any order approximate analytical solution of the nonlinear Volterra's integral equation for accelerator dynamic systems

International Nuclear Information System (INIS)

Liu Chunliang; Xie Xi; Chen Yinbao

1991-01-01

The universal nonlinear dynamic system equation is equivalent to its nonlinear Volterra's integral equation, and any order approximate analytical solution of the nonlinear Volterra's integral equation is obtained by exact analytical method, thus giving another derivation procedure as well as another computation algorithm for the solution of the universal nonlinear dynamic system equation

Solution of the Chew-Low equations in the quadratic approximation

International Nuclear Information System (INIS)

Gerdt, V.P.; Zharkov, A.Yu.

1982-01-01

Within the framework of the iteration scheme for constructing the general solution of the Chew-Low equations as suggested earlier the second order power contributions are found. In contrast to the linear approximation obtained before the quadratic approximation includes an infinite number of poles on the complex plane of the uniformizing variable w. It is shown that taking into account the second order corrections in the general solution allows us to select the class of solutions possessing the Born pole at w=0. The most cumbersome part of analytical computations has been carried out by computer using the algebraic system REDUCE-2

Kinetic Theory of the Inner Magnetospheric Plasma

CERN Document Server

Khazanov, George V

2011-01-01

This book provides a broad introduction to the kinetic theory of space plasma physics with the major focus on the inner magnetospheric plasma. It is designed to provide a comprehensive description of the different kinds of transport equations for both plasma particles and waves with an emphasis on the applicability and limitations of each set of equations. The major topics are: Kinetic Theory of Superthermal Electrons, Kinetic Foundation of the Hydrodynamic Description of Space Plasmas (including wave-particle interaction processes), and Kinetic Theory of the Terrestrial Ring Current. Distinguishable features of this book are the analytical solutions of simplified transport equations. Approximate analytic solutions of transport phenomena are very useful because they help us gain physical insight into how the system responds to varying sources of mass, momentum and energy and also to various external boundary conditions. They also provide us a convenient method to test the validity of complicated numerical mod...

BRIEF COMMUNICATION: On the drift kinetic equation driven by plasma flows

Science.gov (United States)

Shaing, K. C.

2010-07-01

A drift kinetic equation that is driven by plasma flows has previously been derived by Shaing and Spong 1990 (Phys. Fluids B 2 1190). The terms that are driven by particle speed that is parallel to the magnetic field B have been neglected. Here, such terms are discussed to examine their importance to the equation and to show that these terms do not contribute to the calculations of plasma viscosity in large aspect ratio toroidal plasmas, e.g. tokamaks and stellarators.

Time-dependent simplified PN approximation to the equations of radiative transfer

International Nuclear Information System (INIS)

Frank, Martin; Klar, Axel; Larsen, Edward W.; Yasuda, Shugo

2007-01-01

The steady-state simplified P N approximation to the radiative transport equation has been successfully applied to many problems involving radiation. This paper presents the derivation of time-dependent simplified P N (SP N ) equations (up to N = 3) via two different approaches. First, we use an asymptotic analysis, similar to the asymptotic derivation of the steady-state SP N equations. Second, we use an approach similar to the original derivation of the steady-state SP N equations and we show that both approaches lead to similar results. Special focus is put on the well-posedness of the equations and the question whether it can be guaranteed that the solution satisfies the correct physical bounds. Several numerical test cases are shown, including an analytical benchmark due to Su and Olson [B. Su, G.L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium, Ann. Nucl. Energy 24 (1997) 1035-1055.

Kinetic description of intense nonneutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov-Maxwell equations

International Nuclear Information System (INIS)

Davidson, R.C.; Chen, C.

1997-08-01

A kinetic description of intense nonneutral beam propagation through a periodic solenoidal focusing field B sol (rvec x) is developed. The analysis is carried out for a thin beam with characteristic beam radius r b much-lt S, and directed axial momentum γ b mβ b c (in the z-direction) large compared with the transverse momentum and axial momentum spread of the beam particles. Making use of the nonlinear Vlasov-Maxwell equations for general distribution function f b (rvec x,rvec p,t) and self-consistent electrostatic field consistent with the thin-beam approximation, the kinetic model is used to investigate detailed beam equilibrium properties for a variety of distribution functions. Examples are presented both for the case of a uniform solenoidal focusing field B z (z) = B 0 = const. and for the case of a periodic solenoidal focusing field B z (z + S) = B z (z). The nonlinear Vlasov-Maxwell equations are simplified in the thin-beam approximation, and an alternative Hamiltonian formulation is developed that is particularly well-suited to intense beam propagation in periodic focusing systems. Based on the present analysis, the Vlasov-Maxwell description of intense nonneutral beam propagation through a periodic solenoidal focusing field rvec B sol (rvec x) is found to be remarkably tractable and rich in physics content. The Vlasov-Maxwell formalism developed here can be extended in a straightforward manner to investigate detailed stability behavior for perturbations about specific choices of beam equilibria

Point kinetics modeling

International Nuclear Information System (INIS)

Kimpland, R.H.

1996-01-01

A normalized form of the point kinetics equations, a prompt jump approximation, and the Nordheim-Fuchs model are used to model nuclear systems. Reactivity feedback mechanisms considered include volumetric expansion, thermal neutron temperature effect, Doppler effect and void formation. A sample problem of an excursion occurring in a plutonium solution accidentally formed in a glovebox is presented

Computer models for kinetic equations of magnetically confined plasmas

International Nuclear Information System (INIS)

Killeen, J.; Kerbel, G.D.; McCoy, M.G.; Mirin, A.A.; Horowitz, E.J.; Shumaker, D.E.

1987-01-01

This paper presents four working computer models developed by the computational physics group of the National Magnetic Fusion Energy Computer Center. All of the models employ a kinetic description of plasma species. Three of the models are collisional, i.e., they include the solution of the Fokker-Planck equation in velocity space. The fourth model is collisionless and treats the plasma ions by a fully three-dimensional particle-in-cell method

Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations

Science.gov (United States)

Kuzemsky, A. L.

2018-01-01

We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.

Neutron wave reflexions in interface media with transport equation P1 approximation

International Nuclear Information System (INIS)

Oliveira Vellozo, S. de.

1977-01-01

The propagation of neutron waves in non multiplying media is investigated employing the Telegrapher's equation obtained from the P 1 approximation of the time, space and energy dependent Boltzmann equation. Solution of the problem of propagation of sinusoidally modulated source incident on one face of the medium is obtained by analysing the Fourier component of a pulsed source introduced, for the corresponding frequency. The amplitude and the phase of the flux are computed as a function of frequency in media consisting of one, two and three regions in order to study the effects of reflection at the interfaces. The results are compared with those from the Diffusion approximation obtained by neglecting the term involving the second order time derivative. (author)

Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method

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Shaheed N. Huseen

2013-01-01

Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

Relations between the kinetic equation and the Langevin models in two-phase flow modelling

International Nuclear Information System (INIS)

Minier, J.P.; Pozorski, J.

1997-05-01

The purpose of this paper is to discuss PDF and stochastic models which are used in two-phase flow modelling. The aim of the present analysis is essentially to try to determine relations and consistency between different models. It is first recalled that different approaches actually correspond to PDF models written either in terms of the process trajectories or in terms of the PDF itself. The main difference lies in the choice of the independent variables which are retained. Two particular models are studied, the Kinetic Equation and the Langevin Equation model. The latter uses a Langevin equation to model the fluid velocities seen along particle trajectories. The Langevin model is more general since it contains an additional variable. It is shown that, in certain cases, this variable can be summed up exactly to retrieve the Kinetic Equation model as a marginal PDF. A joint fluid and solid particle PDF which includes the characteristics of both phases is proposed at the end of the paper. (author)

Temperature waves and the Boltzmann kinetic equation for phonons

International Nuclear Information System (INIS)

Urushev, D.; Borisov, M.; Vavrek, A.

1988-01-01

The ordinary parabolic equation for thermal conduction based on the Fourier empiric law as well as the generalized thermal conduction equation based on the Maxwell law have been derived from the Boltzmann equation for the phonons within the relaxation time approximation. The temperature waves of the so-called second sound in crystals at low temperatures are transformed into Fourier waves at low frequencies with respect to the characteristic frequency of the U-processes. These waves are transformed into temperature waves similar to the second sound waves in He II at frequences higher than the U-processes characteristic. 1 fig., 19 refs

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

Science.gov (United States)

Singh, Brajesh K; Srivastava, Vineet K

2015-04-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

Perturbed invariant subspaces and approximate generalized functional variable separation solution for nonlinear diffusion-convection equations with weak source

Science.gov (United States)

Xia, Ya-Rong; Zhang, Shun-Li; Xin, Xiang-Peng

2018-03-01

In this paper, we propose the concept of the perturbed invariant subspaces (PISs), and study the approximate generalized functional variable separation solution for the nonlinear diffusion-convection equation with weak source by the approximate generalized conditional symmetries (AGCSs) related to the PISs. Complete classification of the perturbed equations which admit the approximate generalized functional separable solutions (AGFSSs) is obtained. As a consequence, some AGFSSs to the resulting equations are explicitly constructed by way of examples.

Lumping of degree-based mean-field and pair-approximation equations for multistate contact processes

Science.gov (United States)

Kyriakopoulos, Charalampos; Grossmann, Gerrit; Wolf, Verena; Bortolussi, Luca

2018-01-01

Contact processes form a large and highly interesting class of dynamic processes on networks, including epidemic and information-spreading networks. While devising stochastic models of such processes is relatively easy, analyzing them is very challenging from a computational point of view, particularly for large networks appearing in real applications. One strategy to reduce the complexity of their analysis is to rely on approximations, often in terms of a set of differential equations capturing the evolution of a random node, distinguishing nodes with different topological contexts (i.e., different degrees of different neighborhoods), such as degree-based mean-field (DBMF), approximate-master-equation (AME), or pair-approximation (PA) approaches. The number of differential equations so obtained is typically proportional to the maximum degree kmax of the network, which is much smaller than the size of the master equation of the underlying stochastic model, yet numerically solving these equations can still be problematic for large kmax. In this paper, we consider AME and PA, extended to cope with multiple local states, and we provide an aggregation procedure that clusters together nodes having similar degrees, treating those in the same cluster as indistinguishable, thus reducing the number of equations while preserving an accurate description of global observables of interest. We also provide an automatic way to build such equations and to identify a small number of degree clusters that give accurate results. The method is tested on several case studies, where it shows a high level of compression and a reduction of computational time of several orders of magnitude for large networks, with minimal loss in accuracy.

AIREK-MOD, Time Dependent Reactor Kinetics with Feedback Differential Equation

International Nuclear Information System (INIS)

Tamagnini, C.

1984-01-01

1 - Nature of physical problem solved: Solves the reactor kinetic equations with respect to time. A standard form for the reactivity behaviour has been introduced in which the reactivity is given by the sum of a polynomial, sine, cosine and exponential expansion. Tabular form is also included. The presence of feedback differential equations in which the dependence on variables different from the considered one is considered enables many heat-exchange problems to be dealt with. 2 - Method of solution: The method employed for the solution of the differential equations is the one developed by E.R. Cohen (Geneva Conference, 1958). 3 - Restrictions on the complexity of the problem: The maximum number of differential equations that can be solved simultaneously is 50. Within this limitation there may be n delayed neutron groups (n less than or equal to 25), on m other linear feedback equations (n+m less than or equal to 49). CDC 1604 version was offered by EIR (Institut Federal de Recherches en matiere de reacteurs, Switzerland)

Production of a sterile species: Quantum kinetics

Science.gov (United States)

Boyanovsky, D.; Ho, C. M.

2007-10-01

Production of a sterile species is studied within an effective model of active-sterile neutrino mixing in a medium in thermal equilibrium. The quantum kinetic equations for the distribution functions and coherences are obtained from two independent methods: the effective action and the quantum master equation. The decoherence time scale for active-sterile oscillations is τdec=2/Γaa, but the evolution of the distribution functions is determined by the two different time scales associated with the damping rates of the quasiparticle modes in the medium: Γ1=Γaacos⁡2θm; Γ2=Γaasin⁡2θm where Γaa is the interaction rate of the active species in the absence of mixing and θm the mixing angle in the medium. These two time scales are widely different away from Mikheyev-Smirnov-Wolfenstein resonances and preclude the kinetic description of active-sterile production in terms of a simple rate equation. We give the complete set of quantum kinetic equations for the active and sterile populations and coherences and discuss in detail the various approximations. A generalization of the active-sterile transition probability in a medium is provided via the quantum master equation. We derive explicitly the usual quantum kinetic equations in terms of the “polarization vector” and show their equivalence to those obtained from the quantum master equation and effective action.

Analytical approaches for the approximate solution of a nonlinear fractional ordinary differential equation

International Nuclear Information System (INIS)

Basak, K C; Ray, P C; Bera, R K

2009-01-01

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0≤t≤2 and 1<α≤2.

Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method

Directory of Open Access Journals (Sweden)

De-Gang Wang

2012-01-01

Full Text Available A novel algorithm, called variable weight fuzzy marginal linearization (VWFML method, is proposed. This method can supply approximate analytic and numerical solutions to Lane-Emden equations. And it is easy to be implemented and extended for solving other nonlinear differential equations. Numerical examples are included to demonstrate the validity and applicability of the developed technique.

Zeno dynamics and high-temperature master equations beyond secular approximation

International Nuclear Information System (INIS)

Militello, B; Messina, A; Scala, M

2013-01-01

Complete positivity of a class of maps generated by master equations derived beyond the secular approximation is discussed. The connection between such a class of evolutions and the physical properties of the system is analyzed in depth. It is also shown that under suitable hypotheses a Zeno dynamics can be induced because of the high temperature of the bath. (paper)

Approximate solution of the Saha equation - temperature as an explicit function of particle densities

International Nuclear Information System (INIS)

Sato, M.

1991-01-01

The Saha equation for a plasma in thermodynamic equilibrium (TE) is approximately solved to give the temperature as an explicit function of population densities. It is shown that the derived expressions for the Saha temperature are valid approximations to the exact solution. An application of the approximate temperature to the calculation of TE plasma parameters is also described. (orig.)

Multivariate Padé Approximation for Solving Nonlinear Partial Differential Equations of Fractional Order

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Veyis Turut

2013-01-01

Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.

Equations involving Malliavin calculus operators applications and numerical approximation

CERN Document Server

Levajković, Tijana

2017-01-01

This book provides a comprehensive and unified introduction to stochastic differential equations and related optimal control problems. The material is new and the presentation is reader-friendly. A major contribution of the book is the development of generalized Malliavin calculus in the framework of white noise analysis, based on chaos expansion representation of stochastic processes and its application for solving several classes of stochastic differential equations with singular data involving the main operators of Malliavin calculus. In addition, applications in optimal control and numerical approximations are discussed.  The book is divided into four chapters. The first, entitled White Noise Analysis and Chaos Expansions, includes notation and provides the reader with the theoretical background needed to understand the subsequent chapters.  In Chapter 2, Generalized Operators of Malliavin Calculus, the Malliavin derivative operator, the Skorokhod integral and the Ornstein-Uhlenbeck operator are introdu...

Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

Science.gov (United States)

Ford, Neville J.; Connolly, Joseph A.

2009-07-01

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

Polynomial approach method to solve the neutron point kinetics equations with use of the analytic continuation

Energy Technology Data Exchange (ETDEWEB)

Tumelero, Fernanda; Petersen, Claudio Zen; Goncalves, Glenio Aguiar [Universidade Federal de Pelotas, Capao do Leao, RS (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Schramm, Marcelo [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

2016-12-15

In this work, we report a solution to solve the Neutron Point Kinetics Equations applying the Polynomial Approach Method. The main idea is to expand the neutron density and delayed neutron precursors as a power series considering the reactivity as an arbitrary function of the time in a relatively short time interval around an ordinary point. In the first interval one applies the initial conditions and the analytical continuation is used to determine the solutions of the next intervals. A genuine error control is developed based on an analogy with the Rest Theorem. For illustration, we also report simulations for different approaches types (linear, quadratic and cubic). The results obtained by numerical simulations for linear approximation are compared with results in the literature.

A non linear half space problem for radiative transfer equations. Application to the Rosseland approximation

International Nuclear Information System (INIS)

Sentis, R.

1984-07-01

The radiative transfer equations may be approximated by a non linear diffusion equation (called Rosseland equation) when the mean free paths of the photons are small with respect to the size of the medium. Some technical assomptions are made, namely about the initial conditions, to avoid any problem of initial layer terms

An approximation theory for nonlinear partial differential equations with applications to identification and control

Science.gov (United States)

Banks, H. T.; Kunisch, K.

1982-01-01

Approximation results from linear semigroup theory are used to develop a general framework for convergence of approximation schemes in parameter estimation and optimal control problems for nonlinear partial differential equations. These ideas are used to establish theoretical convergence results for parameter identification using modal (eigenfunction) approximation techniques. Results from numerical investigations of these schemes for both hyperbolic and parabolic systems are given.

On the kinetic theory of the one-component plasma

International Nuclear Information System (INIS)

Cohen, J.S.

1984-01-01

In this thesis, kinetic theory is applied to transport phenomena of a one-component plasma. Existing kinetic equations, containing both dynamical screening effects and close binary collisions do not suffer from divergencies. Recently an approximation for the pair correlation function has been proposed that is valid for small values of the plasma collision parameter. Upon insertion of this expression into the general form of the collision integral, one obtains another convergent kinetic equation. This thesis shows that both kinetic equations yield the same coefficient of heat conductivity and viscosity; and that for a hot dilute plasma the arbitrary transport coefficient is rather insensitive to the pair correlation function. In the second part, the author studies the diffusion of a tagged particle in an external magnetic field. It is found that the longitudinal self-diffusion coefficient contra-varies monotonically with the magnetic field strength. (Auth.)

Approximate Treatment of the Dirac Equation with Hyperbolic Potential Function

Science.gov (United States)

Durmus, Aysen

2018-03-01

The time independent Dirac equation is solved analytically for equal scalar and vector hyperbolic potential function in the presence of Greene and Aldrich approximation scheme. The bound state energy equation and spinor wave functions expressed by the hypergeometric function have been obtained in detail with asymptotic iteration approach. In order to indicate the accuracy of this different approach proposed to solve second order linear differential equations, we present that in the non-relativistic limit, analytical solutions of the Dirac equation converge to those of the Schrödinger one. We introduce numerical results of the theoretical analysis for hyperbolic potential function. Bound states corresponding to arbitrary values of n and l are reported for potential parameters covering a wide range of interaction. Also, we investigate relativistic vibrational energy spectra of alkali metal diatomic molecules in the different electronic states. It is observed that theoretical vibrational energy values are consistent with experimental Rydberg-Klein-Rees (RKR) results and vibrational energies of NaK, K_2 and KRb diatomic molecules interacting with hyperbolic potential smoothly converge to the experimental dissociation limit D_e=2508cm^{-1}, 254cm^{-1} and 4221cm^{-1}, respectively.

Non-relativistic and relativistic quantum kinetic equations in nuclear physics

International Nuclear Information System (INIS)

Botermans, W.M.M.

1989-01-01

In this thesis an attempt is made to draw up a quantummechanical tranport equation for the explicit calculation oof collision processes between two (heavy) ions, by making proper approaches of the exact equations (non-rel.: N-particles Schroedinger equation; rel.: Euler-Lagrange field equations.). An important starting point in the drag-up of the theory is the behaviour of nuclear matter in equilibrium which is determined by individual as well as collective effects. The central point in this theory is the effective interaction between two nucleons both surrounded by other nucleons. In the derivation of the tranport equations use is made of the green's function formalism as developed by Schwinger and Keldys. For the Green's function kinematic equations are drawn up and are solved by choosing a proper factorization of three- and four-particle Green's functions in terms of one- and two-particle Green's functions. The necessary boundary condition is obtained by explicitly making use of Boltzmann's assumption that colliding particles are statistically uncorrelated. Finally a transport equation is obtained in which the mean field as well as the nucleon-nucleon collisions are given by the same (medium dependent) interaction. This interaction is the non-equilibrium extension of the interaction as given in the Brueckner theory of nuclear matter. Together, kinetic equation and interaction, form a self-consistent set of equations for the case of a non-relativistic as well as for the case of a relativistic starting point. (H.W.) 148 refs.; 6 figs.; 411 schemes

Approximate method for solving the velocity dependent transport equation in a slab lattice

International Nuclear Information System (INIS)

Ferrari, A.

1966-01-01

A method is described that is intended to provide an approximate solution of the transport equation in a medium simulating a water-moderated plate filled reactor core. This medium is constituted by a periodic array of water channels and absorbing plates. The velocity dependent transport equation in slab geometry is included. The computation is performed in a water channel: the absorbing plates are accounted for by the boundary conditions. The scattering of neutrons in water is assumed isotropic, which allows the use of a double Pn approximation to deal with the angular dependence. This method is able to represent the discontinuity of the angular distribution at the channel boundary. The set of equations thus obtained is dependent only on x and v and the coefficients are independent on x. This solution suggests to try solutions involving Legendre polynomials. This scheme leads to a set of equations v dependent only. To obtain an explicit solution, a thermalization model must now be chosen. Using the secondary model of Cadilhac a solution of this set is easy to get. The numerical computations were performed with a particular secondary model, the well-known model of Wigner and Wilkins. (author) [fr

Point kinetics equations for subcritical systems based on the importance function associated to an external neutron source

International Nuclear Information System (INIS)

Carvalho Gonçalves, Wemerson de; Martinez, Aquilino Senra; Carvalho da Silva, Fernando

2015-01-01

Highlights: • We define the new function importance. • We calculate the kinetic parameters Λ, β, Γ and Q to: 0.95, 0.96, 0.97, 0.98 and 0.99. • We compared the results with those obtained by the main important functions. • We found that the calculated kinetic parameters are physically consistent. - Abstract: This paper aims to determine the parameters for a new set of equations of point kinetic subcritical systems, based on the concept of importance of Heuristic Generalized Perturbation Theory (HGPT). The importance function defined here is related to both the subcriticality and the external neutron source worth (which keeps the system at steady state). The kinetic parameters defined in this work are compared with the corresponding parameters when adopting the importance functions proposed by Gandini and Salvatores (2002), Dulla et al. (2006) and Nishihara et al. (2003). Furthermore, the point kinetics equations developed here are solved for two different transients, considering the parameters obtained with different importance functions. The results collected show that there is a similar behavior of the solution of the point kinetics equations, when used with the parameters obtained by the importance functions proposed by Gandini and Salvatores (2002) and Dulla et al. (2006), specially near the criticality. However, this is not verified as the system gets farther from criticality

Blood-organ transfer kinetics

International Nuclear Information System (INIS)

Skrable, K.W.; Chabot, G.E.; French, C.S.; Wrenn, M.E.; Lipsztein, J.; Sasso, T.L.; Durbin, P.W.

1980-01-01

Exact and approximate kinetics equations relating to the transfer and elimination of radionuclides from the blood and various organs in the body are presented. These expressions may be used to estimate the instantaneous activity or the total number of disintegrations of a radionuclide in the blood or various organs of reference in the body, hence, also the respective dose rates and doses. The exact kinetics equations may be used to relate measurements of radionuclides in excreta to burdens in the body. They do give better results for exposure intervals long compared to the effective mean lives of the radionuclide in the various organs of reference, and they yield the exact steady state expressions. Fortunately, this condition is often satisfied for the relatively long standard exposure interval of 50 years that is applied to occupational exposure. In addition, the steady state expressions may be used along with metabolic data of the distribution of elements in the body, diet and excreta to estimate values of the rate constants used in both the exact and approximate expressions. A comparison of the exact and approximate expressions is given for the uranium metabolic model of Wrenn et al. and a comparison is made with current ICRP models. (author)

Initial value problem for the equations of reactor kinetics

International Nuclear Information System (INIS)

Kyncl, J.

1987-08-01

The initial value problem for the equations of reactor kinetics is solved while taking temperature feedback into account. The space where the problem is solved is chosen such as to correspond to the mathematical properties of cross-section models. The local solution is found by the iterative method, its uniqueness is proved and it is also shown that the existence of global solution is ensured in most cases. Finally, the problem of a weak solution is discussed. (author). 5 refs

One-dimensional free-electron laser equations without the slowly varying envelope approximation

Directory of Open Access Journals (Sweden)

C. Maroli

2011-07-01

Full Text Available A set of one-dimensional equations has been deduced in the time domain from the Maxwell-Lorentz system with the aim of describing the free-electron laser radiation without using the slowly varying envelope approximation (SVEA. These equations are valid even in the case of arbitrarily short electron bunches and of current distributions with ripples on the scale of or shorter than the wavelength. Numerical examples are presented, showing that for long homogeneous bunches the new set of equations gives results in agreement with the SVEA free-electron laser theory and that the use of short or prebunched electron beams leads to a decrease of the emission lethargy. Furthermore, we demonstrate that in all cases in which the backward low frequency wave has negligible effects, these equations can be reduced to a form similar to the usual 1D SVEA equations but with a different definition of the bunching term.

New numerical approximation for solving fractional delay differential equations of variable order using artificial neural networks

Science.gov (United States)

Zúñiga-Aguilar, C. J.; Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Alvarado-Martínez, V. M.; Romero-Ugalde, H. M.

2018-02-01

In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.

Asymptotic analysis of the local potential approximation to the Wetterich equation

Science.gov (United States)

Bender, Carl M.; Sarkar, Sarben

2018-06-01

This paper reports a study of the nonlinear partial differential equation that arises in the local potential approximation to the Wetterich formulation of the functional renormalization group equation. A cut-off-dependent shift of the potential in this partial differential equation is performed. This shift allows a perturbative asymptotic treatment of the differential equation for large values of the infrared cut-off. To leading order in perturbation theory the differential equation becomes a heat equation, where the sign of the diffusion constant changes as the space-time dimension D passes through 2. When D    2 one obtains a backward heat equation whose initial-value problem is ill-posed. For the special case D  =  1 the asymptotic series for cubic and quartic models is extrapolated to the small infrared-cut-off limit by using Padé techniques. The effective potential thus obtained from the partial differential equation is then used in a Schrödinger-equation setting to study the stability of the ground state. For cubic potentials it is found that this Padé procedure distinguishes between a -symmetric theory and a conventional Hermitian theory (g real). For an theory the effective potential is nonsingular and has a stable ground state but for a conventional theory the effective potential is singular. For a conventional Hermitian theory and a -symmetric theory (g  >  0) the results are similar; the effective potentials in both cases are nonsingular and possess stable ground states.

A nodal collocation approximation for the multi-dimensional PL equations - 2D applications

International Nuclear Information System (INIS)

Capilla, M.; Talavera, C.F.; Ginestar, D.; Verdu, G.

2008-01-01

A classical approach to solve the neutron transport equation is to apply the spherical harmonics method obtaining a finite approximation known as the P L equations. In this work, the derivation of the P L equations for multi-dimensional geometries is reviewed and a nodal collocation method is developed to discretize these equations on a rectangular mesh based on the expansion of the neutronic fluxes in terms of orthogonal Legendre polynomials. The performance of the method and the dominant transport Lambda Modes are obtained for a homogeneous 2D problem, a heterogeneous 2D anisotropic scattering problem, a heterogeneous 2D problem and a benchmark problem corresponding to a MOX fuel reactor core

Black hole equations of motion in the quasistationary approximation

International Nuclear Information System (INIS)

Zhdanov, V.I.; Shtelen', V.M.

1980-01-01

Black hole motion is considered under the effect of external actions from the point of view of a remoted observer. The shift of the black hole and the metrix structure are found at the presence of other gravitational bodies using the Zerilli equation. It is shown that in the region, where the space curvature is small, the contribution of the field of the black hole, moving with acceleration, coincides in configuration with the field of usual body, black hole motion in quasistationary approximation occuring according to laws of Newtonian dynamics

Filtering of sound from the Navier-Stokes equations. [An approximation for describing thermal convection in a compressible fluid

Energy Technology Data Exchange (ETDEWEB)

Paolucci, S.

1982-12-01

An approximation leading to anelastic equations capable of describing thermal convection in a compressible fluid is given. These equations are more general than the Oberbeck-Boussinesq equations and different than the standard anelastic equations in that they can be used for the computation of convection in a fluid with large density gradients present. We show that the equations do not contain acoustic waves, while at the same time they can still describe the propagation of internal waves. Throughout we show that the filtering of acoustic waves, within the limits of the approximation, does not appreciably alter the description of the physics.

Evaluation of Lagergren Kinetics Equation by Using Novel Kinetics Expression of Sorption of Zn2+ onto Horse Dung Humic Acid (HD-HA

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Bambang Rusdiarso

2016-12-01

Full Text Available Extraction and purification of humic acid from dry horse dung powder (HD-HA was performed successfully and the purified HD-HA was then applied as sorbent to adsorb Zn2+. Extraction and purification were performed based on procedure of Stevenson (1994 under atmospheric air. Parameters investigated in this work consist of effect of medium sorption acidity, sorption rate (ka and desorption rate constant (kd, Langmuir (monolayer and Freundlich (multilayer sorption capacities, and energy (E of sorption. The ka and kd were determined according to the kinetic model of second order sorption reaching equilibrium, monolayer sorption capacity (b and energy (E were determined according to Langmuir isotherm model, and multilayer sorption capacity (B was determined based on Freundlich isotherm model. Sorption of Zn2+ on purified HD-HA was maximum at pH 5.0. The novel kinetic expression resulted from proposed kinetic model has been shown to be more applicable than the commonly known Lagergren equation obtained from the pseudo-first order sorption model. The application of the equation revealed that the intercept of Lagergren equation, ln qe was more complex function of initial concentration of Zn2+ (a, Langmuir sorption capacity (b, and sorbed Zn2+ at equilibrium (xe.

Kinetic theory of tearing instability

International Nuclear Information System (INIS)

Hazeltine, R.D.; Dobrott, D.; Wang, T.S.

1975-01-01

The guiding-center kinetic equation with Fokker-Planck collision term is used to study, in cylindrical geometry, a class of dissipative instabilities of which the classical tearing mode is an archetype. Variational solution of the kinetic equation obviates the use of an approximate Ohm's law or adiabatic assumption, as used in previous studies, and it provides a dispersive relation which is uniformly valid for any ratio of wave frequency to collision frequency. One result of using the rigorous collision operator is the prediction of a new instability. This instability, driven by the electron temperature gradient, is predicted to occur under the long mean-free path conditions of present tokamak experiments, and has significant features in common with the kink-like oscillations observed in such experiments

Variational estimates of point-kinetics parameters

International Nuclear Information System (INIS)

Favorite, J.A.; Stacey, W.M. Jr.

1995-01-01

Variational estimates of the effect of flux shifts on the integral reactivity parameter of the point-kinetics equations and on regional power fractions were calculated for a variety of localized perturbations in two light water reactor (LWR) model problems representing a small, tightly coupled core and a large, loosely coupled core. For the small core, the flux shifts resulting from even relatively large localized reactivity changes (∼600 pcm) were small, and the standard point-kinetics approximation estimates of reactivity were in error by only ∼10% or less, while the variational estimates were accurate to within ∼1%. For the larger core, significant (>50%) flux shifts occurred in response to local perturbations, leading to errors of the same magnitude in the standard point-kinetics approximation of the reactivity worth. For positive reactivity, the error in the variational estimate of reactivity was only a few percent in the larger core, and the resulting transient power prediction was 1 to 2 orders of magnitude more accurate than with the standard point-kinetics approximation. For a large, local negative reactivity insertion resulting in a large flux shift, the accuracy of the variational estimate broke down. The variational estimate of the effect of flux shifts on reactivity in point-kinetics calculations of transients in LWR cores was found to generally result in greatly improved accuracy, relative to the standard point-kinetics approximation, the exception being for large negative reactivity insertions with large flux shifts in large, loosely coupled cores

Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches.

Science.gov (United States)

Xu, Zhenli; Ma, Manman; Liu, Pei

2014-07-01

We propose a modified Poisson-Nernst-Planck (PNP) model to investigate charge transport in electrolytes of inhomogeneous dielectric environment. The model includes the ionic polarization due to the dielectric inhomogeneity and the ion-ion correlation. This is achieved by the self energy of test ions through solving a generalized Debye-Hückel (DH) equation. We develop numerical methods for the system composed of the PNP and DH equations. Particularly, toward the numerical challenge of solving the high-dimensional DH equation, we developed an analytical WKB approximation and a numerical approach based on the selective inversion of sparse matrices. The model and numerical methods are validated by simulating the charge diffusion in electrolytes between two electrodes, for which effects of dielectrics and correlation are investigated by comparing the results with the prediction by the classical PNP theory. We find that, at the length scale of the interface separation comparable to the Bjerrum length, the results of the modified equations are significantly different from the classical PNP predictions mostly due to the dielectric effect. It is also shown that when the ion self energy is in weak or mediate strength, the WKB approximation presents a high accuracy, compared to precise finite-difference results.

Evaluation of time correlation functions from a generalized Enskog equation

Energy Technology Data Exchange (ETDEWEB)

Yip, S.; Alley, W.E.; Alder, B.J.

1982-01-01

Numerical results for the density and current correlation functions in dense hard-shape fluids are obtained from a kinetic equation which is the extension of the linearized Enskog equation to finite wavelengths in order to demonstrate the convergence of the method of solution. Comparison is made to a previously proposed approximate solution.

Evaluation of time correlation functions from a generalized Enskog equation

International Nuclear Information System (INIS)

Yip, S.; Alley, W.E.; Alder, B.J.

1982-01-01

Numerical results for the density and current correlation functions in dense hard-shape fluids are obtained from a kinetic equation which is the extension of the linearized Enskog equation to finite wavelengths in order to demonstrate the convergence of the method of solution. Comparison is made to a previously proposed approximate solution

Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients

CERN Document Server

Hutzenthaler, Martin

2015-01-01

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation method

Unified implicit kinetic scheme for steady multiscale heat transfer based on the phonon Boltzmann transport equation

Science.gov (United States)

Zhang, Chuang; Guo, Zhaoli; Chen, Songze

2017-12-01

An implicit kinetic scheme is proposed to solve the stationary phonon Boltzmann transport equation (BTE) for multiscale heat transfer problem. Compared to the conventional discrete ordinate method, the present method employs a macroscopic equation to accelerate the convergence in the diffusive regime. The macroscopic equation can be taken as a moment equation for phonon BTE. The heat flux in the macroscopic equation is evaluated from the nonequilibrium distribution function in the BTE, while the equilibrium state in BTE is determined by the macroscopic equation. These two processes exchange information from different scales, such that the method is applicable to the problems with a wide range of Knudsen numbers. Implicit discretization is implemented to solve both the macroscopic equation and the BTE. In addition, a memory reduction technique, which is originally developed for the stationary kinetic equation, is also extended to phonon BTE. Numerical comparisons show that the present scheme can predict reasonable results both in ballistic and diffusive regimes with high efficiency, while the memory requirement is on the same order as solving the Fourier law of heat conduction. The excellent agreement with benchmark and the rapid converging history prove that the proposed macro-micro coupling is a feasible solution to multiscale heat transfer problems.

Rarefied gas flow simulations using high-order gas-kinetic unified algorithms for Boltzmann model equations

Science.gov (United States)

Li, Zhi-Hui; Peng, Ao-Ping; Zhang, Han-Xin; Yang, Jaw-Yen

2015-04-01

This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2-20 and Knudsen numbers 0.0001-5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere-cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N-S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive

Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium.

Science.gov (United States)

Ceccato, Alessandro; Frezzato, Diego

2018-02-14

The chemical Langevin equation and the associated chemical Fokker-Planck equation are well-known continuous approximations of the discrete stochastic evolution of reaction networks. In this work, we show that these approximations suffer from a physical inconsistency, namely, the presence of nonphysical probability currents at the thermal equilibrium even for closed and fully detailed-balanced kinetic schemes. An illustration is given for a model case.

Fractional approximations for linear first order differential equation with polynomial coefficients-application to E1(x) and Z(s)

International Nuclear Information System (INIS)

Martin, P.; Zamudio-Cristi, J.

1982-01-01

A method is described to obtain fractional approximations for linear first order differential equations with polynomial coefficients. This approximation can give good accuracy in a large region of the complex variable plane that may include all the real axis. The parameters of the approximation are solutions of algebraic equations obtained through the coefficients of the highest and lowest power of the variable after the sustitution of the fractional approximation in the differential equation. The method is more general than the asymptotical Pade method, and it is not required to determine the power series or asymptotical expansion. A simple approximation for the exponential integral is found, which give three exact digits for most of the real values of the variable. Approximations of higher accuracy and of the same degree than other authors are also obtained. (Author) [pt

The Pade approximate method for solving problems in plasma kinetic theory

International Nuclear Information System (INIS)

Jasperse, J.R.; Basu, B.

1992-01-01

The method of Pade Approximates has been a powerful tool in solving for the time dependent propagator (Green function) in model quantum field theories. We have developed a modified Pade method which we feel has promise for solving linearized collisional and weakly nonlinear problems in plasma kinetic theory. In order to illustrate the general applicability of the method, in this paper we discuss Pade solutions for the linearized collisional propagator and the collisional dielectric function for a model collisional problem. (author) 3 refs., 2 tabs

Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients

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Nauman Raza

2016-01-01

Full Text Available The nonlinear Klein-Gordon equation (KGE models many nonlinear phenomena. In this paper, we propose a scheme for numerical approximation of solutions of the one-dimensional nonlinear KGE. A common approach to find a solution of a nonlinear system is to first linearize the equations by successive substitution or the Newton iteration method and then solve a linear least squares problem. Here, we show that it can be advantageous to form a sum of squared residuals of the nonlinear problem and then find a zero of the gradient. Our scheme is based on the Sobolev gradient method for solving a nonlinear least square problem directly. The numerical results are compared with Lattice Boltzmann Method (LBM. The L2, L∞, and Root-Mean-Square (RMS values indicate better accuracy of the proposed method with less computational effort.

Tracer kinetics: Modelling by partial differential equations of inhomogeneous compartments with age-dependent elimination rates. Pt. 1

International Nuclear Information System (INIS)

Winkler, E.

1991-01-01

Mathematical models in tracer kinetics are usually based on ordinary differential equations which correspond to a system of kinetically homogeneous compartments (standard compartments). A generalization is possible by the admission of inhomogeneities in the behaviour of the elements belonging to a compartment. The important special case of the age-dependence of elimination rates is treated in its deterministic version. It leads to partial different equations (i.e., systems with distributed coefficients) with the 'age' or the 'residence time' of an element of the compartment as a variable additional to 'time'. The basic equations for one generalized compartment and for systems of such compartments are given together with their general solutions. (orig.) [de

On the exact solution for the multi-group kinetic neutron diffusion equation in a rectangle

International Nuclear Information System (INIS)

Petersen, C.Z.; Vilhena, M.T.M.B. de; Bodmann, B.E.J.

2011-01-01

In this work we consider the two-group bi-dimensional kinetic neutron diffusion equation. The solution procedure formalism is general with respect to the number of energy groups, neutron precursor families and regions with different chemical compositions. The fast and thermal flux and the delayed neutron precursor yields are expanded in a truncated double series in terms of eigenfunctions that, upon insertion into the kinetic equation and upon taking moments, results in a first order linear differential matrix equation with source terms. We split the matrix appearing in the transformed problem into a sum of a diagonal matrix plus the matrix containing the remaining terms and recast the transformed problem into a form that can be solved in the spirit of Adomian's recursive decomposition formalism. Convergence of the solution is guaranteed by the Cardinal Interpolation Theorem. We give numerical simulations and comparisons with available results in the literature. (author)

An integral equation method for discrete and continuous distribution of centres in thermoluminescence kinetics

International Nuclear Information System (INIS)

Kantorovich, L.N.; Fogel, G.M.; Gotlib, V.I.

1990-01-01

Thermoluminescence kinetics is discussed within the framework of a band model containing an arbitrary number of types of recombination and trapping centres at an arbitrary correlation of all centre parameters. It is shown that the initial system of kinetic equations is reduced to an equivalent system consisting of two integro-differential equations which permit one to perform an accurate generalisation, in the case of a continuous centre distribution, to their parameters for the description of irradiation and thermoluminescence, taking into account charge carrier redistribution to both types of centre. In addition, if only one electron (hole) channel is taken into account, only one integro-differential equation is obtained. On the basis of this equation a precise algebraic equation is obtained for calculation of the area of an arbitrary part of the thermoluminescence curve (TLC), consisting of one or several peaks, which slightly overlap with other peaks. It is shown that at doses which are less than the saturation dose, when the centres are not completely filled by the charge carriers, the dose dependences of such a part of the TLC may have a non-linear character at a simultaneous linear dependence of the area of the whole TLC. At doses which are greater than the saturation dose, the dose dependences of the area of the whole TLC, as well as of its separate parts, undergo breaks at the saturation doses. (author)

The strong running coupling from an approximate gluon Dyson-Schwinger equation

International Nuclear Information System (INIS)

Alkofer, R.; Hauck, A.

1996-01-01

Using Mandelstam's approximation to the gluon Dyson-Schwinger equation we calculate the gluon self-energy in a renormalisation group invariant fashion. We obtain a non-perturbative Β function. The scaling behavior near the ultraviolet stable fixed point is in good agreement with perturbative QCD. No further fixed point for positive values of the coupling is found: α S increases without bound in the infrared

Quasi-linear equation for magnetoplasma oscillations in the weakly relativistic approximation

International Nuclear Information System (INIS)

Rizzato, F.B.

1985-01-01

Some limitations which are present in the dynamical equations for collisionless plasmas are discussed. Some elementary corrections to the linear theories are obtained in a heuristic form, which directly lead to the so-called quasi-linear theories in its non-relativistic and relativistic forms. The effect of the relativistic variation of the gyrofrequency on the diffusion coefficient is examined in a typically perturbative approximation. (author)

The approximate inverse in action: IV. Semi-discrete equations in a Banach space setting

International Nuclear Information System (INIS)

Schuster, T; Schöpfer, F; Rieder, A

2012-01-01

This article concerns the method of approximate inverse to solve semi-discrete, linear operator equations in Banach spaces. Semi-discrete means that we search for a solution in an infinite-dimensional Banach space having only a finite number of data available. In this sense the situation is applicable to a large variety of applications where a measurement process delivers a discretization of an infinite-dimensional data space. The method of approximate inverse computes scalar products of the data with pre-computed reconstruction kernels which are associated with mollifiers and the dual of the model operator. The convergence, approximation power and regularization property of this method when applied to semi-discrete operator equations in Hilbert spaces has been investigated in three prequels to this paper. Here we extend these results to a Banach space setting. We prove convergence and stability for general Banach spaces and reproduce the results specifically for the integration operator acting on the space of continuous functions. (paper)

-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations

Directory of Open Access Journals (Sweden)

Lee HyunYoung

2010-01-01

Full Text Available We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.

Generalized multivariate Fokker-Planck equations derived from kinetic transport theory and linear nonequilibrium thermodynamics

International Nuclear Information System (INIS)

Frank, T.D.

2002-01-01

We study many particle systems in the context of mean field forces, concentration-dependent diffusion coefficients, generalized equilibrium distributions, and quantum statistics. Using kinetic transport theory and linear nonequilibrium thermodynamics we derive for these systems a generalized multivariate Fokker-Planck equation. It is shown that this Fokker-Planck equation describes relaxation processes, has stationary maximum entropy distributions, can have multiple stationary solutions and stationary solutions that differ from Boltzmann distributions

Recent developments in thermoluminescence kinetics: applications to other thermally stimulated processes

International Nuclear Information System (INIS)

Levy, P.W.

1985-01-01

Recent thermoluminescence (TL) studies indicate that many kinetic properties are not in accord with the well known 1st and 2nd order TL kinetic equations. For example, the usual equations do not describe: (1) the shape of certain single glow peaks. (2) The shape of glow peaks in many glow curves containing more than one glow peak. (3) The dependence of the peak temperature, the FWHM, the shape, and other properties on the pre-measurement dose. However, the properties of some single glow peaks are precisely described, or closely approximated by, the more general basic equation from which the usual 1st and 2nd order equations are obtained as special cases. Furthermore, glow curves containing more than one glow peak are described by a system of equations that includes interactions between different types of traps and is a straightforward extension of the general one peak equation. This system - called Interactive Kinetics - accounts for most properties, and explains many anomalies, associated with glow curves containing more than one glow peak. It is particularly convenient for computerized analysis procedures. Lastly, it is suggested that other thermally stimulated processes depend on analogous interactions and are describable by similar sets of kinetic equations that are convenient for computer analysis. 11 refs., 2 figs

Kinetic equations and fluctuations in μspace of one-component dilute plasmas

International Nuclear Information System (INIS)

Tokuyama, Michio; Mori, Hazime

1977-01-01

Kinetic equations for a spatially coarse-grained electron density in μ phase space A(p, r; t) with a length cutoff b and for its fluctuations are studied by a scaling method and a time-convolutionless approach developed by the present authors. An electron gas with a small plasma parameter epsilon=1/c (lambda sub(D)) 3 has three characteristic lengths; the Landau cutoff r sub(L)=epsilon lambda sub(D), the Debye length lambda sub(D)=√k sub(B)T/4πe 2 c and the mean free path l sub(f)=lambda sub(D)/epsilon, e and c being electronic charge and mean electron density, respectively. It is shown that there are two characteristic regions of the length cutoff b. One is a coherent region where r sub(L)<< b<< lambda sub(D). Its characteristic scaling is c→0, b→infinity, t→infinity with b√c and t√c being kept constant. The Vlasov equation is derived in this limit. The other is a kinetic region where lambda sub(D)<< b<< l sub(f). Its characteristic scaling is c→0, b→infinity, t→infinity with bc and tc being kept constant. The Vlasov term disappears and the Balescu-Lenard-Boltzmann-Landau equation, which is free of divergence for both close and distant collisions, is derived in this limit. It is shown that the fluctuations of A(p, r; t) obey a Markov process with scaling exponents α=0, β=1/2 in the coherent region near thermal equilibrium, while they obey a Gaussian Markov process with α=0, β=1 in the kinetic region. The present theory does not need the factorization ansatz and Bogoliubov's functional ansatz. (auth.)

Finite difference approximation of control via the potential in a 1-D Schrodinger equation

Directory of Open Access Journals (Sweden)

K. Kime

2000-04-01

Full Text Available We consider the problem of steering given initial data to given terminal data via a time-dependent potential, the control, in a 1-D Schrodinger equation. We determine a condition for existence of a transferring potential within our approximation. Using Maple, we give equations for the control and also examples in which the potential is restricted to be centralized and to be a step potential.

Modeling capsid kinetics assembly from the steady state distribution of multi-sizes aggregates

Energy Technology Data Exchange (ETDEWEB)

Hozé, Nathanaël; Holcman, David

2014-01-24

The kinetics of aggregation for particles of various sizes depends on their diffusive arrival and fusion at a specific nucleation site. We present here a mean-field approximation and a stochastic jump model for aggregates at equilibrium. This approach is an alternative to the classical Smoluchowski equations that do not have a close form and are not solvable in general. We analyze these mean-field equations and obtain the kinetics of a cluster formation. Our approach provides a simplified theoretical framework to study the kinetics of viral capsid formation, such as HIV from the self-assembly of the structural proteins Gag.

The neutron's Dirac-equation: Its rigorous solution at slab-like magnetic fields, non-relativistic approximation, energy spectra and statistical characteristics

International Nuclear Information System (INIS)

Zhang Yongde.

1987-03-01

In this paper, the neutron Dirac-equation is presented. After decoupling it into two equations of the simple spinors, the rigorous solution of this equation is obtained in the case of slab-like uniform magnetic fields at perpendicular incidence. At non-relativistic approximation and first order approximation of weak field (NRWFA), our results have included all results that have been obtained in references for this case up to now. The corresponding transformations of the neutron's spin vectors are given. The single particle spectrum and its approximate expression are obtained. The characteristics of quantum statistics with the approximate expression of energy spectrum are studied. (author). 15 refs

Numerical solution of the kinetic equation in reactor shielding

International Nuclear Information System (INIS)

Germogenova, T.A.

1975-01-01

A review is made of methods of solving marginal problems of multi-group systems of equations of neutron and γ radiation transfer. The first stage of the solution - the quantification of the basic task, is determined by the qualitative behaviour of the solution - is the nature of its performance and asymptotics. In the second stage - solution of the approximating system, various modifications of the iterative method are as a rule used. A description is given of the features of the major Soviet complexes of programmes (ROZ and RADUGA) for the solution of multi-group systems of transfer equations and some methodological research findings are presented. (author)

Application of the reactor kinetics equations to the reactor safety analysis

International Nuclear Information System (INIS)

Sdouz, G.

1976-01-01

The reactor kinetics equations which can be solved by the computer program AIREK-III are used to describe the behavior of fast reactivity transients. By supplementing this computer program it was possible to solve additional safety problems, e.g. the course of reactor excursions induced by any form of reactivity input, the control of reactivity input as a function of a threshold-energy and the computation of produced energy. (author)

Application of the fractional neutron point kinetic equation: Start-up of a nuclear reactor

International Nuclear Information System (INIS)

Polo-Labarrios, M.-A.; Espinosa-Paredes, G.

2012-01-01

Highlights: ► Neutron density behavior at reactor start up with fractional neutron point kinetics. ► There is a relaxation time associated with a rapid variation in the neutron flux. ► Physical interpretation of the fractional order is related with non-Fickian effects. ► Effect of the anomalous diffusion coefficient and the relaxation time is analyzed. ► Neutron density is related with speed and duration of the control rods lifting. - Abstract: In this paper we present the behavior of the variation of neutron density when the nuclear reactor power is increased using the fractional neutron point kinetic (FNPK) equation with a single-group of delayed neutron precursor. It is considered that there is a relaxation time associated with a rapid variation in the neutron flux and its physical interpretation of the fractional order is related with non-Fickian effects from the neutron diffusion equation point of view. We analyzed the case of increase the nuclear reactor power when reactor is cold start-up which is a process of inserting reactivity by lifting control rods discontinuously. The results show that for short time scales of the start-up the neutronic density behavior with FNPK shows sub-diffusive effects whose absorption are government by control rods velocity. For large times scale, the results shows that the classical equation of the neutron point kinetics over predicted the neutron density regarding to FNPK.

Solving kinetic equations with adaptive mesh in phase space for rarefied gas dynamics and plasma physics (Invited)

International Nuclear Information System (INIS)

Kolobov, Vladimir; Arslanbekov, Robert; Frolova, Anna

2014-01-01

The paper describes an Adaptive Mesh in Phase Space (AMPS) technique for solving kinetic equations with deterministic mesh-based methods. The AMPS technique allows automatic generation of adaptive Cartesian mesh in both physical and velocity spaces using a Tree-of-Trees data structure. We illustrate advantages of AMPS for simulations of rarefied gas dynamics and electron kinetics on low temperature plasmas. In particular, we consider formation of the velocity distribution functions in hypersonic flows, particle kinetics near oscillating boundaries, and electron kinetics in a radio-frequency sheath. AMPS provide substantial savings in computational cost and increased efficiency of the mesh-based kinetic solvers

Solving kinetic equations with adaptive mesh in phase space for rarefied gas dynamics and plasma physics (Invited)

Energy Technology Data Exchange (ETDEWEB)

Kolobov, Vladimir [CFD Research Corporation, Huntsville, AL 35805, USA and The University of Alabama in Huntsville, Huntsville, AL 35805 (United States); Arslanbekov, Robert [CFD Research Corporation, Huntsville, AL 35805 (United States); Frolova, Anna [Computing Center of the Russian Academy of Sciences, Moscow, 119333 (Russian Federation)

2014-12-09

The paper describes an Adaptive Mesh in Phase Space (AMPS) technique for solving kinetic equations with deterministic mesh-based methods. The AMPS technique allows automatic generation of adaptive Cartesian mesh in both physical and velocity spaces using a Tree-of-Trees data structure. We illustrate advantages of AMPS for simulations of rarefied gas dynamics and electron kinetics on low temperature plasmas. In particular, we consider formation of the velocity distribution functions in hypersonic flows, particle kinetics near oscillating boundaries, and electron kinetics in a radio-frequency sheath. AMPS provide substantial savings in computational cost and increased efficiency of the mesh-based kinetic solvers.

Numerical approximation of null controls for the heat equation: Ill-posedness and remedies

International Nuclear Information System (INIS)

Münch, Arnaud; Zuazua, Enrique

2010-01-01

The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski–Lions in the nineties, because of the anomalous behavior of the high-frequency spurious numerical waves. Various efficient remedies have been developed and analyzed in the last decade to filter out these high-frequency components: Fourier filtering, Tychonoff's regularization, mixed finite-element methods, multi-grid strategies, etc. Recently convergence rate results have also been obtained. This work is devoted to analyzing this issue for the heat equation, which is the opposite paradigm because of its strong dissipativity and smoothing properties. The existing analytical results guarantee that, at least in some simple situations, as in the finite-difference scheme in 1 − d, the null or trajectory controls for numerical approximation schemes converge. This is due to the intrinsic high-frequency damping of the heat equation that is inherited by its numerical approximation schemes. But when developing numerical simulations the topic appears to be much more subtle and difficult. In fact, efficiently computing the null control for a numerical approximation scheme of the heat equation is a difficult problem in itself. The difficulty is strongly related to the regularizing effect of the heat kernel. The controls of minimal L 2 -norm are characterized as minima of quadratic functionals on the solutions of the adjoint heat equation, or its numerical versions. These functionals are shown to be coercive in very large spaces of solutions, sufficient to guarantee the L 2 character of controls, but very far from being identifiable as energy spaces for the adjoint system. The very weak coercivity of the functionals under consideration makes the approximation problem exponentially ill-posed and the functional framework far from being well adapted to standard techniques in numerical analysis. In practice, the controls of the

High-field transport of electrons and radiative effects using coupled force-balance and Fokker-Planck equations beyond the relaxation-time approximation

International Nuclear Information System (INIS)

Huang, Danhong; Apostolova, T.; Alsing, P.M.; Cardimona, D.A.

2004-01-01

The dynamics of a many-electron system under both dc and infrared fields is separated into a center-of-mass and a relative motion. The first-order force-balance equation is employed for the slow center-of-mass motion of electrons, and the Fokker-Planck equation is used for the ultrafast relative scattering motion of degenerate electrons. This approach allows us to include the anisotropic energy-relaxation process which has been neglected in the energy-balance equation in the past. It also leads us to include the anisotropic coupling to the incident infrared field with different polarizations. Based on this model, the transport of electrons is explored under strong dc and infrared fields by going beyond the relaxation-time approximation. The anisotropic dependence of the electron distribution function on the parallel and perpendicular kinetic energies of electrons is displayed with respect to the dc field direction, and the effect of anisotropic coupling to an incident infrared field with polarizations parallel and perpendicular to the applied dc electric field is shown. The heating of electrons is more accurately described beyond the energy-balance equation with the inclusion of an anisotropic coupling to the infrared field. The drift velocity of electrons is found to increase with the amplitude of the infrared field due to a suppressed momentum-relaxation process (or frictional force) under parallel polarization but decreases with the amplitude due to an enhanced momentum-relaxation process under perpendicular polarization

Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method

KAUST Repository

Louaked, Mohammed; Seloula, Nour; Trabelsi, Saber

2017-01-01

In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method

KAUST Repository

Louaked, Mohammed

2017-07-20

In this work, we propose and analyze the pressure stabilization method for the unsteady incompressible Brinkman-Forchheimer equations. We present a time discretization scheme which can be used with any consistent finite element space approximation. Second-order error estimate is proven. Some numerical results are also given.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach

KAUST Repository

Collier, Nathan

2011-05-14

We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, in the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation.

Quasiclassical Theory of Spin Dynamics in Superfluid ^3He: Kinetic Equations in the Bulk and Spin Response of Surface Majorana States

Science.gov (United States)

Silaev, M. A.

2018-06-01

We develop a theory based on the formalism of quasiclassical Green's functions to study the spin dynamics in superfluid ^3He. First, we derive kinetic equations for the spin-dependent distribution function in the bulk superfluid reproducing the results obtained earlier without quasiclassical approximation. Then, we consider spin dynamics near the surface of fully gapped ^3He-B-phase taking into account spin relaxation due to the transitions in the spectrum of localized fermionic states. The lifetimes of longitudinal and transverse spin waves are calculated taking into account the Fermi-liquid corrections which lead to a crucial modification of fermionic spectrum and spin responses.

Comparative analysis of approximations used in the methods of Faddeev equations and hyperspherical harmonics

International Nuclear Information System (INIS)

Mukhtarova, M.I.

1988-01-01

Comparative analysis of approximations, used in the methods of Faddeev equations and hyperspherical harmonics (MHH) was conducted. The differences in solutions of these methods, related with introduction of approximation of sufficient partial states into the three-nucleon problem, is shown. MHH method is preferred. It is shown that MHH advantage can be manifested clearly when studying new classes of interactions: three-particle, Δ-isobar, nonlocal and other interactions

Analytic solution of vector model kinetic equations with constant kernel and their applications

International Nuclear Information System (INIS)

Latyshev, A.V.

1993-01-01

For the first time exact solutions the heif-space boundary value problems for model kinetic equations is obtained. Here x > 0, μ is an element of (-∞, 0) union (0, +∞), Σ = diag {σ 1 , σ 2 }, C = [c ij ] - 2 x 2-matrix, Ψ (x, μ) is vector-column with elements ψ 1 and ψ 2 . Exact solution of the diffusion slip flow of the binary gas mixture as a application for the model Boltzmann equation with collision operator in the McCormack's form is found. 18 refs

The O(N) model at nonzero temperature: renormalization of the gap equations in Hartree and large-N approximations

International Nuclear Information System (INIS)

Lenaghan, J.T.; Rischke, D.H.

2000-01-01

The temperature dependence of the sigma meson and pion masses is studied in the framework of the O(N ) model. The Cornwall-Jackiw-Tomboulis formalism is applied to derive gap equations for the masses in the Hartree and large-N approximations. Renormalization of the gap equations is carried out within the cut-off and counter-term renormalization schemes. A consistent renormalization of the gap equations within the cut-off scheme is found to be possible only in the large-N approximation and for a finite value of the cut-off. On the other hand, the counter-term scheme allows for a consistent renormalization of both the large-N and Hartree approximations. In these approximations, the meson masses at a given nonzero temperature depend in general on the choice of the cut-off or renormalization scale. As an application, we also discuss the in-medium on-shell decay widths for sigma mesons and pions at rest. (author)

Numerical solution of multi groups point kinetic equations by simulink toolbox of Matlab software

International Nuclear Information System (INIS)

Hadad, K.; Mohamadi, A.; Sabet, H.; Ayobian, N.; Khani, M.

2004-01-01

The simulink toolbox of Matlab Software was employed to solve the point kinetics equation with six group delayed neutrons. The method of Adams-Bash ford showed a good convergence in solving the system of simultaneous equations and the obtained results showed good agreements with other numerical schemes. The flexibility of the package in changing the system parameters and the user friendly interface makes this approach a reliable educational package in revealing the affects of reactivity changes on power incursions

Kinetic coefficients for quark-antiquark plasma with quantal treatment of color

International Nuclear Information System (INIS)

Dyrek, A.; Florkowski, W.

1986-07-01

We discuss the near-equilibrium state of the q-bar q plasma treated as a system of classical particles with quantized color charges. The matrix of the kinetic coefficients is calculated (in the relaxation approximation of the transport equation) and compared with its classical version. The color Ohm law is recovered but the structure of the kinetic matrix is different. 5 refs. (author)

Solution of the two-dimensional space-time reactor kinetics equation by a locally one-dimensional method

International Nuclear Information System (INIS)

Chen, G.S.; Christenson, J.M.

1985-01-01

In this paper, the authors present some initial results from an investigation of the application of a locally one-dimensional (LOD) finite difference method to the solution of the two-dimensional, two-group reactor kinetics equations. Although the LOD method is relatively well known, it apparently has not been previously applied to the space-time kinetics equations. In this investigation, the LOD results were benchmarked against similar computational results (using the same computing environment, the same programming structure, and the same sample problems) obtained by the TWIGL program. For all of the problems considered, the LOD method provided accurate results in one-half to one-eight of the time required by the TWIGL program

The wave equation: From eikonal to anti-eikonal approximation

Directory of Open Access Journals (Sweden)

Luis Vázquez

2016-06-01

Full Text Available When the refractive index changes very slowly compared to the wave-length we may use the eikonal approximation to the wave equation. In the opposite case, when the refractive index highly variates over the distance of one wave-length, we have what can be termed as the anti-eikonal limit. This situation is addressed in this work. The anti-eikonal limit seems to be a relevant tool in the modelling and design of new optical media. Besides, it describes a basic universal behaviour, independent of the actual values of the refractive index and, thus, of the media, for the components of a wave with wave-length much greater than the characteristic scale of the refractive index.

Nonperturbative confinement in quantum chromodynamics : I. Study of an approximate equation of Mandelstam

NARCIS (Netherlands)

Atkinson, D.; Drohm, J. K.; Johnson, P. W.; Stam, K.

1981-01-01

An approximated form of the Dyson–Schwinger equation for the gluon propagator in quarkless QCD is subjected to nonlinear functional and numerical analysis. It is found that solutions exist, and that these have a double pole at the origin of the square of the propagator momentum, together with an

A highly accurate algorithm for the solution of the point kinetics equations

International Nuclear Information System (INIS)

Ganapol, B.D.

2013-01-01

Highlights: • Point kinetics equations for nuclear reactor transient analysis are numerically solved to extreme accuracy. • Results for classic benchmarks found in the literature are given to 9-digit accuracy. • Recent results of claimed accuracy are shown to be less accurate than claimed. • Arguably brings a chapter of numerical evaluation of the PKEs to a close. - Abstract: Attempts to resolve the point kinetics equations (PKEs) describing nuclear reactor transients have been the subject of numerous articles and texts over the past 50 years. Some very innovative methods, such as the RTS (Reactor Transient Simulation) and CAC (Continuous Analytical Continuation) methods of G.R. Keepin and J. Vigil respectively, have been shown to be exceptionally useful. Recently however, several authors have developed methods they consider accurate without a clear basis for their assertion. In response, this presentation will establish a definitive set of benchmarks to enable those developing PKE methods to truthfully assess the degree of accuracy of their methods. Then, with these benchmarks, two recently published methods, found in this journal will be shown to be less accurate than claimed and a legacy method from 1984 will be confirmed

Symbolic computation of analytic approximate solutions for nonlinear differential equations with initial conditions

Science.gov (United States)

Lin, Yezhi; Liu, Yinping; Li, Zhibin

2012-01-01

The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions of nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, and the two-step Adomian decomposition method (TSADM) combined with the Padé technique, a new algorithm is proposed to construct accurate analytic approximations of nonlinear differential equations with initial conditions. Furthermore, a MAPLE package is developed, which is user-friendly and efficient. One only needs to input a system, initial conditions and several necessary parameters, then our package will automatically deliver analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the validity of the package. Our program provides a helpful and easy-to-use tool in science and engineering to deal with initial value problems. Program summaryProgram title: NAPA Catalogue identifier: AEJZ_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEJZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4060 No. of bytes in distributed program, including test data, etc.: 113 498 Distribution format: tar.gz Programming language: MAPLE R13 Computer: PC Operating system: Windows XP/7 RAM: 2 Gbytes Classification: 4.3 Nature of problem: Solve nonlinear differential equations with initial conditions. Solution method: Adomian decomposition method and Padé technique. Running time: Seconds at most in routine uses of the program. Special tasks may take up to some minutes.

Picard Approximation of Stochastic Differential Equations and Application to LIBOR Models

DEFF Research Database (Denmark)

Papapantoleon, Antonis; Skovmand, David

The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our...... exponential to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements....

Statistical approach to LHCD modeling using the wave kinetic equation

International Nuclear Information System (INIS)

Kupfer, K.; Moreau, D.; Litaudon, X.

1993-04-01

Recent work has shown that for parameter regimes typical of many present day current drive experiments, the orbits of the launched LH rays are chaotic (in the Hamiltonian sense), so that wave energy diffuses through the stochastic layer and fills the spectral gap. We have analyzed this problem using a statistical approach, by solving the wave kinetic equation for the coarse-grained spectral energy density. An interesting result is that the LH absorption profile is essentially independent of both the total injected power and the level of wave stochastic diffusion

The application of Legendre-tau approximation to parameter identification for delay and partial differential equations

Science.gov (United States)

Ito, K.

1983-01-01

Approximation schemes based on Legendre-tau approximation are developed for application to parameter identification problem for delay and partial differential equations. The tau method is based on representing the approximate solution as a truncated series of orthonormal functions. The characteristic feature of the Legendre-tau approach is that when the solution to a problem is infinitely differentiable, the rate of convergence is faster than any finite power of 1/N; higher accuracy is thus achieved, making the approach suitable for small N.

Gas-kinetic unified algorithm for hypersonic flows covering various flow regimes solving Boltzmann model equation in nonequilibrium effect

International Nuclear Information System (INIS)

Li, Zhihui; Ma, Qiang; Wu, Junlin; Jiang, Xinyu; Zhang, Hanxin

2014-01-01

Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body

Wave equation dispersion inversion using a difference approximation to the dispersion-curve misfit gradient

KAUST Repository

Zhang, Zhendong; Schuster, Gerard T.; Liu, Yike; Hanafy, Sherif M.; Li, Jing

2016-01-01

We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh wave dispersion curve using a difference approximation to the gradient of the misfit function. We call this wave equation inversion of skeletonized

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

Science.gov (United States)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

Kinetic coefficients for quark-antiquark plasma

International Nuclear Information System (INIS)

Czyz, W.; Florkowski, W.

1986-03-01

The quark-antiquark plasma near equilibrium is studied. The results are based on the Heinz kinetic equations with the Boltzmann collision operator approximated by a relaxation term with the relaxation time, τ, treated as a small parameter. Linear in τ solutions of these equations are used to calculate the transport coefficients: the non-abelian version of Ohm's law, and the shear and volume viscosities. We introduce new chemical potentials which determine the color density matrix of quarks (antiquarks). Gradients of these potentials generate color currents. 12 refs. (author)

Using trees to compute approximate solutions to ordinary differential equations exactly

Science.gov (United States)

Grossman, Robert

1991-01-01

Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

An equation of state for purely kinetic k-essence inspired by cosmic topological defects

Energy Technology Data Exchange (ETDEWEB)

Cordero, Ruben; Gonzalez, Eduardo L.; Queijeiro, Alfonso [Instituto Politecnico Nacional, Departamento de Fisica, Escuela Superior de Fisica y Matematicas, Ciudad de Mexico (Mexico)

2017-06-15

We investigate the physical properties of a purely kinetic k-essence model with an equation of state motivated in superconducting membranes. We compute the equation of state parameter w and discuss its physical evolution via a nonlinear equation of state. Using the adiabatic speed of sound and energy density, we restrict the range of parameters of the model in order to have an acceptable physical behavior. We study the evolution of the scale factor and address the question of the possible existence of finite-time future singularities. Furthermore, we analyze the evolution of the luminosity distance d{sub L} with redshift z by comparing (normalizing) it with the ΛCDM model. Since the equation of state parameter is z-dependent the evolution of the luminosity distance is also analyzed using the Alcock-Paczynski test. (orig.)

Controlled Nonlinear Stochastic Delay Equations: Part II: Approximations and Pipe-Flow Representations

International Nuclear Information System (INIS)

Kushner, Harold J.

2012-01-01

This is the second part of a work dealing with key issues that have not been addressed in the modeling and numerical optimization of nonlinear stochastic delay systems. We consider new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. Part I was concerned with issues concerning the class of admissible controls and their approximations, since the classical definitions are inadequate for our models. This part is concerned with transportation equation representations and their approximations. Such representations of nonlinear stochastic delay models have been crucial in the development of numerical algorithms with much reduced memory and computational requirements. The representations for the new models are not obvious and are developed. They also provide a template for the adaptation of the Markov chain approximation numerical methods.

Approximate solution of space and time fractional higher order phase field equation

Science.gov (United States)

Shamseldeen, S.

2018-03-01

This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions

International Nuclear Information System (INIS)

Tarvainen, Tanja; Vauhkonen, Marko; Kolehmainen, Ville; Arridge, Simon R; Kaipio, Jari P

2005-01-01

In this paper, a coupled radiative transfer equation and diffusion approximation model is extended for light propagation in turbid medium with low-scattering and non-scattering regions. The light propagation is modelled with the radiative transfer equation in sub-domains in which the assumptions of the diffusion approximation are not valid. The diffusion approximation is used elsewhere in the domain. The two equations are coupled through their boundary conditions and they are solved simultaneously using the finite element method. The streamline diffusion modification is used to avoid the ray-effect problem in the finite element solution of the radiative transfer equation. The proposed method is tested with simulations. The results of the coupled model are compared with the finite element solutions of the radiative transfer equation and the diffusion approximation and with results of Monte Carlo simulation. The results show that the coupled model can be used to describe photon migration in turbid medium with low-scattering and non-scattering regions more accurately than the conventional diffusion model

A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations

Energy Technology Data Exchange (ETDEWEB)

Liu, Chang, E-mail: cliuaa@ust.hk [Department of Mathematics and Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon (Hong Kong); Xu, Kun, E-mail: makxu@ust.hk [Department of Mathematics and Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon (Hong Kong); Sun, Quanhua, E-mail: qsun@imech.ac.cn [State Key Laboratory of High-temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, No. 15 Beisihuan Xi Rd, Beijing 100190 (China); Cai, Qingdong, E-mail: caiqd@mech.pku.edu.cn [Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871 (China)

2016-06-01

Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier–Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region

Magnetoresistance in organic semiconductors: Including pair correlations in the kinetic equations for hopping transport

Science.gov (United States)

Shumilin, A. V.; Kabanov, V. V.; Dediu, V. I.

2018-03-01

We derive kinetic equations for polaron hopping in organic materials that explicitly take into account the double occupation possibility and pair intersite correlations. The equations include simplified phenomenological spin dynamics and provide a self-consistent framework for the description of the bipolaron mechanism of the organic magnetoresistance. At low applied voltages, the equations can be reduced to those for an effective resistor network that generalizes the Miller-Abrahams network and includes the effect of spin relaxation on the system resistivity. Our theory discloses the close relationship between the organic magnetoresistance and the intersite correlations. Moreover, in the absence of correlations, as in an ordered system with zero Hubbard energy, the magnetoresistance vanishes.

LOCFES-B: Solving the one-dimensional transport equation with user-selected spatial approximations

International Nuclear Information System (INIS)

Jarvis, R.D.; Nelson, P.

1993-01-01

Closed linear one-cell functional (CLOF) methods constitute an abstractly defined class of spatial approximations to the one-dimensional discrete ordinates equations of linear particle transport that encompass, as specific instances, the vast majority of the spatial approximations that have been either used or suggested in the computational solution of these equations. A specific instance of the class of CLOF methods is defined by a (typically small) number of functions of the cell width, total cross section, and direction cosine of particle motion. The LOCFES code takes advantage of the latter observation by permitting the use, within a more-or-less standard source iteration solution process, of an arbitrary CLOF method as defined by a user-supplied subroutine. The design objective of LOCFES was to provide automated determination of the order of accuracy (i.e., order of the discretization error) in the fine-mesh limit for an arbitrary user-selected CLOF method. This asymptotic order of accuracy is one widely used measure of the merit of a spatial approximation. This paper discusses LOCFES-B, which is a code that uses methods developed in LOCFES to solve one-dimensional linear particle transport problems with any user-selected CLOF method. LOCFES-B provides automatic solution of a given problem to within an accuracy specified by user input and provides comparison of the computational results against results from externally provided benchmark results

Accuracy of approximations of solutions to Fredholm equations by kernel methods

Czech Academy of Sciences Publication Activity Database

Gnecco, G.; Kůrková, Věra; Sanguineti, M.

2012-01-01

Roč. 218, č. 14 (2012), s. 7481-7497 ISSN 0096-3003 R&D Projects: GA ČR GAP202/11/1368; GA MŠk OC10047 Grant - others:CNR-AV ČR(CZ-IT) Project 2010–2012 “Complexity of Neural -Network and Kernel Computational Models Institutional research plan: CEZ:AV0Z10300504 Keywords : approximate solutions to integral equations * radial and kernel-based networks * Gaussian kernels * model complexity * analysis of algorithms Subject RIV: IN - Informatics, Computer Science Impact factor: 1.349, year: 2012

Electric Conductivity of Hot and Dense Quark Matter in a Magnetic Field with Landau Level Resummation via Kinetic Equations

Science.gov (United States)

Fukushima, Kenji; Hidaka, Yoshimasa

2018-04-01

We compute the electric conductivity of quark matter at finite temperature T and a quark chemical potential μ under a magnetic field B beyond the lowest Landau level approximation. The electric conductivity transverse to B is dominated by the Hall conductivity σH. For the longitudinal conductivity σ∥, we need to solve kinetic equations. Then, we numerically find that σ∥ has only a mild dependence on μ and the quark mass mq. Moreover, σ∥ first decreases and then linearly increases as a function of B , leading to an intermediate B region that looks consistent with the experimental signature for the chiral magnetic effect. We also point out that σ∥ at a nonzero B remains within the range of the lattice-QCD estimate at B =0 .

Electron energy distribution function in the positive column of a neon glow discharge using the black wall approximation

International Nuclear Information System (INIS)

Al-Hawat, Sh; Naddaf, M

2005-01-01

The electron energy distribution function (EEDF) was determined from the second derivative of the I-V Langmuir probe characteristics and, thereafter, theoretically calculated by solving the plasma kinetic equation, using the black wall (BW) approximation, in the positive column of a neon glow discharge. The pressure has been varied from 0.5 to 4 Torr and the current from 10 to 30 mA. The measured electron temperature, density and electric field strength were used as input data for solving the kinetic equation. Comparisons were made between the EEDFs obtained from experiment, the BW approach, the Maxwellian distribution and the Rutcher solution of the kinetic equation in the elastic energy range. The best conditions for the BW approach are found to be under the discharge conditions: current density j d = 4.45 mA cm -2 and normalized electric field strength E/p = 1.88 V cm -1 Torr -1

Electron energy distribution function in the positive column of a neon glow discharge using the black wall approximation

Science.gov (United States)

Al-Hawat, Sh; Naddaf, M.

2005-04-01

The electron energy distribution function (EEDF) was determined from the second derivative of the I-V Langmuir probe characteristics and, thereafter, theoretically calculated by solving the plasma kinetic equation, using the black wall (BW) approximation, in the positive column of a neon glow discharge. The pressure has been varied from 0.5 to 4 Torr and the current from 10 to 30 mA. The measured electron temperature, density and electric field strength were used as input data for solving the kinetic equation. Comparisons were made between the EEDFs obtained from experiment, the BW approach, the Maxwellian distribution and the Rutcher solution of the kinetic equation in the elastic energy range. The best conditions for the BW approach are found to be under the discharge conditions: current density jd = 4.45 mA cm-2 and normalized electric field strength E/p = 1.88 V cm-1 Torr-1.

Application of a nodal collocation approximation for the multidimensional PL equations to the 3D Takeda benchmark problems

International Nuclear Information System (INIS)

Capilla, M.; Talavera, C.F.; Ginestar, D.; Verdú, G.

2012-01-01

Highlights: ► The multidimensional P L approximation to the nuclear transport equation is reviewed. ► A nodal collocation method is developed for the spatial discretization of P L equations. ► Advantages of the method are lower dimension and good characterists of the associated algebraic eigenvalue problem. ► The P L nodal collocation method is implemented into the computer code SHNC. ► The SHNC code is verified with 2D and 3D benchmark eigenvalue problems from Takeda and Ikeda, giving satisfactory results. - Abstract: P L equations are classical approximations to the neutron transport equations, which are obtained expanding the angular neutron flux in terms of spherical harmonics. These approximations are useful to study the behavior of reactor cores with complex fuel assemblies, for the homogenization of nuclear cross-sections, etc., and most of these applications are in three-dimensional (3D) geometries. In this work, we review the multi-dimensional P L equations and describe a nodal collocation method for the spatial discretization of these equations for arbitrary odd order L, which is based on the expansion of the spatial dependence of the fields in terms of orthonormal Legendre polynomials. The performance of the nodal collocation method is studied by means of obtaining the k eff and the stationary power distribution of several 3D benchmark problems. The solutions are obtained are compared with a finite element method and a Monte Carlo method.

A Padé approximant approach to two kinds of transcendental equations with applications in physics

International Nuclear Information System (INIS)

Luo, Qiang; Wang, Zhidan; Han, Jiurong

2015-01-01

In this paper, we obtain the analytical solutions of two kinds of transcendental equations with numerous applications in college physics by means of the Lagrange inversion theorem. Afterwards we rewrite them in the form of a ratio of rational polynomials by a second-order Padé approximant from a practical and instructional perspective. Our method is illustrated in a pedagogical manner for the benefit of students at the undergraduate level. The approximate formulas introduced in the paper can be applied to abundant examples in physics textbooks, such as Fraunhofer single-slit diffraction, Wien’s displacement law, and the Schrödinger equation with single- or double-δ potential. These formulas, consequently, can reach considerable accuracies according to the numerical results; therefore, they promise to act as valuable ingredients in the standard teaching curriculum. (paper)

Compactness and robustness: Applications in the solution of integral equations for chemical kinetics and electromagnetic scattering

Science.gov (United States)

Zhou, Yajun

This thesis employs the topological concept of compactness to deduce robust solutions to two integral equations arising from chemistry and physics: the inverse Laplace problem in chemical kinetics and the vector wave scattering problem in dielectric optics. The inverse Laplace problem occurs in the quantitative understanding of biological processes that exhibit complex kinetic behavior: different subpopulations of transition events from the "reactant" state to the "product" state follow distinct reaction rate constants, which results in a weighted superposition of exponential decay modes. Reconstruction of the rate constant distribution from kinetic data is often critical for mechanistic understandings of chemical reactions related to biological macromolecules. We devise a "phase function approach" to recover the probability distribution of rate constants from decay data in the time domain. The robustness (numerical stability) of this reconstruction algorithm builds upon the continuity of the transformations connecting the relevant function spaces that are compact metric spaces. The robust "phase function approach" not only is useful for the analysis of heterogeneous subpopulations of exponential decays within a single transition step, but also is generalizable to the kinetic analysis of complex chemical reactions that involve multiple intermediate steps. A quantitative characterization of the light scattering is central to many meteoro-logical, optical, and medical applications. We give a rigorous treatment to electromagnetic scattering on arbitrarily shaped dielectric media via the Born equation: an integral equation with a strongly singular convolution kernel that corresponds to a non-compact Green operator. By constructing a quadratic polynomial of the Green operator that cancels out the kernel singularity and satisfies the compactness criterion, we reveal the universality of a real resonance mode in dielectric optics. Meanwhile, exploiting the properties of

Wave equation of a nonlinear triatomic molecule and the adiabatic correction to the Born--Oppenheimer approximation

International Nuclear Information System (INIS)

Bardo, R.D.; Wolfsberg, M.

1977-01-01

The wave equation for a nonlinear polyatomic molecule is formulated in molecule-fixed coordinates by a method originally due to Hirschfelder and Wigner. Application is made to a triatomic molecule, and the wave equation is explicitly presented in a useful molecule-fixed coordinate system. The formula for the adiabatic correction to the Born--Oppenheimer approximation for a triatomic molecule is obtained. The extension of the present formulation to larger polyatomic molecules is pointed out. Some terms in the triatomic molecule wave equation are discussed in detail

Application of Littlewood-Paley decomposition to the regularity of Boltzmann type kinetic equations

International Nuclear Information System (INIS)

EL Safadi, M.

2007-03-01

We study the regularity of kinetic equations of Boltzmann type.We use essentially Littlewood-Paley method from harmonic analysis, consisting mainly in working with dyadics annulus. We shall mainly concern with the homogeneous case, where the solution f(t,x,v) depends only on the time t and on the velocities v, while working with realistic and singular cross-sections (non cutoff). In the first part, we study the particular case of Maxwellian molecules. Under this hypothesis, the structure of the Boltzmann operator and his Fourier transform write in a simple form. We show a global C ∞ regularity. Then, we deal with the case of general cross-sections with 'hard potential'. We are interested in the Landau equation which is limit equation to the Boltzmann equation, taking in account grazing collisions. We prove that any weak solution belongs to Schwartz space S. We demonstrate also a similar regularity for the case of Boltzmann equation. Let us note that our method applies directly for all dimensions, and proofs are often simpler compared to other previous ones. Finally, we finish with Boltzmann-Dirac equation. In particular, we adapt the result of regularity obtained in Alexandre, Desvillettes, Wennberg and Villani work, using the dissipation rate connected with Boltzmann-Dirac equation. (author)

Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces

Directory of Open Access Journals (Sweden)

Mourad Kerboua

2014-12-01

Full Text Available We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.

Energy transfer in structured and unstructured environments: Master equations beyond the Born-Markov approximations

Energy Technology Data Exchange (ETDEWEB)

Iles-Smith, Jake, E-mail: Jakeilessmith@gmail.com [Controlled Quantum Dynamics Theory, Imperial College London, London SW7 2PG (United Kingdom); Photon Science Institute and School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL (United Kingdom); Department of Photonics Engineering, DTU Fotonik, Ørsteds Plads, 2800 Kongens Lyngby (Denmark); Dijkstra, Arend G. [Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg (Germany); Lambert, Neill [CEMS, RIKEN, Saitama 351-0198 (Japan); Nazir, Ahsan, E-mail: ahsan.nazir@manchester.ac.uk [Photon Science Institute and School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL (United Kingdom)

2016-01-28

We explore excitonic energy transfer dynamics in a molecular dimer system coupled to both structured and unstructured oscillator environments. By extending the reaction coordinate master equation technique developed by Iles-Smith et al. [Phys. Rev. A 90, 032114 (2014)], we go beyond the commonly used Born-Markov approximations to incorporate system-environment correlations and the resultant non-Markovian dynamical effects. We obtain energy transfer dynamics for both underdamped and overdamped oscillator environments that are in perfect agreement with the numerical hierarchical equations of motion over a wide range of parameters. Furthermore, we show that the Zusman equations, which may be obtained in a semiclassical limit of the reaction coordinate model, are often incapable of describing the correct dynamical behaviour. This demonstrates the necessity of properly accounting for quantum correlations generated between the system and its environment when the Born-Markov approximations no longer hold. Finally, we apply the reaction coordinate formalism to the case of a structured environment comprising of both underdamped (i.e., sharply peaked) and overdamped (broad) components simultaneously. We find that though an enhancement of the dimer energy transfer rate can be obtained when compared to an unstructured environment, its magnitude is rather sensitive to both the dimer-peak resonance conditions and the relative strengths of the underdamped and overdamped contributions.

Existence and discrete approximation for optimization problems governed by fractional differential equations

Science.gov (United States)

Bai, Yunru; Baleanu, Dumitru; Wu, Guo-Cheng

2018-06-01

We investigate a class of generalized differential optimization problems driven by the Caputo derivative. Existence of weak Carathe ´odory solution is proved by using Weierstrass existence theorem, fixed point theorem and Filippov implicit function lemma etc. Then a numerical approximation algorithm is introduced, and a convergence theorem is established. Finally, a nonlinear programming problem constrained by the fractional differential equation is illustrated and the results verify the validity of the algorithm.

A New Algorithm for the Approximation of the Schrödinger Equation

Directory of Open Access Journals (Sweden)

LIN Rong-an

2016-01-01

Full Text Available In this paper a four stages twelfth algebraic order symmetric two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives is developed for the first time in the literature. For the new proposed method: (1 we will study the phase-lag analysis, (2 we will present the development of the new method, (3 the local truncation error (LTE analysis will be studied. The analysis is based on a test problem which is the radial time independent Schrödinger equation, (4 the stability and the interval of periodicity analysis will be presented, (5 stepsize control technique will also be presented, (6 the examination of the accuracy and computational cost of the proposed algorithm which is based on the approximation of the Schrödinger equation.

Yosida approximations of stochastic differential equations in infinite dimensions and applications

CERN Document Server

Govindan, T E

2016-01-01

This research monograph brings together, for the first time, the varied literature on Yosida approximations of stochastic differential equations (SDEs) in infinite dimensions and their applications into a single cohesive work. The author provides a clear and systematic introduction to the Yosida approximation method and justifies its power by presenting its applications in some practical topics such as stochastic stability and stochastic optimal control. The theory assimilated spans more than 35 years of mathematics, but is developed slowly and methodically in digestible pieces. The book begins with a motivational chapter that introduces the reader to several different models that play recurring roles throughout the book as the theory is unfolded, and invites readers from different disciplines to see immediately that the effort required to work through the theory that follows is worthwhile. From there, the author presents the necessary prerequisite material, and then launches the reader into the main discussi...

KINETIC-J: A computational kernel for solving the linearized Vlasov equation applied to calculations of the kinetic, configuration space plasma current for time harmonic wave electric fields

Science.gov (United States)

Green, David L.; Berry, Lee A.; Simpson, Adam B.; Younkin, Timothy R.

2018-04-01

We present the KINETIC-J code, a computational kernel for evaluating the linearized Vlasov equation with application to calculating the kinetic plasma response (current) to an applied time harmonic wave electric field. This code addresses the need for a configuration space evaluation of the plasma current to enable kinetic full-wave solvers for waves in hot plasmas to move beyond the limitations of the traditional Fourier spectral methods. We benchmark the kernel via comparison with the standard k →-space forms of the hot plasma conductivity tensor.

Kinetic behaviour of the adsorption and desorption of phosphorus-32 on aluminium hydroxide

International Nuclear Information System (INIS)

Ribeiro, E.M.G.

1993-01-01

Great amount of phosphate fertilizers are used in agriculture. Soil fertility have been studied using fertilizer labelled with phosphorus 32 to improve agronomic practices by increasing the efficient use of phosphate fertilizer. Previous research work have been published suggesting the potential use of kinetics parameters to characterize phosphorus in soil and to diagnosis the phosphate level. In this work the kinetic behaviour of the absorption and desorption of phosphorus-32 on a synthetic aluminium hydroxide was studied attempting to detect the formation of a precipitated phase on the hydroxide surface. The kinetic data for adsorption was adjusted with the Elovich and Fardeau equations for isotopic exchange. It was verified a change in the kinetic behaviour when the surface was approximately 80% saturated. This change suggested the formation of a precipitate. The kinetic data for desorption was fitted with the Fardeau equation, and it was verified the desorption kinetics slower than the desorption. (B.C.A.). 40 refs, 17 figs, 5 tabs

Analytic Approximations for Soliton Solutions of Short-Wave Models for Camassa-Holm and Degasperis-Procesi Equations

International Nuclear Information System (INIS)

Yang Pei; Li Zhibin; Chen Yong

2010-01-01

In this paper, the short-wave model equations are investigated, which are associated with the Camassa-Holm (CH) and Degasperis-Procesi (DP) shallow-water wave equations. Firstly, by means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformations back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems. (general)

Q-branch Raman scattering and modern kinetic thoery

Energy Technology Data Exchange (ETDEWEB)

Monchick, L. [The Johns Hopkins Univ., Laurel, MD (United States)

1993-12-01

The program is an extension of previous APL work whose general aim was to calculate line shapes of nearly resonant isolated line transitions with solutions of a popular quantum kinetic equation-the Waldmann-Snider equation-using well known advanced solution techniques developed for the classical Boltzmann equation. The advanced techniques explored have been a BGK type approximation, which is termed the Generalized Hess Method (GHM), and conversion of the collision operator to a block diagonal matrix of symmetric collision kernels which then can be approximated by discrete ordinate methods. The latter method, which is termed the Collision Kernel method (CC), is capable of the highest accuracy and has been used quite successfully for Q-branch Raman scattering. The GHM method, not quite as accurate, is applicable over a wider range of pressures and has proven quite useful.

The solution of the point kinetics equations via converged accelerated Taylor series (CATS)

Energy Technology Data Exchange (ETDEWEB)

Ganapol, B.; Picca, P. [Dept. of Aerospace and Mechanical Engineering, Univ. of Arizona (United States); Previti, A.; Mostacci, D. [Laboratorio di Montecuccolino, Alma Mater Studiorum - Universita di Bologna (Italy)

2012-07-01

This paper deals with finding accurate solutions of the point kinetics equations including non-linear feedback, in a fast, efficient and straightforward way. A truncated Taylor series is coupled to continuous analytical continuation to provide the recurrence relations to solve the ordinary differential equations of point kinetics. Non-linear (Wynn-epsilon) and linear (Romberg) convergence accelerations are employed to provide highly accurate results for the evaluation of Taylor series expansions and extrapolated values of neutron and precursor densities at desired edits. The proposed Converged Accelerated Taylor Series, or CATS, algorithm automatically performs successive mesh refinements until the desired accuracy is obtained, making use of the intermediate results for converged initial values at each interval. Numerical performance is evaluated using case studies available from the literature. Nearly perfect agreement is found with the literature results generally considered most accurate. Benchmark quality results are reported for several cases of interest including step, ramp, zigzag and sinusoidal prescribed insertions and insertions with adiabatic Doppler feedback. A larger than usual (9) number of digits is included to encourage honest benchmarking. The benchmark is then applied to the enhanced piecewise constant algorithm (EPCA) currently being developed by the second author. (authors)

On the use of analytical approximate expressions for the transfer rate in excitation transfer kinetics

International Nuclear Information System (INIS)

Kusba, J.; Sipp, B.

1985-01-01

We present a discussion about the range of validity of the usual approximate transfer rate expressions used in the description of the kinetics of diffusion-modulated excitation transfer, for a reactive interaction of exponential functional form. We simulate the features of energy transfer by a numerical inversion of the exact Laplace transform of the transfer rate. It is shown that for high diffusion coefficients of the order of 10 -5 cm 2 s -1 , the kinetics may be well reproduced, even at short times, by the asymptotic form of the transfer rate. For slow molecular displacements, the short time static regime is brought to direct observation, but the transfer rate approaches is asymptotic value at a much later time

Mathematical treatment of isotopologue and isotopomer speciation and fractionation in biochemical kinetics

Science.gov (United States)

Maggi, Federico; Riley, William J.

2010-03-01

We present a mathematical treatment of the kinetic equations that describe isotopologue and isotopomer speciation and fractionation during enzyme-catalyzed biochemical reactions. These equations, presented here with the name GEBIK (general equations for biochemical isotope kinetics) and GEBIF (general equations for biochemical isotope fractionation), take into account microbial biomass and enzyme dynamics, reaction stoichiometry, isotope substitution number, and isotope location within each isotopologue and isotopomer. In addition to solving the complete GEBIK and GEBIF, we also present and discuss two approximations to the full solutions under the assumption of biomass-free and enzyme steady-state, and under the quasi-steady-state assumption as applied to the complexation rate. The complete and approximate approaches are applied to observations of biological denitrification in soils. Our analysis highlights that the full GEBIK and GEBIF provide a more accurate description of concentrations and isotopic compositions of substrates and products throughout the reaction than do the approximate forms. We demonstrate that the isotopic effects of a biochemical reaction depend, in the most general case, on substrate and complex concentrations and, therefore, the fractionation factor is a function of time. We also demonstrate that inverse isotopic effects can occur for values of the fractionation factor smaller than 1, and that reactions that do not discriminate isotopes do not necessarily imply a fractionation factor equal to 1.

Hamiltonian structure of isospectral deformation equation and semi-classical approximation to factorized S-matrices

International Nuclear Information System (INIS)

Chudnovsky, D.V.; Chudnovsky, G.V.

1980-01-01

We consider semi-classical approximation to factorized S-matrices. We show that this new class of matrices, called s-matrices, defines Hamiltonian structures for isospectral deformation equations. Concrete examples of factorized s-matrices are constructed and they are used to define Hamiltonian structure for general two-dimensional isospectral deformation systems. (orig.)

A method for the approximate solutions of the unsteady boundary layer equations

International Nuclear Information System (INIS)

Abdus Sattar, Md.

1990-12-01

The approximate integral method proposed by Bianchini et al. to solve the unsteady boundary layer equations is considered here with a simple modification to the scale function for the similarity variable. This is done by introducing a time dependent length scale. The closed form solutions, thus obtained, give satisfactory results for the velocity profile and the skin friction to a limiting case in comparison with the results of the past investigators. (author). 7 refs, 2 figs

Electron energy distribution function in the positive column of a neon glow discharge using the black wall approximation

Energy Technology Data Exchange (ETDEWEB)

Al-Hawat, Sh; Naddaf, M [Physics Department, Atomic Energy Commission, PO Box 6091, Damascus (Syrian Arab Republic)

2005-04-21

The electron energy distribution function (EEDF) was determined from the second derivative of the I-V Langmuir probe characteristics and, thereafter, theoretically calculated by solving the plasma kinetic equation, using the black wall (BW) approximation, in the positive column of a neon glow discharge. The pressure has been varied from 0.5 to 4 Torr and the current from 10 to 30 mA. The measured electron temperature, density and electric field strength were used as input data for solving the kinetic equation. Comparisons were made between the EEDFs obtained from experiment, the BW approach, the Maxwellian distribution and the Rutcher solution of the kinetic equation in the elastic energy range. The best conditions for the BW approach are found to be under the discharge conditions: current density j{sub d} = 4.45 mA cm{sup -2} and normalized electric field strength E/p = 1.88 V cm{sup -1} Torr{sup -1}.

Enhancement accuracy of approximated solutions of the nonlinear singular integral equations of Chew-Low type

International Nuclear Information System (INIS)

Zhidkov, E.P.; Nguen Mong; Khoromskij, B.N.

1979-01-01

The ways of enhancement of the accuracy of approximate solutions of the Chew-Low type equation are considered. Difference schemes are proposed which allow one to obtain solution expansion in degrees of lattice step. On the basis of the expansion by the Richardson method the refinement of approximated solutions is made. Besides, the iteration process is constructed which reduces immediately to the solution of enhanced accuracy. The efficiency of the methods proposed is illustrated by numerical examples

L2-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations

Directory of Open Access Journals (Sweden)

Hyun Young Lee

2010-01-01

Full Text Available We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson method. We adopt an appropriate elliptic-type projection, which leads to optimal ℓ∞(L2 error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.

Strong convergence and convergence rates of approximating solutions for algebraic Riccati equations in Hilbert spaces

Science.gov (United States)

Ito, Kazufumi

1987-01-01

The linear quadratic optimal control problem on infinite time interval for linear time-invariant systems defined on Hilbert spaces is considered. The optimal control is given by a feedback form in terms of solution pi to the associated algebraic Riccati equation (ARE). A Ritz type approximation is used to obtain a sequence pi sup N of finite dimensional approximations of the solution to ARE. A sufficient condition that shows pi sup N converges strongly to pi is obtained. Under this condition, a formula is derived which can be used to obtain a rate of convergence of pi sup N to pi. The results of the Galerkin approximation is demonstrated and applied for parabolic systems and the averaging approximation for hereditary differential systems.

Solution for the multigroup neutron space kinetics equations by the modified Picard algorithm

International Nuclear Information System (INIS)

Tavares, Matheus G.; Petersen, Claudio Z.; Schramm, Marcelo; Zanette, Rodrigo

2017-01-01

In this work, we used a modified Picards method to solve the Multigroup Neutron Space Kinetics Equations (MNSKE) in Cartesian geometry. The method consists in assuming an initial guess for the neutron flux and using it to calculate a fictitious source term in the MNSKE. A new source term is calculated applying its solution, and so on, iteratively, until a stop criterion is satisfied. For the solution of the fast and thermal neutron fluxes equations, the Laplace Transform technique is used in time variable resulting in a rst order linear differential matrix equation, which are solved by classical methods in the literature. After each iteration, the scalar neutron flux and the delayed neutron precursors are reconstructed by polynomial interpolation. We obtain the fluxes and precursors through Numerical Inverse Laplace Transform using the Stehfest method. We present numerical simulations and comparisons with available results in literature. (author)

Solution for the multigroup neutron space kinetics equations by the modified Picard algorithm

Energy Technology Data Exchange (ETDEWEB)

Tavares, Matheus G.; Petersen, Claudio Z., E-mail: matheus.gulartetavares@gmail.com [Universidade Federal de Pelotas (UFPEL), Capao do Leao, RS (Brazil). Departamento de Matematica e Estatistica; Schramm, Marcelo, E-mail: schrammmarcelo@gmail.com [Universidade Federal de Pelotas (UFPEL), RS (Brazil). Centro de Engenharias; Zanette, Rodrigo, E-mail: rodrigozanette@hotmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Instituto de Matematica e Estatistica

2017-07-01

In this work, we used a modified Picards method to solve the Multigroup Neutron Space Kinetics Equations (MNSKE) in Cartesian geometry. The method consists in assuming an initial guess for the neutron flux and using it to calculate a fictitious source term in the MNSKE. A new source term is calculated applying its solution, and so on, iteratively, until a stop criterion is satisfied. For the solution of the fast and thermal neutron fluxes equations, the Laplace Transform technique is used in time variable resulting in a rst order linear differential matrix equation, which are solved by classical methods in the literature. After each iteration, the scalar neutron flux and the delayed neutron precursors are reconstructed by polynomial interpolation. We obtain the fluxes and precursors through Numerical Inverse Laplace Transform using the Stehfest method. We present numerical simulations and comparisons with available results in literature. (author)

A separable approximation of the NN-Paris-potential in the framework of the Bethe-Salpeter equation

International Nuclear Information System (INIS)

Schwarz, K.; Haidenbauer, J.; Froehlich, J.

1985-09-01

The Bethe-Salpeter equation is solved with a separable kernel for the most important nucleon-nucleon partial wave states. We employ the Ernst Shakin-Thaler method in the framework of minimal relativity (Blankenbeckler-Sugar equation) to generate a separable representation of the meson-theoretical Paris potential. These separable interactions, which closely approximate the on-shell- and half-off-shell behaviour of the Paris potential, are then cast into a covariant form for application in the Bethe-Salpeter equation. The role of relativistic effects is discussed with respect to on-shell and off-shell properties of the NN-system. (Author)

Approximated transport-of-intensity equation for coded-aperture x-ray phase-contrast imaging.

Science.gov (United States)

Das, Mini; Liang, Zhihua

2014-09-15

Transport-of-intensity equations (TIEs) allow better understanding of image formation and assist in simplifying the "phase problem" associated with phase-sensitive x-ray measurements. In this Letter, we present for the first time to our knowledge a simplified form of TIE that models x-ray differential phase-contrast (DPC) imaging with coded-aperture (CA) geometry. The validity of our approximation is demonstrated through comparison with an exact TIE in numerical simulations. The relative contributions of absorption, phase, and differential phase to the acquired phase-sensitive intensity images are made readily apparent with the approximate TIE, which may prove useful for solving the inverse phase-retrieval problem associated with these CA geometry based DPC.

NATO Advanced Research Workshop on Approximation by Solutions of Partial Differential Equations, Quadrature Formulae, and Related Topics

CERN Document Server

Goldstein, M; Haussmann, W; Hayman, W; Rogge, L

1992-01-01

This volume consists of the proceedings of the NATO Advanced Research Workshop on Approximation by Solutions of Partial Differential Equations, Quadrature Formulae, and Related Topics, which was held at Hanstholm, Denmark. These proceedings include the main invited talks and contributed papers given during the workshop. The aim of these lectures was to present a selection of results of the latest research in the field. In addition to covering topics in approximation by solutions of partial differential equations and quadrature formulae, this volume is also concerned with related areas, such as Gaussian quadratures, the Pompelu problem, rational approximation to the Fresnel integral, boundary correspondence of univalent harmonic mappings, the application of the Hilbert transform in two dimensional aerodynamics, finely open sets in the limit set of a finitely generated Kleinian group, scattering theory, harmonic and maximal measures for rational functions and the solution of the classical Dirichlet problem. In ...

The KASY synthesis programme for the approximative solution of the 3-dimensional neutron diffusion equation

International Nuclear Information System (INIS)

Buckel, G.; Wouters, R. de; Pilate, S.

1977-01-01

The synthesis code KASY for an approximate solution of the three-dimensional neutron diffusion equation is described; the state of the art as well as envisaged program extensions and the application to tasks from the field of reactor designing are dealt with. (RW) [de

Analytical approximate equations for the resistivity and its temperature coefficient in thin polycrystalline metallic films

International Nuclear Information System (INIS)

Tellier, C.R.; Tosser, A.J.

1977-01-01

In the usual thickness range of sputtered metallic films, analytical linearized approximate expressions of polycrystalline film resistivity and its t.c.r. are deduced from the Mayadas-Shatzkes theoretical equations. A good experimental fit is observed for Al rf sputtered metal films. (orig.) [de

Plasma kinetic theory

International Nuclear Information System (INIS)

Elliott, J.A.

1993-01-01

Plasma kinetic theory is discussed and a comparison made with the kinetic theory of gases. The plasma is described by a modified set of fluid equations and it is shown how these fluid equations can be derived. (UK)

On the equivalence of convergent kinetic equations for hot dilute plasmas: Generating functions for collision brackets

NARCIS (Netherlands)

Cohen, J.S.; Suttorp, L.G.

1982-01-01

The generating functions for the collision brackets associated with two alternative convergent kinetic equations are derived for small values of the plasma parameter. It is shown that the first few terms in the asymptotic expansions of these generating functions are identical. Consequently, both

Deviation from the kinetic law of mass action for reactions induced by binary encounters in liquid solutions

International Nuclear Information System (INIS)

Doktorov, Alexander B; Kipriyanov, Alexey A

2007-01-01

In considering the irreversible chemical reaction A+B→ C+B in liquid solutions two many-particle approaches to the derivation of binary non-Markovian kinetic equations are compared: simple superposition decoupling and a method of extracting 'pair' channels from three-particle correlation evolution. It is shown that both methods provide an almost identical description of this reaction. However, in studies of reversible reactions in liquid solutions only the channel extraction method gives a correct physically clear description of the reaction though it consists of a sequence of steps: the development of integral encounter theory (IET), effective pairs approximation (EPA), modified encounter theory (MET), and the final regular form (RF) of kinetic equations. It is shown that the rate equations often encountered in the literature correspond to the independence of transient channels of 'scattering' in the bimolecular reversible reaction (A+B -B), while the independent transient channel of 'decay' in the reversible reactionA+B -C is defined solely by time integral convolution. In the general case transient channels in non-Markovian theory are not independent, and their interference manifests itself as a non-Markovian inhomogeneous source in binary non-Markovian kinetic equations in regular form. Based on the derived equations new universal kinetics (independent of models) of chemical equilibrium attainment have been obtained. It is shown that these kinetics can differ essentially from the kinetics corresponding to the kinetic law of mass action of formal chemical kinetics

Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

International Nuclear Information System (INIS)

Schnoerr, David; Grima, Ramon; Sanguinetti, Guido

2017-01-01

Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the chemical master equation. Despite its simple structure, no analytic solutions to the chemical master equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient approximation and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several approximation methods, including the chemical Langevin equation, the system size expansion, moment closure approximations, time-scale separation approximations and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, approximations and inference methods for stochastic chemical kinetics. (topical review)

Developpement of a numerical method for Navier-Stokes equations in anelastic approximation: application to Rayleigh-Taylor instabilities

International Nuclear Information System (INIS)

Hammouch, Z.

2012-01-01

The 'anelastic' approximation allows us to filter the acoustic waves thanks to an asymptotic development of the Navier-Stokes equations, so increasing the averaged time step, during the numerical simulation of hydrodynamic instabilities development. So, the anelastic equations for a two fluid mixture in case of Rayleigh-Taylor instability are established.The linear stability of Rayleigh-Taylor flow is studied, for the first time, for perfect fluids in the anelastic approximation. We define the Stokes problem resulting from Navier-Stokes equations without the non linear terms (a part of the buoyancy is considered); the ellipticity is demonstrated, the Eigenmodes and the invariance related to the pressure are detailed. The Uzawa's method is extended to the anelastic approximation and shows the decoupling speeds in 3D, the particular case k = 0 and the spurious modes of pressure. Passing to multi-domain allowed to establish the transmission conditions.The algorithms and the implementation in the existing program are validated by comparing the Uzawa's operator in Fortran and Mathematica languages, to an experiment with incompressible fluids and results from anelastic and compressible numerical simulations. The study of the influence of the initial stratification of both fluids on the development of the Rayleigh-Taylor instability is initiated. (author) [fr

Application of Elovich equation on uptake kinetics of 137Cs by living freshwater macrophytes - a short duration laboratory study

International Nuclear Information System (INIS)

Jaison, T.J.; Patra, A.K.; Ravi, P.M.; Tripathi, R.M.

2014-01-01

Application of Elovich equation on uptake kinetics of 137 Cs by two living macrophytes during controlled experiments on short duration exposure is studied. Compliance to 2 nd order kinetics indicates the mechanism could be chemi-sorption, involving polar functional groups present on the extracelluar surface of the macrophytes. Data analysis suggests that Myriophyllum s. exhibits faster adsorption rate than Hydrilla v. As Myriophyllum s. exhibits better kinetics than Hydrilla v., former could be a better natural adsorbing media for 137 Cs. (author)

Numerical simulation of flood inundation using a well-balanced kinetic scheme for the shallow water equations with bulk recharge and discharge

Science.gov (United States)

Ersoy, Mehmet; Lakkis, Omar; Townsend, Philip

2016-04-01

The flow of water in rivers and oceans can, under general assumptions, be efficiently modelled using Saint-Venant's shallow water system of equations (SWE). SWE is a hyperbolic system of conservation laws (HSCL) which can be derived from a starting point of incompressible Navier-Stokes. A common difficulty in the numerical simulation of HSCLs is the conservation of physical entropy. Work by Audusse, Bristeau, Perthame (2000) and Perthame, Simeoni (2001), proposed numerical SWE solvers known as kinetic schemes (KSs), which can be shown to have desirable entropy-consistent properties, and are thus called well-balanced schemes. A KS is derived from kinetic equations that can be integrated into the SWE. In flood risk assessment models the SWE must be coupled with other equations describing interacting meteorological and hydrogeological phenomena such as rain and groundwater flows. The SWE must therefore be appropriately modified to accommodate source and sink terms, so kinetic schemes are no longer valid. While modifications of SWE in this direction have been recently proposed, e.g., Delestre (2010), we depart from the extant literature by proposing a novel model that is "entropy-consistent" and naturally extends the SWE by respecting its kinetic formulation connections. This allows us to derive a system of partial differential equations modelling flow of a one-dimensional river with both a precipitation term and a groundwater flow model to account for potential infiltration and recharge. We exhibit numerical simulations of the corresponding kinetic schemes. These simulations can be applied to both real world flood prediction and the tackling of wider issues on how climate and societal change are affecting flood risk.

Theoretical study of the countercurrent in an ultracentrifuge-approximate solution of the countercurrent equations

Energy Technology Data Exchange (ETDEWEB)

Jacques, R.

1975-03-15

Integrating the linearized Navier-Stokes equations linearized along the whole length of the centrifuge, we get a differential relation between the mean axial velocity and the centrifugal and viscosity forces on the ends. Then, these equations are integrated near the ends by a boundary layer approximation method. We assume that outside the boundary layer, the axial velocity reaches its mean value. So we obtain on the first hand the repartition of all physical quantities in the boundary layer, on the second hand a differential equation between the mean axial velocity and the boundary conditions imposed on the ends. This equation, valid both for the mechanical and thermal counter-current is solved numerically. Its solution shows the existence of a second boundary layer close to the wall of the tube. The present theory extends Martin's one in that it takes into account: (1) the action of pressure forces; (2) zero velocity on the wall with no transport; (3) the interaction between mechanical and thermal effects which tend to decrease the efficiency and the intensity of the counter-current. (author)

Hydrodynamic limits of kinetic equations for polyatomic and reactive gases

Directory of Open Access Journals (Sweden)

Bisi M.

2017-03-01

Full Text Available Starting from a kinetic BGK-model for a rarefied polyatomic gas, based on a molecular structure of discrete internal energy levels, an asymptotic Chapman-Enskog procedure is developed in the asymptotic continuum limit in order to derive consistent fluid-dynamic equations for macroscopic fields at Navier-Stokes level. In this way, the model allows to treat the gas as a mixture of mono-atomic species. Explicit expressions are given not only for dynamical pressure, but also for shear stress, diffusion velocities, and heat flux. The analysis is shown to deal properly also with a mixture of reactive gases, endowed for simplicity with translational degrees of freedom only, in which frame analogous results can be achieved.

A Gas-kinetic Discontinuous Galerkin Method for Viscous Flow Equations

International Nuclear Information System (INIS)

Liu, Hongwei; Xu, Kun

2007-01-01

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at the cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gaskinetic BGK model and the Navier-Stokes equations, the Navier-Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (10) and two dimensional(20) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method

Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

Directory of Open Access Journals (Sweden)

Hua Yang

2012-01-01

Full Text Available We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs. Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.

An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations

International Nuclear Information System (INIS)

Sun, Wenjun; Jiang, Song; Xu, Kun

2015-01-01

The solutions of radiative transport equations can cover both optical thin and optical thick regimes due to the large variation of photon's mean-free path and its interaction with the material. In the small mean free path limit, the nonlinear time-dependent radiative transfer equations can converge to an equilibrium diffusion equation due to the intensive interaction between radiation and material. In the optical thin limit, the photon free transport mechanism will emerge. In this paper, we are going to develop an accurate and robust asymptotic preserving unified gas kinetic scheme (AP-UGKS) for the gray radiative transfer equations, where the radiation transport equation is coupled with the material thermal energy equation. The current work is based on the UGKS framework for the rarefied gas dynamics [14], and is an extension of a recent work [12] from a one-dimensional linear radiation transport equation to a nonlinear two-dimensional gray radiative system. The newly developed scheme has the asymptotic preserving (AP) property in the optically thick regime in the capturing of diffusive solution without using a cell size being smaller than the photon's mean free path and time step being less than the photon collision time. Besides the diffusion limit, the scheme can capture the exact solution in the optical thin regime as well. The current scheme is a finite volume method. Due to the direct modeling for the time evolution solution of the interface radiative intensity, a smooth transition of the transport physics from optical thin to optical thick can be accurately recovered. Many numerical examples are included to validate the current approach

Kinetic stability analyses in a bumpy cylinder

International Nuclear Information System (INIS)

Dominguez, R.R.; Berk, H.L.

1981-01-01

Recent interest in the ELMO Bumpy Torus (EBT) has prompted a number of stability analyses of both the hot electron rings and the toroidal plasma. Typically these works employ the local approximation, neglecting radial eigenmode structure and ballooning effects to perform the stability analysis. In the present work we develop a fully kinetic formalism for performing nonlocal stability analyses in a bumpy cylinder. We show that the Vlasov-Maxwell integral equations (with one ignorable coordinate) are self-adjoint and hence amenable to analysis using numerical techniques developed for self-adjoint systems of equations. The representation we obtain for the kernel of the Vlasov-Maxwell equations is a differential operator of arbitrarily high order. This form leads to a manifestly self-adjoint system of differential equations for long wavelength modes

Diffusion Coefficient Calculations With Low Order Legendre Polynomial and Chebyshev Polynomial Approximation for the Transport Equation in Spherical Geometry

International Nuclear Information System (INIS)

Yasa, F.; Anli, F.; Guengoer, S.

2007-01-01

We present analytical calculations of spherically symmetric radioactive transfer and neutron transport using a hypothesis of P1 and T1 low order polynomial approximation for diffusion coefficient D. Transport equation in spherical geometry is considered as the pseudo slab equation. The validity of polynomial expansionion in transport theory is investigated through a comparison with classic diffusion theory. It is found that for causes when the fluctuation of the scattering cross section dominates, the quantitative difference between the polynomial approximation and diffusion results was physically acceptable in general

Landau fluid equations for electromagnetic and electrostatic fluctuations

International Nuclear Information System (INIS)

Hedrick, C.L.; Leboeuf, J.

1992-01-01

Closure relations are developed to allow approximate treatment of Landau damping and growth using fluid equations for both electrostatic and electromagnetic modes. The coefficients in these closure relations are related to approximations of the plasma dispersion function by ratios of polynomials. Thirteen different numerical sets of coefficients are given and explicitly related to previous fits to the plasma dispersion function. The application of the techniques presented in this paper is illustrated with the specific example of resistive g modes. Comparisons of full kinetic and approximate results are made for the solutions to the dispersion relation, radially resolved modes in sheared magnetic geometry, and the plasma dispersion function itself

From quantum to semiclassical kinetic equations: Nuclear matter estimates

International Nuclear Information System (INIS)

Galetti, D.; Mizrahi, S.S.; Nemes, M.C.; Toledo Piza, A.F.R. de

1985-01-01

Starting from the exact microscopic time evolution of the quantum one body density associated with a many fermion system semiclassical approximations are derived to it. In the limit where small momentum transfer two body collisions are dominant we get a Fokker-Planck equation and work out friction and diffusion tensors explicitly for nuclear matter. If arbitrary momentum transfers are considered a Boltzmann equation is derived and used to calculate the viscosity coefficient of nuclear matter. A derivation is given of the collision term used by Landau to describe the damping of zero sound waves at low temperature in Plasmas. Memory effects are essential for this. The damping of zero sound waves in nuclear matter is also calculated and the value so obtained associated with the bulk value of the damping of giant resonances in finite nuclei. The bulk value is estimated to be quite small indicating the importance of the nuclear surface for the damping. (Author) [pt

Born approximation to a perturbative numerical method for the solution of the Schrodinger equation

International Nuclear Information System (INIS)

Adam, Gh.

1978-05-01

A perturbative numerical (PN) method is given for the solution of a regular one-dimensional Cauchy problem arising from the Schroedinger equation. The present method uses a step function approximation for the potential. Global, free of scaling difficulty, forward and backward PN algorithms are derived within first order perturbation theory (Born approximation). A rigorous analysis of the local truncation errors is performed. This shows that the order of accuracy of the method is equal to four. In between the mesh points, the global formula for the wavefunction is accurate within O(h 4 ), while that for the first order derivative is accurate within O(h 3 ). (author)

Quantal Brownian Motion from RPA dynamics: The master and Fokker-Planck equations

International Nuclear Information System (INIS)

Yannouleas, C.

1984-05-01

From the purely quantal RPA description of the damped harmonic oscillator and of the corresponding Brownian Motion within the full space (phonon subspace plus reservoir), a master equation (as well as a Fokker-Planck equation) for the reduced density matrix (for the reduced Wigner function, respectively) within the phonon subspace is extracted. The RPA master equation agrees with the master equation derived by the time-dependent perturbative approaches which utilize Tamm-Dancoff Hilbert spaces and invoke the rotating wave approximation. Since the RPA yields a full, as well as a contracted description, it can account for both the kinetic and the unperturbed oscillator momenta. The RPA description of the quantal Brownian Motion contrasts with the descriptions provided by the time perturbative approaches whether they invoke or not the rotating wave approximation. The RPA description also contrasts with the phenomenological phase space quantization. (orig.)

Distributed Approximating Functional Approach to Burgers' Equation ...

African Journals Online (AJOL)

This equation is similar to, but simpler than, the Navier-Stokes equation in fluid dynamics. To verify this advantage through some comparison studies, an exact series solution are also obtained. In addition, the presented scheme has numerically stable behavior. After demonstrating the convergence and accuracy of the ...

The Incorporation of Truncated Fourier Series into Finite Difference Approximations of Structural Stability Equations

Science.gov (United States)

Hannah, S. R.; Palazotto, A. N.

1978-01-01

A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.

The auxiliary field method and approximate analytical solutions of the Schroedinger equation with exponential potentials

Energy Technology Data Exchange (ETDEWEB)

Silvestre-Brac, Bernard [LPSC Universite Joseph Fourier, Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, Avenue des Martyrs 53, F-38026 Grenoble-Cedex (France); Semay, Claude; Buisseret, Fabien [Groupe de Physique Nucleaire Theorique, Universite de Mons-Hainaut, Academie universitaire Wallonie-Bruxelles, Place du Parc 20, B-7000 Mons (Belgium)], E-mail: silvestre@lpsc.in2p3.fr, E-mail: claude.semay@umh.ac.be, E-mail: fabien.buisseret@umh.ac.be

2009-06-19

The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies of the Schroedinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schroedinger equation with exponential potentials of the form -{alpha}r{sup {lambda}}exp(-{beta}r) can also be analytically solved by using the auxiliary field method. Closed formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn to the Yukawa potential and the pure exponential potential.

The auxiliary field method and approximate analytical solutions of the Schroedinger equation with exponential potentials

International Nuclear Information System (INIS)

Silvestre-Brac, Bernard; Semay, Claude; Buisseret, Fabien

2009-01-01

The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies of the Schroedinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schroedinger equation with exponential potentials of the form -αr λ exp(-βr) can also be analytically solved by using the auxiliary field method. Closed formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn to the Yukawa potential and the pure exponential potential

Kinetic computer modeling of microwave surface-wave plasma production

International Nuclear Information System (INIS)

Ganachev, Ivan P.

2004-01-01

Kinetic computer plasma modeling occupies an intermediate position between the time consuming rigorous particle dynamic simulation and the fast but rather rough cold- or warm-plasma fluid models. The present paper reviews the kinetic modeling of microwave surface-wave discharges with accent on recent kinetic self-consistent models, where the external input parameters are reduced to the necessary minimum (frequency and intensity of the applied microwave field and pressure and geometry of the discharge vessel). The presentation is limited to low pressures, so that Boltzmann equation is solved in non-local approximation and collisional electron heating is neglected. The numerical results reproduce correctly the bi-Maxwellian electron energy distribution functions observed experimentally. (author)

On kinetic description of electromagnetic processes in a quantum plasma

International Nuclear Information System (INIS)

Tyshetskiy, Yu.; Vladimirov, S. V.; Kompaneets, R.

2011-01-01

A nonlinear kinetic equation for nonrelativistic quantum plasma with electromagnetic interaction of particles is obtained in the Hartree's mean-field approximation. It is cast in a convenient form of Vlasov-Boltzmann-type equation with ''quantum interference integral'', which allows for relatively straightforward modification of existing classical Vlasov codes to incorporate quantum effects (quantum statistics and quantum interference of overlapping particles wave functions), without changing the bulk of the codes. Such modification (upgrade) of existing Vlasov codes may provide a direct and effective path to numerical simulations of nonlinear electrostatic and electromagnetic phenomena in quantum plasmas, especially of processes where kinetic effects are important (e.g., modulational interactions and stimulated scattering phenomena involving plasma modes at short wavelengths or high-order kinetic modes, dynamical screening and interaction of charges in quantum plasma, etc.) Moreover, numerical approaches involving such modified Vlasov codes would provide a useful basis for theoretical analyses of quantum plasmas, as quantum and classical effects can be easily separated there.

Numerical study to represent non-isothermal melt-crystallization kinetics at laser-powder cladding

CSIR Research Space (South Africa)

Niziev, VG

2013-04-01

Full Text Available for volume fraction of new phase with the known nucleation and growth rates. This approximate approach is convenient when associated heat and phase transition kinetics equations are solved together. Earlier, this approach was used, for example, for phase... Copyright © 2013 SciRes. MNSMS V. G. NIZIEV ET AL. 64 evolution. Phase change is taken as kinetic process which is associated with nucleation and growth of nuclei of liquid...

The relationship between the Wigner-Weyl kinetic formalism and the complex geometrical optics method

OpenAIRE

Maj, Omar

2004-01-01

The relationship between two different asymptotic techniques developed in order to describe the propagation of waves beyond the standard geometrical optics approximation, namely, the Wigner-Weyl kinetic formalism and the complex geometrical optics method, is addressed. More specifically, a solution of the wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained which yields the same wavefield intensity as the complex geometrical optics method. Such a relationship is also disc...

Analytical representation for solution of the neutron point kinetics equation with time-dependent reactivity and free of the stiffness character

International Nuclear Information System (INIS)

Silva, Milena Wollmann da

2013-01-01

In this work, we report a genuine analytical representation for the solution of the neutron point kinetics equation free of the stiffness character, assuming that the reactivity is a continuous and sectionally continuous function of time. To this end, we initially cast the point kinetics equation in a first order linear differential equation. Next, we split the corresponding matrix as a sum of a diagonal matrix with a matrix, whose components contain the off-diagonal elements. Next, expanding the neutron density and the delayed neutron precursors concentrations in a truncated series, and replacing these expansions in the matrix equation, we come out with an equation, which allows to construct a recursive system, a first order matrix differential equation with source. The fundamental characteristic of this system relies on the fact that the corresponding matrix is diagonal, meanwhile the source term is written in terms of the matrix with the off-diagonal components. Further, the first equation of the recursive system has no source and satisfies the initial conditions. On the other hand, the remaining equations satisfy the null initial condition. Due to the diagonal feature of the matrix, we attain analytical solutions for these recursive equations. We also mention that we evaluate the results for any time value, without the analytical continuity because the purposed solution is free on the stiffness character. Finally, we present numerical simulations and comparisons against literature results, considering specific the applications for the following reactivity functions: constant, step, ramp, and sine. (author)

On an analytical formulation for the mono-energetic neutron space-kinetic equation in full cylinder symmetry

International Nuclear Information System (INIS)

Oliveira, F.R.; Bodmann, B.E.J.; Vilhena, M.T.; Carvalho, F.

2017-01-01

Highlights: • The present work presents an exact solution to neutron spatial kinetic equation. • It is an exact solution in a heterogeneous cylinder with temporal dependence. • The solution was constructed through the separation of variables method. - Abstract: In the present work we discuss a system of partial differential equations that model neutron space-kinetics in cylindrical geometry and are defined by two sectionally homogeneous cylinder cells, mono-energetic neutrons and one group of delayed neutron precursors. The solution is determined using the technique of variable separation. The associated complete spectra with respect to each variable separation are analysed and truncated such as to allow a parameterized global solution. For the obtained solution we present some numerical results for the scalar neutron flux and its time dependence and projection on the cylinder axis z and the radial and cylinder axis projection. As a case study we consider an insertion of an absorbing medium in the upper cylinder cell. Continuity of the scalar flux at the interface between the two cylinder elements and conserved current density is explained and related to scale invariance of the partial differential equation system together with the initial and boundary conditions. Some numerical results for the scalar angular neutron flux and associated current densities are shown.

Approximate Forward Difference Equations for the Lower Order Non-Stationary Statistics of Geometrically Non-Linear Systems subject to Random Excitation

DEFF Research Database (Denmark)

Köylüoglu, H. U.; Nielsen, Søren R. K.; Cakmak, A. S.

Geometrically non-linear multi-degree-of-freedom (MDOF) systems subject to random excitation are considered. New semi-analytical approximate forward difference equations for the lower order non-stationary statistical moments of the response are derived from the stochastic differential equations...... of motion, and, the accuracy of these equations is numerically investigated. For stationary excitations, the proposed method computes the stationary statistical moments of the response from the solution of non-linear algebraic equations....

An approximate analysis of the diffusing flow in a self-controlled heat pipe.

Science.gov (United States)

Somogyi, D.; Yen, H. H.

1973-01-01

Constant-density two-dimensional axisymmetric equations are presented for the diffusing flow of a class of self-controlled heat pipes. The analysis is restricted to the vapor space. Condensation of the vapor is related to its mass fraction at the wall by the gas kinetic formula. The Karman-Pohlhausen integral method is applied to obtain approximate solutions. Solutions are presented for a water heat pipe with neon control gas.

Interpreting the Coulomb-field approximation for generalized-Born electrostatics using boundary-integral equation theory.

Science.gov (United States)

Bardhan, Jaydeep P

2008-10-14

The importance of molecular electrostatic interactions in aqueous solution has motivated extensive research into physical models and numerical methods for their estimation. The computational costs associated with simulations that include many explicit water molecules have driven the development of implicit-solvent models, with generalized-Born (GB) models among the most popular of these. In this paper, we analyze a boundary-integral equation interpretation for the Coulomb-field approximation (CFA), which plays a central role in most GB models. This interpretation offers new insights into the nature of the CFA, which traditionally has been assessed using only a single point charge in the solute. The boundary-integral interpretation of the CFA allows the use of multiple point charges, or even continuous charge distributions, leading naturally to methods that eliminate the interpolation inaccuracies associated with the Still equation. This approach, which we call boundary-integral-based electrostatic estimation by the CFA (BIBEE/CFA), is most accurate when the molecular charge distribution generates a smooth normal displacement field at the solute-solvent boundary, and CFA-based GB methods perform similarly. Conversely, both methods are least accurate for charge distributions that give rise to rapidly varying or highly localized normal displacement fields. Supporting this analysis are comparisons of the reaction-potential matrices calculated using GB methods and boundary-element-method (BEM) simulations. An approximation similar to BIBEE/CFA exhibits complementary behavior, with superior accuracy for charge distributions that generate rapidly varying normal fields and poorer accuracy for distributions that produce smooth fields. This approximation, BIBEE by preconditioning (BIBEE/P), essentially generates initial guesses for preconditioned Krylov-subspace iterative BEMs. Thus, iterative refinement of the BIBEE/P results recovers the BEM solution; excellent agreement

ABOUT SOME APPROXIMATIONS TO THE CLOSED SET OF NOT TRIVIAL SOLUTIONS OF THE EQUATIONS OF GINZBURG - LANDAU

Directory of Open Access Journals (Sweden)

A. A. Fonarev

2014-01-01

Full Text Available Possibility of use of a projective iterative method for search of approximations to the closed set of not trivial generalised solutions of a boundary value problem for Ginzburg - Landau's equations of the phenomenological theory of superconduction is investigated. The projective iterative method combines a projective method and iterative process. The generalised solutions of a boundary value problem for Ginzburg - Landau's equations are critical points of a functional of a superconductor free energy.

Solution of two-dimensional equations of neutron transport in 4P0-approximation of spherical harmonics method

International Nuclear Information System (INIS)

Polivanskij, V.P.

1989-01-01

The method to solve two-dimensional equations of neutron transport using 4P 0 -approximation is presented. Previously such approach was efficiently used for the solution of one-dimensional problems. New an attempt is made to apply the approach to solution of two-dimensional problems. Algorithm of the solution is given, as well as results of test neutron-physical calculations. A considerable as compared with diffusion approximation is shown. 11 refs

Approximate analytical solution of diffusion equation with fractional time derivative using optimal homotopy analysis method

Directory of Open Access Journals (Sweden)

S. Das

2013-12-01

Full Text Available In this article, optimal homotopy-analysis method is used to obtain approximate analytic solution of the time-fractional diffusion equation with a given initial condition. The fractional derivatives are considered in the Caputo sense. Unlike usual Homotopy analysis method, this method contains at the most three convergence control parameters which describe the faster convergence of the solution. Effects of parameters on the convergence of the approximate series solution by minimizing the averaged residual error with the proper choices of parameters are calculated numerically and presented through graphs and tables for different particular cases.

Kinetics parameters of a slurry remediation process in rotating drum bioreactors

International Nuclear Information System (INIS)

Esquivel-Rios, I.; Rodriguez-Meza, M. A.; Barrera-Cortes, J.

2009-01-01

The knowledge of biotransformation pollution dynamics in any systems is important for design and optimization purposes of biochemical processes involved. this is focus to the determination of kinetics parameters such as the maximum specific growth rate (μMAX), saturation constant (Ks), biomass yield (YX/S; X: biomass, S: substrate) and oxygen consumption (YO 2 /S; O 2 : oxygen). Several approximations, based on Monod equation, have been developed for estimating kinetics parameters in terms of concentration and type of substrate, bioprocess type and microflora available. (Author)

Microscopic theory of warm ionized gases: equation of state and kinetic Schottky anomaly

International Nuclear Information System (INIS)

Capolupo, A; Giampaolo, S M; Illuminati, F

2013-01-01

Based on accurate Lennard-Jones type interaction potentials, we derive a closed set of state equations for the description of warm atomic gases in the presence of ionization processes. The specific heat is predicted to exhibit peaks in correspondence to single and multiple ionizations. Such kinetic analogue in atomic gases of the Schottky anomaly in solids is enhanced at intermediate and low atomic densities. The case of adiabatic compression of noble gases is analyzed in detail and the implications on sonoluminescence are discussed.

Tracer kinetics: Modelling by partial differential equations of inhomogeneous compartments with age-dependent elimination rates. Pt. 2

International Nuclear Information System (INIS)

Winkler, E.

1991-01-01

The general theory of inhomogeneous compartments with age-dependent elimination rates is illustrated by examples. Mathematically, it turns out that models consisting of partial differential equations include ordinary, delayed and integro-differential equations, a general fact which is treated here in the context of linear tracer kinetics. The examples include standard compartments as a degenerate case, systems of standard compartments (compartment blocks), models resulting in special residence time distributions, models with pipes, and systems with heterogeneous particles. (orig./BBR) [de

Resonance transport and kinetic entropy

International Nuclear Information System (INIS)

Ivanov, Yu.B.; Knoll, J.; Voskresensky, D.N.

2000-01-01

We continue the description of the dynamics of unstable particles within the real-time formulation of nonequilibrium field theory initiated in a previous paper . There we suggest to use Baym's PHI-functional method in order to achieve approximation schemes with 'built in' consistency with respect to conservation laws and thermodynamics even in the case of particles with finite damping width. Starting from Kadanoff-Baym equations we discuss a consistent first order gradient approach to transport which preserves the PHI-derivable properties. The validity conditions for the resulting quantum four-phase-space kinetic theory are discussed under the perspective to treat particles with broad damping widths. This non-equilibrium dynamics naturally includes all those quantum features already inherent in the corresponding equilibrium limit (e.g. Matsubara formalism) at the same level of PHI-derivable approximation. Various collision-term diagrams are discussed including those of higher order which lead to memory effects. As an important novel part we derive a generalized nonequilibrium expression for the kinetic entropy flow, which includes contributions from fluctuations and mass-width effects. In special cases an H-theorem is derived implying that the entropy can only increase with time. Memory effects in the kinetic terms provide contributions to the kinetic entropy flow that in the equilibrium limit recover the famous bosonic type T 3 lnT correction to the specific heat in the case of Fermi liquids like Helium-3

Time adaptivity in the diffusive wave approximation to the shallow water equations

KAUST Repository

Collier, Nathan; Radwan, Hany; Dalcí n, Lisandro D.; Calo, Victor M.

2013-01-01

We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. © 2011 Elsevier B.V.

Time adaptivity in the diffusive wave approximation to the shallow water equations

KAUST Repository

Collier, Nathan

2013-05-01

We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. © 2011 Elsevier B.V.

Electron kinetics with attachment and ionization from higher order solutions of Boltzmann's equation

International Nuclear Information System (INIS)

Winkler, R.; Wilhelm, J.; Braglia, G.L.

1989-01-01

An appropriate approach is presented for solving the Boltzmann equation for electron swarms and nonstationary weakly ionized plasmas in the hydrodynamic stage, including ionization and attachment processes. Using a Legendre-polynomial expansion of the electron velocity distribution function the resulting eigenvalue problem has been solved at any even truncation-order. The technique has been used to study velocity distribution, mean collision frequencies, energy transfer rates, nonstationary behaviour and power balance in hydrodynamic stage, of electrons in a model plasma and a plasma of pure SF 6 . The calculations have been performed for increasing approximation-orders, up to the converged solution of the problem. In particular, the transition from dominant attachment to prevailing ionization when increasing the field strength has been studied. Finally the establishment of the hydrodynamic stage for a selected case in the model plasma has been investigated by solving the nonstationary, spatially homogeneous Boltzmann equation in twoterm approximation. (author)

ASPECTS OF KINETICS AUTOTHERMAL THERMOPHILIC AEROBIC DIGESTION OF SEWAGE SLUDGE - THE USE OF EQUATIONS OF VARIOUS ORDERS

Directory of Open Access Journals (Sweden)

Magdalena Filkiewicz

2016-12-01

Work to identify the kinetics of the process are aimed at, among others, creating a model describing the speed of the process, including obtaining an answer whether the above equations can be the basis for further work on identifying the factors influencing the stabilization process.

Charge exchange of muons in gases. Kinetic equations

International Nuclear Information System (INIS)

Turner, R.E.

1983-01-01

Kinetic equations for the spin-density operators of the diamagnetic and paramagnetic states of the positive muon are obtained for the description of the slowing-down process encountered when high-energy muons thermalize in a single-component gas. The motion of this two-species system is generated by the Liouville superoperators associated with the diamagnetic and paramagnetic spin Hamiltonians and by time-dependent rate superoperators which depict the probabilities per collision that an electron is captured or lost. These rates are translational averages of the appropriate Boltzmann collision operators. That is, they are momentum and position integrals of the product of either the electron capture or loss total cross section with the single-particle translational density operators for the muon (or muonium) and a gas particle. These rates are time dependent because the muon (or muonium) translational density operator is time dependent. The initial amplitudes and phases of the observed thermal spin polarization in muon-spin-rotation (μSR) experiments are then obtained in terms of the spin-density operators emerging from the stopping regime

Solution of fractional kinetic equation by a class of integral transform of pathway type

Science.gov (United States)

Kumar, Dilip

2013-04-01

Solutions of fractional kinetic equations are obtained through an integral transform named Pα-transform introduced in this paper. The Pα-transform is a binomial type transform containing many class of transforms including the well known Laplace transform. The paper is motivated by the idea of pathway model introduced by Mathai [Linear Algebra Appl. 396, 317-328 (2005), 10.1016/j.laa.2004.09.022]. The composition of the transform with differential and integral operators are proved along with convolution theorem. As an illustration of applications to the general theory of differential equations, a simple differential equation is solved by the new transform. Being a new transform, the Pα-transform of some elementary functions as well as some generalized special functions such as H-function, G-function, Wright generalized hypergeometric function, generalized hypergeometric function, and Mittag-Leffler function are also obtained. The results for the classical Laplace transform is retrieved by letting α → 1.

The nonlocal electron kinetics for a low-pressure glow discharge dusty plasma

Science.gov (United States)

Liang, Yonggan; Wang, Ying; Li, Hui; Tian, Ruihuan; Yuan, Chengxun; Kudryavtsev, A. A.; Rabadanov, K. M.; Wu, Jian; Zhou, Zhongxiang; Tian, Hao

2018-05-01

The nonlocal electron kinetic model based on the Boltzmann equation is developed in low-pressure argon glow discharge dusty plasmas. The additional electron-dust elastic and inelastic collision processes are considered when solving the kinetic equation numerically. The orbital motion limited theory and collision enhanced collection approximation are employed to calculate the dust surface potential. The electron energy distribution function (EEDF), effective electron temperature Teff, and dust surface potential are investigated under different plasma and dust conditions by solving the Boltzmann and the dust charging current balance equations self-consistently. A comparison of the calculation results obtained from nonlocal and local kinetic models is made. It is shown that the appearance of dust particles leads to a deviation of the EEDF from its original profile for both nonlocal and local kinetic models. With the increase in dust density and size, the effective electron temperature and dust surface potential decrease due to the high-energy electron loss on the dust surface. Meanwhile, the nonlocal and local results differ much from each other under the same calculation condition. It is concluded that, for low-pressure (PR ≤ 1 cm*Torr) glow discharge dusty plasmas, the existence of dust particles will amplify the difference of local and nonlocal EEDFs, which makes the local kinetic model more improper to determine the main parameters of the positive column. The nonlocal kinetic model should be used for the calculation of the EEDFs and dusty plasma parameters.

Inverse kinetics equations for on line measurement of reactivity using personal computer

International Nuclear Information System (INIS)

Ratemi, Wajdi; El Gadamsi, Walied; Beleid, Abdul Kariem

1993-01-01

Computer with their astonishing speed of calculations along with their easy connection to real systems, are very appropriate for digital measurements of real system variables. In the nuclear industry, such computer application will produce compact control rooms of real power plants, where information and results display can be obtained through push button concept. In our study, we use two personal computers for the purpose of simulation and measurement. One of them is used as a digital simulator to a real reactor, where we effectively simulate the reactor power through a cross talk network. The computed power is passed at certain chosen sampling time to the other computer. The purpose of the other computer is to use the inverse kinetics equations to calculate the reactivity parameter based on the received power and then it performs on line display of the power curve and the reactivity curve using color graphics. In this study, we use the one group version of the inverse kinetics algorithm which can easily be extended to larger group version. The language of programming used in Turbo BASIC, which is very comparable, in terms of efficiency, to FORTRAN language, besides its effective graphics routines. With the use of the extended version of the Inverse Kinetics algorithm, we can effectively apply this techniques of measurement for the purpose of on line display of the reactivity of the Tajoura Research Reactor. (author)

Direct simulation Monte Carlo method for the Uehling-Uhlenbeck-Boltzmann equation.

Science.gov (United States)

Garcia, Alejandro L; Wagner, Wolfgang

2003-11-01

In this paper we describe a direct simulation Monte Carlo algorithm for the Uehling-Uhlenbeck-Boltzmann equation in terms of Markov processes. This provides a unifying framework for both the classical Boltzmann case as well as the Fermi-Dirac and Bose-Einstein cases. We establish the foundation of the algorithm by demonstrating its link to the kinetic equation. By numerical experiments we study its sensitivity to the number of simulation particles and to the discretization of the velocity space, when approximating the steady-state distribution.

Iterative approximation of the solution of a monotone operator equation in certain Banach spaces

International Nuclear Information System (INIS)

Chidume, C.E.

1988-01-01

Let X=L p (or l p ), p ≥ 2. The solution of the equation Ax=f, f is an element of X is approximated in X by an iteration process in each of the following two cases: (i) A is a bounded linear mapping of X into itself which is also bounded below; and, (ii) A is a nonlinear Lipschitz mapping of X into itself and satisfies ≥ m |x-y| 2 , for some constant m > 0 and for all x, y in X, where j is the single-valued normalized duality mapping of X into X* (the dual space of X). A related result deals with the iterative approximation of the fixed point of a Lipschitz strictly pseudocontractive mapping in X. (author). 12 refs

Space distribution and energy straggling of charged particles via Fokker-Planck equation

International Nuclear Information System (INIS)

Manservisi, S.; Molinari, V.; Nespoli, A.

1996-01-01

The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-know Bohr of Bethe formula is found as a first-order approximation of the Fokker-Planck equation

Solutions to the linearized Navier-Stokes equations for channel flow via the WKB approximation

Science.gov (United States)

Leonard, Anthony

2017-11-01

Progress on determining semi-analytical solutions to the linearized Navier-Stokes equations for incompressible channel flow, laminar and turbulent, is reported. Use of the WKB approximation yields, e.g., solutions to initial-value problem for the inviscid Orr-Sommerfeld equation in terms of the Bessel functions J+ 1 / 3 ,J- 1 / 3 ,J1 , and Y1 and their modified counterparts for any given wave speed c = ω /kx and k⊥ ,(k⊥2 =kx2 +kz2) . Of particular note to be discussed is a sequence i = 1 , 2 , . . . of homogeneous inviscid solutions with complex k⊥ i for each speed c, (0 < c <=Umax), in the downstream direction. These solutions for the velocity component normal to the wall v are localized in the plane parallel to the wall. In addition, for limited range of negative c, (- c * <= c <= 0) , we have found upstream-traveling homogeneous solutions with real k⊥(c) . In both cases the solutions for v serve as a source for corresponding solutions to the inviscid Squire equation for the vorticity component normal to the wall ωy.

Wave equation dispersion inversion using a difference approximation to the dispersion-curve misfit gradient

KAUST Repository

Zhang, Zhendong

2016-07-26

We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh wave dispersion curve using a difference approximation to the gradient of the misfit function. We call this wave equation inversion of skeletonized surface waves because the skeletonized dispersion curve for the fundamental-mode Rayleigh wave is inverted using finite-difference solutions to the multi-dimensional elastic wave equation. The best match between the predicted and observed dispersion curves provides the optimal S-wave velocity model. Our method can invert for lateral velocity variations and also can mitigate the local minimum problem in full waveform inversion with a reasonable computation cost for simple models. Results with synthetic and field data illustrate the benefits and limitations of this method. © 2016 Elsevier B.V.

Intermediate modeling between kinetic equations and hydrodynamic limits: derivation, analysis and simulations

International Nuclear Information System (INIS)

Parisot, M.

2011-01-01

This work is dedicated study of a problem resulting from plasma physics: the thermal transfer of electrons in a plasma close to equilibrium Maxwellian. Firstly, a dimensional study of the Vlasov-Fokker-Planck-Maxwell system is performed, allowing one hand to identify a physically relevant parameter of scale and also to define mathematically the contours of validity domain. The asymptotic regime called Spitzer-Harm is studied for a relatively general class of collision operator. The following part of this work is devoted to the derivation and study of the hydrodynamic limit of the system of Vlasov-Maxwell-Landau outside the strictly asymptotic. A model proposed by Schurtz and Nicolais located in this context and analyzed. The particularity of this model lies in the application of a delocalization operation in the heat flux. The link with non-local models of Luciani and Mora is established as well as mathematics properties as the principle of maximum and entropy dissipation. Then a formal derivation from the Vlasov equations with a simplified collision operator, is proposed. The derivation, inspired by the recent work of D. Levermore, involves decomposition methods according to the spherical harmonics and methods of closing called diffusion methods. A hierarchy of intermediate models between the kinetic equations and the hydrodynamic limit is described. In particular a new hydrodynamic system integro-differential by nature, is proposed. The Schurtz and Nicolai model appears as a simplification of the system resulting from the derivation, assuming a steady flow of heat. The above results are then generalized to account for the internal energy dependence which appears naturally in the equation establishment. The existence and uniqueness of the solution of the nonstationary system are established in a simplified framework. The last part is devoted was the implementation of a specific numerical scheme to solve these models. We propose a finite volume approach can be

Effective evolution equations from many-body quantum mechanics

International Nuclear Information System (INIS)

Benedikter, Niels Patriz

2014-01-01

Systems of interest in physics often consist of a very large number of interacting particles. In certain physical regimes, effective non-linear evolution equations are commonly used as an approximation for making predictions about the time-evolution of such systems. Important examples are Bose-Einstein condensates of dilute Bose gases and degenerate Fermi gases. While the effective equations are well-known in physics, a rigorous justification is very difficult. However, a rigorous derivation is essential to precisely understand the range and the limits of validity and the quality of the approximation. In this thesis, we prove that the time evolution of Bose-Einstein condensates in the Gross-Pitaevskii regime can be approximated by the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schroedinger equation. We then turn to fermionic systems and prove that the evolution of a degenerate Fermi gas can be approximated by the time-dependent Hartree-Fock equation (TDHF) under certain assumptions on the semiclassical structure of the initial data. Finally, we extend the latter result to fermions with relativistic kinetic energy. All our results provide explicit bounds on the error as the number of particles becomes large. A crucial methodical insight on bosonic systems is that correlations can be modeled by Bogolyubov transformations. We construct initial data appropriate for the Gross-Pitaevskii regime using a Bogolyubov transformation acting on a coherent state, which amounts to studying squeezed coherent states. As a crucial insight for fermionic systems, we point out a semiclassical structure in states close to the ground state of fermions in a trap. As a convenient language for studying the dynamics of fermionic systems, we use particle-hole transformations.

Space-time reactor kinetics for heterogeneous reactor structure

Energy Technology Data Exchange (ETDEWEB)

Raisic, N [Boris Kidric Institute of nuclear sciences Vinca, Belgrade (Yugoslavia)

1969-11-15

An attempt is made to formulate time dependent diffusion equation based on Feinberg-Galanin theory in the from analogue to the classical reactor kinetic equation. Parameters of these equations could be calculated using the existing codes for static reactor calculation based on the heterogeneous reactor theory. The obtained kinetic equation could be analogues in form to the nodal kinetic equation. Space-time distribution of neutron flux in the reactor can be obtained by solving these equations using standard methods.

Solutions to aggregation-diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

OpenAIRE

Fagioli, Simone; Radici, Emanuela

2018-01-01

We investigate the existence of weak type solutions for a class of aggregation-diffusion PDEs with nonlinear mobility obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to nonnegative initial data in $L^{\\infty} \\cap BV$ away from vacuum and supported in a closed interval with zero-velocity boundary conditions. The main novelti...

Theory of warm ionized gases: equation of state and kinetic Schottky anomaly.

Science.gov (United States)

Capolupo, A; Giampaolo, S M; Illuminati, F

2013-10-01

Based on accurate Lennard-Jones-type interaction potentials, we derive a closed set of state equations for the description of warm atomic gases in the presence of ionization processes. The specific heat is predicted to exhibit peaks in correspondence to single and multiple ionizations. Such kinetic analog in atomic gases of the Schottky anomaly in solids is enhanced at intermediate and low atomic densities. The case of adiabatic compression of noble gases is analyzed in detail and the implications on sonoluminescence are discussed. In particular, the predicted plasma electron density in a sonoluminescent bubble turns out to be in good agreement with the value measured in recent experiments.

Improvement of neutron kinetics module in TRAC-BF1code: one-dimensional nodal collocation method

Energy Technology Data Exchange (ETDEWEB)

Jambrina, Ana; Barrachina, Teresa; Miro, Rafael; Verdu, Gumersindo, E-mail: ajambrina@iqn.upv.es, E-mail: tbarrachina@iqn.upv.es, E-mail: rmiro@iqn.upv.es, E-mail: gverdu@iqn.upv.es [Universidade Politecnica de Valencia (UPV), Valencia (Spain); Soler, Amparo, E-mail: asoler@iberdrola.es [SEA Propulsion S.L., Madrid (Spain); Concejal, Alberto, E-mail: acbe@iberdrola.es [Iberdrola Ingenieria y Construcion S.A.U., Madrid (Spain)

2013-07-01

The TRAC-BF1 one-dimensional kinetic model is a formulation of the neutron diffusion equation in the two energy groups' approximation, based on the analytical nodal method (ANM). The advantage compared with a zero-dimensional kinetic model is that the axial power profile may vary with time due to thermal-hydraulic parameter changes and/or actions of the control systems but at has the disadvantages that in unusual situations it fails to converge. The nodal collocation method developed for the neutron diffusion equation and applied to the kinetics resolution of TRAC-BF1 thermal-hydraulics, is an adaptation of the traditional collocation methods for the discretization of partial differential equations, based on the development of the solution as a linear combination of analytical functions. It has chosen to use a nodal collocation method based on a development of Legendre polynomials of neutron fluxes in each cell. The qualification is carried out by the analysis of the turbine trip transient from the NEA benchmark in Peach Bottom NPP using both the original 1D kinetics implemented in TRAC-BF1 and the 1D nodal collocation method. (author)

Simulation of ITG instabilities with fully kinetic ions and drift-kinetic electrons in tokamaks

Science.gov (United States)

Hu, Youjun; Chen, Yang; Parker, Scott

2017-10-01

A turbulence simulation model with fully kinetic ions and drift-kinetic electrons is being developed in the toroidal electromagnetic turbulence code GEM. This is motivated by the observation that gyrokinetic ions are not well justified in simulating turbulence in tokamak edges with steep density profile, where ρi / L is not small enough to be used a small parameter needed by the gyrokinetic ordering (here ρi is the gyro-radius of ions and L is the scale length of density profile). In this case, the fully kinetic ion model may be useful. Our model uses an implicit scheme to suppress high-frequency compressional Alfven waves and waves associated with the gyro-motion of ions. The ion orbits are advanced by using the well-known Boris scheme, which reproduces correct drift-motion even with large time-step comparable to the ion gyro-period. The field equation in this model is Ampere's law with the magnetic field eliminated by using an implicit scheme of Faraday's law. The current contributed by ions are computed by using an implicit δf method. A flux tube approximation is adopted, which makes the field equation much easier to solve. Numerical results of electromagnetic ITG obtained from this model will be presented and compared with the gyrokinetic results. This work is supported by U.S. Department of Energy, Office of Fusion Energy Sciences under Award No. DE-SC0008801.

Solving Two -Dimensional Diffusion Equations with Nonlocal Boundary Conditions by a Special Class of Padé Approximants

Directory of Open Access Journals (Sweden)

Mohammad Siddique

2010-08-01

Full Text Available Parabolic partial differential equations with nonlocal boundary conditions arise in modeling of a wide range of important application areas such as chemical diffusion, thermoelasticity, heat conduction process, control theory and medicine science. In this paper, we present the implementation of positivity- preserving Padé numerical schemes to the two-dimensional diffusion equation with nonlocal time dependent boundary condition. We successfully implemented these numerical schemes for both Homogeneous and Inhomogeneous cases. The numerical results show that these Padé approximation based numerical schemes are quite accurate and easily implemented.

An analytical approximation scheme to two-point boundary value problems of ordinary differential equations

International Nuclear Information System (INIS)

Boisseau, Bruno; Forgacs, Peter; Giacomini, Hector

2007-01-01

A new (algebraic) approximation scheme to find global solutions of two-point boundary value problems of ordinary differential equations (ODEs) is presented. The method is applicable for both linear and nonlinear (coupled) ODEs whose solutions are analytic near one of the boundary points. It is based on replacing the original ODEs by a sequence of auxiliary first-order polynomial ODEs with constant coefficients. The coefficients in the auxiliary ODEs are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. The problem of obtaining the parameters of the global (connecting) solutions, analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the 'connecting parameters' for a number of nonlinear ODEs arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Abrikosov-Nielsen-Olesen vortex, as well as the 't Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODEs coming from the exact renormalization group. The ground-state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision. (fast track communication)

On discontinuous Galerkin and discrete ordinates approximations for neutron transport equation and the critical eigenvalue