Iterative solutions of finite difference diffusion equations

International Nuclear Information System (INIS)

Menon, S.V.G.; Khandekar, D.C.; Trasi, M.S.

1981-01-01

The heterogeneous arrangement of materials and the three-dimensional character of the reactor physics problems encountered in the design and operation of nuclear reactors makes it necessary to use numerical methods for solution of the neutron diffusion equations which are based on the linear Boltzmann equation. The commonly used numerical method for this purpose is the finite difference method. It converts the diffusion equations to a system of algebraic equations. In practice, the size of this resulting algebraic system is so large that the iterative methods have to be used. Most frequently used iterative methods are discussed. They include : (1) basic iterative methods for one-group problems, (2) iterative methods for eigenvalue problems, and (3) iterative methods which use variable acceleration parameters. Application of Chebyshev theorem to iterative methods is discussed. The extension of the above iterative methods to multigroup neutron diffusion equations is also considered. These methods are applicable to elliptic boundary value problems in reactor design studies in particular, and to elliptic partial differential equations in general. Solution of sample problems is included to illustrate their applications. The subject matter is presented in as simple a manner as possible. However, a working knowledge of matrix theory is presupposed. (M.G.B.)

Diffusive instabilities in hyperbolic reaction-diffusion equations

Science.gov (United States)

Zemskov, Evgeny P.; Horsthemke, Werner

2016-03-01

We investigate two-variable reaction-diffusion systems of the hyperbolic type. A linear stability analysis is performed, and the conditions for diffusion-driven instabilities are derived. Two basic types of eigenvalues, real and complex, are described. Dispersion curves for both types of eigenvalues are plotted and their behavior is analyzed. The real case is related to the Turing instability, and the complex one corresponds to the wave instability. We emphasize the interesting feature that the wave instability in the hyperbolic equations occurs in two-variable systems, whereas in the parabolic case one needs three reaction-diffusion equations.

Numeric algorithms for parallel processors computer architectures with applications to the few-groups neutron diffusion equations

International Nuclear Information System (INIS)

Zee, S.K.

1987-01-01

A numeric algorithm and an associated computer code were developed for the rapid solution of the finite-difference method representation of the few-group neutron-diffusion equations on parallel computers. Applications of the numeric algorithm on both SIMD (vector pipeline) and MIMD/SIMD (multi-CUP/vector pipeline) architectures were explored. The algorithm was successfully implemented in the two-group, 3-D neutron diffusion computer code named DIFPAR3D (DIFfusion PARallel 3-Dimension). Numerical-solution techniques used in the code include the Chebyshev polynomial acceleration technique in conjunction with the power method of outer iteration. For inner iterations, a parallel form of red-black (cyclic) line SOR with automated determination of group dependent relaxation factors and iteration numbers required to achieve specified inner iteration error tolerance is incorporated. The code employs a macroscopic depletion model with trace capability for selected fission products' transients and critical boron. In addition to this, moderator and fuel temperature feedback models are also incorporated into the DIFPAR3D code, for realistic simulation of power reactor cores. The physics models used were proven acceptable in separate benchmarking studies

Solutions for a non-Markovian diffusion equation

International Nuclear Information System (INIS)

Lenzi, E.K.; Evangelista, L.R.; Lenzi, M.K.; Ribeiro, H.V.; Oliveira, E.C. de

2010-01-01

Solutions for a non-Markovian diffusion equation are investigated. For this equation, we consider a spatial and time dependent diffusion coefficient and the presence of an absorbent term. The solutions exhibit an anomalous behavior which may be related to the solutions of fractional diffusion equations and anomalous diffusion.

Coarse-grain parallel solution of few-group neutron diffusion equations

International Nuclear Information System (INIS)

Sarsour, H.N.; Turinsky, P.J.

1991-01-01

The authors present a parallel numerical algorithm for the solution of the finite difference representation of the few-group neutron diffusion equations. The targeted architectures are multiprocessor computers with shared memory like the Cray Y-MP and the IBM 3090/VF, where coarse granularity is important for minimizing overhead. Most of the work done in the past, which attempts to exploit concurrence, has concentrated on the inner iterations of the standard outer-inner iterative strategy. This produces very fine granularity. To coarsen granularity, the authors introduce parallelism at the nested outer-inner level. The problem's spatial domain was partitioned into contiguous subregions and assigned a processor to solve for each subregion independent of all other subregions, hence, processors; i.e., each subregion is treated as a reactor core with imposed boundary conditions. Since those boundary conditions on interior surfaces, referred to as internal boundary conditions (IBCs), are not known, a third iterative level, the recomposition iterations, is introduced to communicate results between subregions

Fractal diffusion equations: Microscopic models with anomalous diffusion and its generalizations

International Nuclear Information System (INIS)

Arkhincheev, V.E.

2001-04-01

To describe the ''anomalous'' diffusion the generalized diffusion equations of fractal order are deduced from microscopic models with anomalous diffusion as Comb model and Levy flights. It is shown that two types of equations are possible: with fractional temporal and fractional spatial derivatives. The solutions of these equations are obtained and the physical sense of these fractional equations is discussed. The relation between diffusion and conductivity is studied and the well-known Einstein relation is generalized for the anomalous diffusion case. It is shown that for Levy flight diffusion the Ohm's law is not applied and the current depends on electric field in a nonlinear way due to the anomalous character of Levy flights. The results of numerical simulations, which confirmed this conclusion, are also presented. (author)

Diffusive limits for linear transport equations

International Nuclear Information System (INIS)

Pomraning, G.C.

1992-01-01

The authors show that the Hibert and Chapman-Enskog asymptotic treatments that reduce the nonlinear Boltzmann equation to the Euler and Navier-Stokes fluid equations have analogs in linear transport theory. In this linear setting, these fluid limits are described by diffusion equations, involving familiar and less familiar diffusion coefficients. Because of the linearity extant, one can carry out explicitly the initial and boundary layer analyses required to obtain asymptotically consistent initial and boundary conditions for the diffusion equations. In particular, the effects of boundary curvature and boundary condition variation along the surface can be included in the boundary layer analysis. A brief review of heuristic (nonasymptotic) diffusion description derivations is also included in our discussion

Derivation of the neutron diffusion equation

International Nuclear Information System (INIS)

Mika, J.R.; Banasiak, J.

1994-01-01

We discuss the diffusion equation as an asymptotic limit of the neutron transport equation for large scattering cross sections. We show that the classical asymptotic expansion procedure does not lead to the diffusion equation and present two modified approaches to overcome this difficulty. The effect of the initial layer is also discussed. (authors). 9 refs

Similarity solutions of reaction–diffusion equation with space- and time-dependent diffusion and reaction terms

Energy Technology Data Exchange (ETDEWEB)

Ho, C.-L. [Department of Physics, Tamkang University, Tamsui 25137, Taiwan (China); Lee, C.-C., E-mail: chieh.no27@gmail.com [Center of General Education, Aletheia University, Tamsui 25103, Taiwan (China)

2016-01-15

We consider solvability of the generalized reaction–diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction–diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction–diffusion systems. Several representative examples of exactly solvable reaction–diffusion equations are presented.

Numerical solutions of diffusive logistic equation

International Nuclear Information System (INIS)

Afrouzi, G.A.; Khademloo, S.

2007-01-01

In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains. The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the 'finite difference scheme' will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years

CTD: a computer program to solve the three dimensional multi-group diffusion equation in X, Y, Z, and triangular Z geometries

Energy Technology Data Exchange (ETDEWEB)

Fletcher, J K

1973-05-01

CTD is a computer program written in Fortran 4 to solve the multi-group diffusion theory equations in X, Y, Z and triangular Z geometries. A power print- out neutron balance and breeding gain are also produced. 4 references. (auth)

Multi-diffusive nonlinear Fokker–Planck equation

International Nuclear Information System (INIS)

Ribeiro, Mauricio S; Casas, Gabriela A; Nobre, Fernando D

2017-01-01

Nonlinear Fokker–Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker–Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker–Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker–Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution. (paper)

Amplitude equations for a sub-diffusive reaction-diffusion system

International Nuclear Information System (INIS)

Nec, Y; Nepomnyashchy, A A

2008-01-01

A sub-diffusive reaction-diffusion system with a positive definite memory operator and a nonlinear reaction term is analysed. Amplitude equations (Ginzburg-Landau type) are derived for short wave (Turing) and long wave (Hopf) bifurcation points

Fractional Diffusion Limit for Collisional Kinetic Equations

KAUST Repository

Mellet, Antoine

2010-08-20

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation. © 2010 Springer-Verlag.

The analytical solution to the 1D diffusion equation in heterogeneous media

International Nuclear Information System (INIS)

Ganapol, B.D.; Nigg, D.W.

2011-01-01

The analytical solution to the time-independent multigroup diffusion equation in heterogeneous plane cylindrical and spherical media is presented. The solution features the simplicity of the one-group formulation while addressing the complication of multigroup diffusion in a fully heterogeneous medium. Beginning with the vector form of the diffusion equation, the approach, based on straightforward mathematics, resolves a set of coupled second order ODEs. The analytical form is facilitated through matrix diagonalization of the neutron interaction matrix rendering the multigroup solution as a series of one-group solutions which, when re-assembled, gives the analytical solution. Customized Eigenmode solutions of the one-group diffusion operator then represent the homogeneous solution in a uniform spatial domain. Once the homogeneous solution is known, the particular solution naturally emerges through variation of parameters. The analytical expression is then numerically implemented through recurrence. Finally, we apply the theory to assess the accuracy of a second order finite difference scheme and to a 1D slab BWR reactor in the four-group approximation. (author)

Nodal spectrum method for solving neutron diffusion equation

International Nuclear Information System (INIS)

Sanchez, D.; Garcia, C. R.; Barros, R. C. de; Milian, D.E.

1999-01-01

Presented here is a new numerical nodal method for solving static multidimensional neutron diffusion equation in rectangular geometry. Our method is based on a spectral analysis of the nodal diffusion equations. These equations are obtained by integrating the diffusion equation in X, Y directions and then considering flat approximations for the current. These flat approximations are the only approximations that are considered in this method, as a result the numerical solutions are completely free from truncation errors. We show numerical results to illustrate the methods accuracy for coarse mesh calculations

NESTLE: Few-group neutron diffusion equation solver utilizing the nodal expansion method for eigenvalue, adjoint, fixed-source steady-state and transient problems

International Nuclear Information System (INIS)

Turinsky, P.J.; Al-Chalabi, R.M.K.; Engrand, P.; Sarsour, H.N.; Faure, F.X.; Guo, W.

1994-06-01

NESTLE is a FORTRAN77 code that solves the few-group neutron diffusion equation utilizing the Nodal Expansion Method (NEM). NESTLE can solve the eigenvalue (criticality); eigenvalue adjoint; external fixed-source steady-state; or external fixed-source. or eigenvalue initiated transient problems. The code name NESTLE originates from the multi-problem solution capability, abbreviating Nodal Eigenvalue, Steady-state, Transient, Le core Evaluator. The eigenvalue problem allows criticality searches to be completed, and the external fixed-source steady-state problem can search to achieve a specified power level. Transient problems model delayed neutrons via precursor groups. Several core properties can be input as time dependent. Two or four energy groups can be utilized, with all energy groups being thermal groups (i.e. upscatter exits) if desired. Core geometries modelled include Cartesian and Hexagonal. Three, two and one dimensional models can be utilized with various symmetries. The non-linear iterative strategy associated with the NEM method is employed. An advantage of the non-linear iterative strategy is that NSTLE can be utilized to solve either the nodal or Finite Difference Method representation of the few-group neutron diffusion equation

Fractional diffusion equation for heterogeneous medium

International Nuclear Information System (INIS)

Polo L, M. A.; Espinosa M, E. G.; Espinosa P, G.; Del Valle G, E.

2011-11-01

The asymptotic diffusion approximation for the Boltzmann (transport) equation was developed in 1950 decade in order to describe the diffusion of a particle in an isotropic medium, considers that the particles have a diffusion infinite velocity. In this work is developed a new approximation where is considered that the particles have a finite velocity, with this model is possible to describe the behavior in an anomalous medium. According with these ideas the model was obtained from the Fick law, where is considered that the temporal term of the current vector is not negligible. As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.e., in a general medium. (Author)

The generalized Airy diffusion equation

Directory of Open Access Journals (Sweden)

Frank M. Cholewinski

2003-08-01

Full Text Available Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo.

Solutions of diffusion equations in two-dimensional cylindrical geometry by series expansions

International Nuclear Information System (INIS)

Ohtani, Nobuo

1976-01-01

A solution of the multi-group multi-regional diffusion equation in two-dimensional cylindrical (rho-z) geometry is obtained in the form of a regionwise double series composed of Bessel and trigonometrical functions. The diffusion equation is multiplied by weighting functions, which satisfy the homogeneous part of the diffusion equation, and the products are integrated over the region for obtaining the equations to determine the fluxes and their normal derivatives at the region boundaries. Multiplying the diffusion equation by each function of the set used for the flux expansion, then integrating the products, the coefficients of the double series of the flux inside each region are calculated using the boundary values obtained above. Since the convergence of the series thus obtained is slow especially near the region boundaries, a method for improving the convergence has been developed. The double series of the flux is separated into two parts. The normal derivative at the region boundary of the first part is zero, and that of the second part takes the value which is obtained in the first stage of this method. The second part is replaced by a continuous function, and the flux is represented by the sum of the continuous function and the double series. A sample critical problem of a two-group two-region system is numerically studied. The results show that the present method yields very accurately the flux integrals in each region with only a small number of expansion terms. (auth.)

Symmetry properties of fractional diffusion equations

Energy Technology Data Exchange (ETDEWEB)

Gazizov, R K; Kasatkin, A A; Lukashchuk, S Yu [Ufa State Aviation Technical University, Karl Marx strausse 12, Ufa (Russian Federation)], E-mail: gazizov@mail.rb.ru, E-mail: alexei_kasatkin@mail.ru, E-mail: lsu@mail.rb.ru

2009-10-15

In this paper, nonlinear anomalous diffusion equations with time fractional derivatives (Riemann-Liouville and Caputo) of the order of 0-2 are considered. Lie point symmetries of these equations are investigated and compared. Examples of using the obtained symmetries for constructing exact solutions of the equations under consideration are presented.

Feynman-Kac equations for reaction and diffusion processes

Science.gov (United States)

Hou, Ru; Deng, Weihua

2018-04-01

This paper provides a theoretical framework for deriving the forward and backward Feynman-Kac equations for the distribution of functionals of the path of a particle undergoing both diffusion and reaction processes. Once given the diffusion type and reaction rate, a specific forward or backward Feynman-Kac equation can be obtained. The results in this paper include those for normal/anomalous diffusions and reactions with linear/nonlinear rates. Using the derived equations, we apply our findings to compute some physical (experimentally measurable) statistics, including the occupation time in half-space, the first passage time, and the occupation time in half-interval with an absorbing or reflecting boundary, for the physical system with anomalous diffusion and spontaneous evanescence.

Three-dimensional h-adaptivity for the multigroup neutron diffusion equations

KAUST Repository

Wang, Yaqi; Bangerth, Wolfgang; Ragusa, Jean

2009-01-01

diffusion equation for reactor applications. In order to follow the physics closely, energy group-dependent meshes are employed. We present a novel algorithm for assembling the terms coupling shape functions from different meshes and show how it can be made

One-velocity neutron diffusion calculations based on a two-group reactor model

Energy Technology Data Exchange (ETDEWEB)

Bingulac, S; Radanovic, L; Lazarevic, B; Matausek, M; Pop-Jordanov, J [Boris Kidric Institute of Nuclear Sciences, Vinca, Belgrade (Yugoslavia)

1965-07-01

Many processes in reactor physics are described by the energy dependent neutron diffusion equations which for many practical purposes can often be reduced to one-dimensional two-group equations. Though such two-group models are satisfactory from the standpoint of accuracy, they require rather extensive computations which are usually iterative and involve the use of digital computers. In many applications, however, and particularly in dynamic analyses, where the studies are performed on analogue computers, it is preferable to avoid iterative calculations. The usual practice in such situations is to resort to one group models, which allow the solution to be expressed analytically. However, the loss in accuracy is rather great particularly when several media of different properties are involved. This paper describes a procedure by which the solution of the two-group neutron diffusion. equations can be expressed analytically in the form which, from the computational standpoint, is as simple as the one-group model, but retains the accuracy of the two-group treatment. In describing the procedure, the case of a multi-region nuclear reactor of cylindrical geometry is treated, but the method applied and the results obtained are of more general application. Another approach in approximate solution of diffusion equations, suggested by Galanin is applicable only in special ideal cases.

Neutron transport equation - indications on homogenization and neutron diffusion

International Nuclear Information System (INIS)

Argaud, J.P.

1992-06-01

In PWR nuclear reactor, the practical study of the neutrons in the core uses diffusion equation to describe the problem. On the other hand, the most correct method to describe these neutrons is to use the Boltzmann equation, or neutron transport equation. In this paper, we give some theoretical indications to obtain a diffusion equation from the general transport equation, with some simplifying hypothesis. The work is organised as follows: (a) the most general formulations of the transport equation are presented: integro-differential equation and integral equation; (b) the theoretical approximation of this Boltzmann equation by a diffusion equation is introduced, by the way of asymptotic developments; (c) practical homogenization methods of transport equation is then presented. In particular, the relationships with some general and useful methods in neutronic are shown, and some homogenization methods in energy and space are indicated. A lot of other points of view or complements are detailed in the text or the remarks

Particle Simulation of Fractional Diffusion Equations

KAUST Repository

Allouch, Samer

2017-07-12

This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green\\'s function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.

Particle Simulation of Fractional Diffusion Equations

KAUST Repository

Allouch, Samer; Lucchesi, Marco; Maî tre, O. P. Le; Mustapha, K. A.; Knio, Omar

2017-01-01

This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green's function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.

Diffusion phenomenon for linear dissipative wave equations

KAUST Repository

Said-Houari, Belkacem

2012-01-01

In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.

The Multigroup Neutron Diffusion Equations/1 Space Dimension

Energy Technology Data Exchange (ETDEWEB)

Linde, Sven

1960-06-15

A description is given of a program for the Ferranti Mercury computer which solves the one-dimensional multigroup diffusion equations in plane, cylindrical or spherical geometry, and also approximates automatically a two-dimensional solution by separating the space variables. In section A the method of calculation is outlined and the preparation of data for two group problems is described. The spatial separation of two-dimensional equations is considered in section B. Section C covers the multigroup equations. These parts are self contained and include all information required for the use of the program. Details of the numerical methods are given in section D. Three sample problems are solved in section E. Punching and operating instructions are given in an appendix.

The Multigroup Neutron Diffusion Equations/1 Space Dimension

International Nuclear Information System (INIS)

Linde, Sven

1960-06-01

A description is given of a program for the Ferranti Mercury computer which solves the one-dimensional multigroup diffusion equations in plane, cylindrical or spherical geometry, and also approximates automatically a two-dimensional solution by separating the space variables. In section A the method of calculation is outlined and the preparation of data for two group problems is described. The spatial separation of two-dimensional equations is considered in section B. Section C covers the multigroup equations. These parts are self contained and include all information required for the use of the program. Details of the numerical methods are given in section D. Three sample problems are solved in section E. Punching and operating instructions are given in an appendix

Separating variables in two-way diffusion equations

International Nuclear Information System (INIS)

Fisch, N.J.; Kruskal, M.D.

1979-10-01

It is shown that solutions to a class of diffusion equations of the two-way type may be found by a method akin to separation of variables. The difficulty with such equations is that the boundary data must be specified partly as initial and partly as final conditions. In contrast to the one-way diffusion equation, where the solution separates only into decaying eigenfunctions, the two-way equations separate into both decaying and growing eigenfunctions. Criteria are set forth for the existence of linear eigenfunctions, which may not be found directly by separating variables. A speculation with interesting ramifications is that the growing and decaying eigenfunctions are separately complete in an appropriate half of the problem domain

Two-Dimensional Space-Time Dependent Multi-group Diffusion Equation with SLOR Method

International Nuclear Information System (INIS)

Yulianti, Y.; Su'ud, Z.; Waris, A.; Khotimah, S. N.

2010-01-01

The research of two-dimensional space-time diffusion equations with SLOR (Successive-Line Over Relaxation) has been done. SLOR method is chosen because this method is one of iterative methods that does not required to defined whole element matrix. The research is divided in two cases, homogeneous case and heterogeneous case. Homogeneous case has been inserted by step reactivity. Heterogeneous case has been inserted by step reactivity and ramp reactivity. In general, the results of simulations are agreement, even in some points there are differences.

An integral equation arising in two group neutron transport theory

International Nuclear Information System (INIS)

Cassell, J S; Williams, M M R

2003-01-01

An integral equation describing the fuel distribution necessary to maintain a flat flux in a nuclear reactor in two group transport theory is reduced to the solution of a singular integral equation. The formalism developed enables the physical aspects of the problem to be better understood and its relationship with the corresponding diffusion theory model is highlighted. The integral equation is solved by reducing it to a non-singular Fredholm equation which is then evaluated numerically

Exact analytical solutions for nonlinear reaction-diffusion equations

International Nuclear Information System (INIS)

Liu Chunping

2003-01-01

By using a direct method via the computer algebraic system of Mathematica, some exact analytical solutions to a class of nonlinear reaction-diffusion equations are presented in closed form. Subsequently, the hyperbolic function solutions and the triangular function solutions of the coupled nonlinear reaction-diffusion equations are obtained in a unified way

Speed ot travelling waves in reaction-diffusion equations

International Nuclear Information System (INIS)

Benguria, R.D.; Depassier, M.C.; Mendez, V.

2002-01-01

Reaction diffusion equations arise in several problems of population dynamics, flame propagation and others. In one dimensional cases the systems may evolve into travelling fronts. Here we concentrate on a reaction diffusion equation which arises as a simple model for chemotaxis and present results for the speed of the travelling fronts. (Author)

Iterative method for obtaining the prompt and delayed alpha-modes of the diffusion equation

International Nuclear Information System (INIS)

Singh, K.P.; Degweker, S.B.; Modak, R.S.; Singh, Kanchhi

2011-01-01

Highlights: → A method for obtaining α-modes of the neutron diffusion equation has been developed. → The difference between the prompt and delayed modes is more pronounced for the higher modes. → Prompt and delayed modes differ more in reflector region. - Abstract: Higher modes of the neutron diffusion equation are required in some applications such as second order perturbation theory, and modal kinetics. In an earlier paper we had discussed a method for computing the α-modes of the diffusion equation. The discussion assumed that all neutrons are prompt. The present paper describes an extension of the method for finding the α-modes of diffusion equation with the inclusion of delayed neutrons. Such modes are particularly suitable for expanding the time dependent flux in a reactor for describing transients in a reactor. The method is illustrated by applying it to a three dimensional heavy water reactor model problem. The problem is solved in two and three neutron energy groups and with one and six delayed neutron groups. The results show that while the delayed α-modes are similar to λ-modes they are quite different from prompt modes. The difference gets progressively larger as we go to higher modes.

Non-probabilistic solutions of imprecisely defined fractional-order diffusion equations

International Nuclear Information System (INIS)

Chakraverty, S.; Tapaswini, Smita

2014-01-01

The fractional diffusion equation is one of the most important partial differential equations (PDEs) to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 < α ≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method (ADM) symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases. (general)

A multigroup flux-limited asymptotic diffusion Fokker-Planck equation

International Nuclear Information System (INIS)

Liu Chengan

1987-01-01

A more perfrect flux-limited method is applied to combine with asymptotic diffusion theory of the radiation transpore, and the high peaked component in the scattering angle is treated with Fokker-Planck methods, thus the flux-limited asymptotic diffusion Fokker-Planck equation has been founded. Since the equation is of diffusion form, it retains the simplity and the convenience of the classical diffusion theory, and improves precision in describing radiation transport problems

Diffusion equation and non-holonomy

International Nuclear Information System (INIS)

Gomes, Luiz Carlos; Lobo, R.; Simao, F.R.A.

1980-01-01

The diffusion equation for particles in a Riemannian space subject to a single constraint is discussed. The implications of the holonomy and non-holonomy of this single constraint is also discussed. (L.C.) [pt

Study of ODE limit problems for reaction-diffusion equations

Directory of Open Access Journals (Sweden)

Jacson Simsen

2018-01-01

Full Text Available In this work we study ODE limit problems for reaction-diffusion equations for large diffusion and we study the sensitivity of nonlinear ODEs with respect to initial conditions and exponent parameters. Moreover, we prove continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction-diffusion equations with spatially variable exponents when the exponents go to 2 in \\(L^{\\infty}(\\Omega\\ and the diffusion coefficients go to infinity.

The Dirichlet problem of a conformable advection-diffusion equation

Directory of Open Access Journals (Sweden)

Avci Derya

2017-01-01

Full Text Available The fractional advection-diffusion equations are obtained from a fractional power law for the matter flux. Diffusion processes in special types of porous media which has fractal geometry can be modelled accurately by using these equations. However, the existing nonlocal fractional derivatives seem complicated and also lose some basic properties satisfied by usual derivatives. For these reasons, local fractional calculus has recently been emerged to simplify the complexities of fractional models defined by nonlocal fractional operators. In this work, the conformable, a local, well-behaved and limit-based definition, is used to obtain a local generalized form of advection-diffusion equation. In addition, this study is devoted to give a local generalized description to the combination of diffusive flux governed by Fick’s law and the advection flux associated with the velocity field. As a result, the constitutive conformable advection-diffusion equation can be easily achieved. A Dirichlet problem for conformable advection-diffusion equation is derived by applying fractional Laplace transform with respect to time t and finite sin-Fourier transform with respect to spatial coordinate x. Two illustrative examples are presented to show the behaviours of this new local generalized model. The dependence of the solution on the fractional order of conformable derivative and the changing values of problem parameters are validated using graphics held by MATLcodes.

Reaction diffusion equations with boundary degeneracy

Directory of Open Access Journals (Sweden)

Huashui Zhan

2016-03-01

Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

Science.gov (United States)

Abd-El-Malek, Mina B.; Kassem, Magda M.; Meky, Mohammed L. M.

2002-03-01

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skin-like membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.

Size dependent diffusive parameters and tensorial diffusion equations in neutronic models for optically small nuclear systems

International Nuclear Information System (INIS)

Premuda, F.

1983-01-01

Two lines in improved neutron diffusion theory extending the efficiency of finite-difference diffusion codes to the field of optically small systems, are here reviewed. The firs involves the nodal solution for tensorial diffusion equation in slab geometry and tensorial formulation in parallelepiped and cylindrical gemometry; the dependence of critical eigenvalue from small slab thicknesses is also analitically investigated and finally a regularized tensorial diffusion equation is derived for slab. The other line refer to diffusion models formally unchanged with respect to the classical one, but where new size-dependent RTGB definitions for diffusion parameters are adopted, requiring that they allow to reproduce, in diffusion approach, the terms of neutron transport global balance; the trascendental equation for the buckling, arising in slab, sphere and parallelepiped geometry from the above requirement, are reported and the sizedependence of the new diffusion coefficient and extrapolated end point is investigated

Unconditionally stable diffusion-acceleration of the transport equation

International Nuclear Information System (INIS)

Larsen, E.W.

1982-01-01

The standard iterative procedure for solving fixed-source discrete-ordinates problems converges very slowly for problems in optically large regions with scattering ratios c near unity. The diffusion-synthetic acceleration method has been proposed to make use of the fact that for this class of problems the diffusion equation is often an accurate approximation to the transport equation. However, stability difficulties have historically hampered the implementation of this method for general transport differencing schemes. In this article we discuss a recently developed procedure for obtaining unconditionally stable diffusion-synthetic acceleration methods for various transport differencing schemes. We motivate the analysis by first discussing the exact transport equation; then we illustrate the procedure by deriving a new stable acceleration method for the linear discontinuous transport differencing scheme. We also provide some numerical results

Unconditionally stable diffusion-acceleration of the transport equation

International Nuclear Information System (INIS)

Larson, E.W.

1982-01-01

The standard iterative procedure for solving fixed-source discrete-ordinates problems converges very slowly for problems in optically thick regions with scattering ratios c near unity. The diffusion-synthetic acceleration method has been proposed to make use of the fact that for this class of problems, the diffusion equation is often an accurate approximation to the transport equation. However, stability difficulties have historically hampered the implementation of this method for general transport differencing schemes. In this article we discuss a recently developed procedure for obtaining unconditionally stable diffusion-synthetic acceleration methods for various transport differencing schemes. We motivate the analysis by first discussing the exact transport equation; then we illustrate the procedure by deriving a new stable acceleration method for the linear discontinuous transport differencing scheme. We also provide some numerical results

Hybrid diffusion-P3 equation in N-layered turbid media: steady-state domain.

Science.gov (United States)

Shi, Zhenzhi; Zhao, Huijuan; Xu, Kexin

2011-10-01

This paper discusses light propagation in N-layered turbid media. The hybrid diffusion-P3 equation is solved for an N-layered finite or infinite turbid medium in the steady-state domain for one point source using the extrapolated boundary condition. The Fourier transform formalism is applied to derive the analytical solutions of the fluence rate in Fourier space. Two inverse Fourier transform methods are developed to calculate the fluence rate in real space. In addition, the solutions of the hybrid diffusion-P3 equation are compared to the solutions of the diffusion equation and the Monte Carlo simulation. For the case of small absorption coefficients, the solutions of the N-layered diffusion equation and hybrid diffusion-P3 equation are almost equivalent and are in agreement with the Monte Carlo simulation. For the case of large absorption coefficients, the model of the hybrid diffusion-P3 equation is more precise than that of the diffusion equation. In conclusion, the model of the hybrid diffusion-P3 equation can replace the diffusion equation for modeling light propagation in the N-layered turbid media for a wide range of absorption coefficients.

Vectorized and multitasked solution of the few-group neutron diffusion equations

International Nuclear Information System (INIS)

Zee, S.K.; Turinsky, P.J.; Shayer, Z.

1989-01-01

A numerical algorithm with parallelism was used to solve the two-group, multidimensional neutron diffusion equations on computers characterized by shared memory, vector pipeline, and multi-CPU architecture features. Specifically, solutions were obtained on the Cray X/MP-48, the IBM-3090 with vector facilities, and the FPS-164. The material-centered mesh finite difference method approximation and outer-inner iteration method were employed. Parallelism was introduced in the inner iterations using the cyclic line successive overrelaxation iterative method and solving in parallel across lines. The outer iterations were completed using the Chebyshev semi-iterative method that allows parallelism to be introduced in both space and energy groups. For the three-dimensional model, power, soluble boron, and transient fission product feedbacks were included. Concentrating on the pressurized water reactor (PWR), the thermal-hydraulic calculation of moderator density assumed single-phase flow and a closed flow channel, allowing parallelism to be introduced in the solution across the radial plane. Using a pinwise detail, quarter-core model of a typical PWR in cycle 1, for the two-dimensional model without feedback the measured million floating point operations per second (MFLOPS)/vector speedups were 83/11.7. 18/2.2, and 2.4/5.6 on the Cray, IBM, and FPS without multitasking, respectively. Lower performance was observed with a coarser mesh, i.e., shorter vector length, due to vector pipeline start-up. For an 18 x 18 x 30 (x-y-z) three-dimensional model with feedback of the same core, MFLOPS/vector speedups of --61/6.7 and an execution time of 0.8 CPU seconds on the Cray without multitasking were measured. Finally, using two CPUs and the vector pipelines of the Cray, a multitasking efficiency of 81% was noted for the three-dimensional model

On an analytical evaluation of the flux and dominant eigenvalue problem for the steady state multi-group multi-layer neutron diffusion equation

Energy Technology Data Exchange (ETDEWEB)

Ceolin, Celina; Schramm, Marcelo; Bodmann, Bardo Ernst Josef; Vilhena, Marco Tullio Mena Barreto de [Universidade Federal do Rio Grande do Sul, Porto Alegre (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Bogado Leite, Sergio de Queiroz [Comissao Nacional de Energia Nuclear, Rio de Janeiro (Brazil)

2014-11-15

In this work the authors solved the steady state neutron diffusion equation for a multi-layer slab assuming the multi-group energy model. The method to solve the equation system is based on an expansion in Taylor Series resulting in an analytical expression. The results obtained can be used as initial condition for neutron space kinetics problems. The neutron scalar flux was expanded in a power series, and the coefficients were found by using the ordinary differential equation and the boundary and interface conditions. The effective multiplication factor k was evaluated using the power method. We divided the domain into several slabs to guarantee the convergence with a low truncation order. We present the formalism together with some numerical simulations.

Diffusion-equation representations of landform evolution in the simplest circumstances: Appendix C

Science.gov (United States)

Hanks, Thomas C.

2009-01-01

The diffusion equation is one of the three great partial differential equations of classical physics. It describes the flow or diffusion of heat in the presence of temperature gradients, fluid flow in porous media in the presence of pressure gradients, and the diffusion of molecules in the presence of chemical gradients. [The other two equations are the wave equation, which describes the propagation of electromagnetic waves (including light), acoustic (sound) waves, and elastic (seismic) waves radiated from earthquakes; and LaPlace’s equation, which describes the behavior of electric, gravitational, and fluid potentials, all part of potential field theory. The diffusion equation reduces to LaPlace’s equation at steady state, when the field of interest does not depend on t. Poisson’s equation is LaPlace’s equation with a source term.

Diffusion equations and the time evolution of foreign exchange rates

Energy Technology Data Exchange (ETDEWEB)

Figueiredo, Annibal; Castro, Marcio T. de [Institute of Physics, Universidade de Brasília, Brasília DF 70910-900 (Brazil); Fonseca, Regina C.B. da [Department of Mathematics, Instituto Federal de Goiás, Goiânia GO 74055-110 (Brazil); Gleria, Iram, E-mail: iram@fis.ufal.br [Institute of Physics, Federal University of Alagoas, Brazil, Maceió AL 57072-900 (Brazil)

2013-10-01

We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers–Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.

Diffusion equations and the time evolution of foreign exchange rates

Science.gov (United States)

Figueiredo, Annibal; de Castro, Marcio T.; da Fonseca, Regina C. B.; Gleria, Iram

2013-10-01

We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers-Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.

Diffusion equations and the time evolution of foreign exchange rates

International Nuclear Information System (INIS)

Figueiredo, Annibal; Castro, Marcio T. de; Fonseca, Regina C.B. da; Gleria, Iram

2013-01-01

We investigate which type of diffusion equation is most appropriate to describe the time evolution of foreign exchange rates. We modify the geometric diffusion model assuming a non-exponential time evolution and the stochastic term is the sum of a Wiener noise and a jump process. We find the resulting diffusion equation to obey the Kramers–Moyal equation. Analytical solutions are obtained using the characteristic function formalism and compared with empirical data. The analysis focus on the first four central moments considering the returns of foreign exchange rate. It is shown that the proposed model offers a good improvement over the classical geometric diffusion model.

Cellular automata for spatiotemporal pattern formation from reaction–diffusion partial differential equations

International Nuclear Information System (INIS)

Ohmori, Shousuke; Yamazaki, Yoshihiro

2016-01-01

Ultradiscrete equations are derived from a set of reaction–diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction–diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction–diffusion equations. (author)

On Solution of a Fractional Diffusion Equation by Homotopy Transform Method

International Nuclear Information System (INIS)

Salah, A.; Hassan, S.S.A.

2012-01-01

The homotopy analysis transform method (HATM) is applied in this work in order to find the analytical solution of fractional diffusion equations (FDE). These equations are obtained from standard diffusion equations by replacing a second-order space derivative by a fractional derivative of order α and a first order time derivative by a fractional derivative. Furthermore, some examples are given. Numerical results show that the homotopy analysis transform method is easy to implement and accurate when applied to a fractional diffusion equations.

On the integrability of the generalized Fisher-type nonlinear diffusion equations

International Nuclear Information System (INIS)

Wang Dengshan; Zhang Zhifei

2009-01-01

In this paper, the geometric integrability and Lax integrability of the generalized Fisher-type nonlinear diffusion equations with modified diffusion in (1+1) and (2+1) dimensions are studied by the pseudo-spherical surface geometry method and prolongation technique. It is shown that the (1+1)-dimensional Fisher-type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo-spherical surface of constant curvature -1 only for m = 2, and the generalized Fisher-type nonlinear diffusion equations in (1+1) and (2+1) dimensions are Lax integrable only for m = 2. This paper extends the results in Bindu et al 2001 (J. Phys. A: Math. Gen. 34 L689) and further provides the integrability information of (1+1)- and (2+1)-dimensional Fisher-type nonlinear diffusion equations for m = 2

Finite difference solution of the time dependent neutron group diffusion equations

International Nuclear Information System (INIS)

Hendricks, J.S.; Henry, A.F.

1975-08-01

In this thesis two unrelated topics of reactor physics are examined: the prompt jump approximation and alternating direction checkerboard methods. In the prompt jump approximation it is assumed that the prompt and delayed neutrons in a nuclear reactor may be described mathematically as being instantaneously in equilibrium with each other. This approximation is applied to the spatially dependent neutron diffusion theory reactor kinetics model. Alternating direction checkerboard methods are a family of finite difference alternating direction methods which may be used to solve the multigroup, multidimension, time-dependent neutron diffusion equations. The reactor mesh grid is not swept line by line or point by point as in implicit or explicit alternating direction methods; instead, the reactor mesh grid may be thought of as a checkerboard in which all the ''red squares'' and '' black squares'' are treated successively. Two members of this family of methods, the ADC and NSADC methods, are at least as good as other alternating direction methods. It has been found that the accuracy of implicit and explicit alternating direction methods can be greatly improved by the application of an exponential transformation. This transformation is incompatible with checkerboard methods. Therefore, a new formulation of the exponential transformation has been developed which is compatible with checkerboard methods and at least as good as the former transformation for other alternating direction methods

Linear fractional diffusion-wave equation for scientists and engineers

CERN Document Server

Povstenko, Yuriy

2015-01-01

This book systematically presents solutions to the linear time-fractional diffusion-wave equation. It introduces the integral transform technique and discusses the properties of the Mittag-Leffler, Wright, and Mainardi functions that appear in the solutions. The time-nonlocal dependence between the flux and the gradient of the transported quantity with the “long-tail” power kernel results in the time-fractional diffusion-wave equation with the Caputo fractional derivative. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s laws are considered and different kinds of boundary conditions for this equation are discussed (Dirichlet, Neumann, Robin, perfect contact). The book provides solutions to the fractional diffusion-wave equation with one, two and three space variables in Cartesian, cylindrical and spherical coordinates. The respective sections of the book can be used for university courses on fractional calculus, heat and mass transfer, transport processes in porous media and ...

Intermittent Motion, Nonlinear Diffusion Equation and Tsallis Formalism

Directory of Open Access Journals (Sweden)

Ervin K. Lenzi

2017-01-01

Full Text Available We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching them from the resting to the movement. The results show that in the asymptotic limit of small and long times, the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for intermediate times depends on the rates present in the reaction terms. In this scenario, we show that, in the asymptotic limits, the distributions for this process are given by in terms of power laws which may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a situation characterized by different diffusive regimes, which emerges when the diffusive term is a mixing of linear and nonlinear terms.

Fractional Number Operator and Associated Fractional Diffusion Equations

Science.gov (United States)

Rguigui, Hafedh

2018-03-01

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.

Differential constraints and exact solutions of nonlinear diffusion equations

International Nuclear Information System (INIS)

Kaptsov, Oleg V; Verevkin, Igor V

2003-01-01

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries

Solution to the Diffusion equation for multi groups in X Y geometry using Linear Perturbation theory

International Nuclear Information System (INIS)

Mugica R, C.A.

2004-01-01

Diverse methods exist to solve numerically the neutron diffusion equation for several energy groups in stationary state among those that highlight those of finite elements. In this work the numerical solution of this equation is presented using Raviart-Thomas nodal methods type finite element, the RT0 and RT1, in combination with iterative techniques that allow to obtain the approached solution in a quick form. Nevertheless the above mentioned, the precision of a method is intimately bound to the dimension of the approach space by cell, 5 for the case RT0 and 12 for the RT1, and/or to the mesh refinement, that makes the order of the problem of own value to solve to grow considerably. By this way if it wants to know an acceptable approach to the value of the effective multiplication factor of the system when this it has experimented a small perturbation it was appeal to the Linear perturbation theory with which is possible to determine it starting from the neutron flow and of the effective multiplication factor of the not perturbed case. Results are presented for a reference problem in which a perturbation is introduced in an assemble that simulates changes in the control bar. (Author)

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

Science.gov (United States)

Frank, T. D.

2008-02-01

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.

Traveling waves and the renormalization group improvedBalitsky-Kovchegov equation

Energy Technology Data Exchange (ETDEWEB)

Enberg, Rikard

2006-12-01

I study the incorporation of renormalization group (RG)improved BFKL kernels in the Balitsky-Kovchegov (BK) equation whichdescribes parton saturation. The RG improvement takes into accountimportant parts of the next-to-leading and higher order logarithmiccorrections to the kernel. The traveling wave front method for analyzingthe BK equation is generalized to deal with RG-resummed kernels,restricting to the interesting case of fixed QCD coupling. The resultsshow that the higher order corrections suppress the rapid increase of thesaturation scale with increasing rapidity. I also perform a "diffusive"differential equation approximation, which illustrates that someimportant qualitative properties of the kernel change when including RGcorrections.

A nonlinear equation for ionic diffusion in a strong binary electrolyte

Science.gov (United States)

Ghosal, Sandip; Chen, Zhen

2010-01-01

The problem of the one-dimensional electro-diffusion of ions in a strong binary electrolyte is considered. The mathematical description, known as the Poisson–Nernst–Planck (PNP) system, consists of a diffusion equation for each species augmented by transport owing to a self-consistent electrostatic field determined by the Poisson equation. This description is also relevant to other important problems in physics, such as electron and hole diffusion across semiconductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here, we derive a more general theory by exploiting the ratio of the Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear partial differential equation that provides a better approximation than the classical linear equation for ambipolar diffusion, but reduces to it in the appropriate limit. PMID:21818176

Solution of the two dimensional diffusion and transport equations in a rectangular lattice with an elliptical fuel element using Fourier transform methods: One and two group cases

International Nuclear Information System (INIS)

Williams, M.M.R.; Hall, S.K.; Eaton, M.D.

2014-01-01

Highlights: • A rectangular reactor cell with an elliptical fuel element. • Solution of transport and diffusion equations by Fourier expansion. • Numerical examples showing convergence. • Two group cell problems. - Abstract: A method for solving the diffusion and transport equations in a rectangular lattice cell with an elliptical fuel element has been developed using a Fourier expansion of the neutron flux. The method is applied to a one group model with a source in the moderator. The cell flux is obtained and also the associated disadvantage factor. In addition to the one speed case, we also consider the two group equations in the cell which now become an eigenvalue problem for the lattice multiplication factor. The method of solution relies upon an efficient procedure to solve a large set of simultaneous linear equations and for this we use the IMSL library routines. Our method is compared with the results from a finite element code. The main drawback of the problem arises from the very large number of terms required in the Fourier series which taxes the storage and speed of the computer. Nevertheless, useful solutions are obtained in geometries that would normally require the use of finite element or analogous methods, for this reason the Fourier method is useful for comparison with that type of numerical approach. Extension of the method to more intricate fuel shapes, such as stars and cruciforms as well as superpositions of these, is possible

Numerical solution of a reaction-diffusion equation

International Nuclear Information System (INIS)

Moyano, Edgardo A.; Scarpettini, Alberto F.

2000-01-01

The purpose of the present work to continue the observations and the numerical experiences on a reaction-diffusion model, that is a simplified form of the neutronic flux equation. The model is parabolic, nonlinear, with Dirichlet boundary conditions. The purpose is to approximate non trivial solutions, asymptotically stables for t → ∞, that is solutions that tend to the elliptic problem, in the Lyapunov sense. It belongs to the so-called reaction-diffusion equations of semi linear kind, that is, linear equations in the heat operator and they have a nonlinear reaction function, in this case f (u, a, b) = u (a - b u), being u concentration, a and b parameters. The study of the incidence of these parameters take an interest to the neutronic flux physics. So that we search non trivial, positive and bounded solutions. The used algorithm is based on the concept of monotone and ordered sequences, and on the existence theorem of Amann and Sattinger. (author)

Derivation of a volume-averaged neutron diffusion equation; Atomos para el desarrollo de Mexico

Energy Technology Data Exchange (ETDEWEB)

Vazquez R, R.; Espinosa P, G. [UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, Mexico D.F. 09340 (Mexico); Morales S, Jaime B. [UNAM, Laboratorio de Analisis en Ingenieria de Reactores Nucleares, Paseo Cuauhnahuac 8532, Jiutepec, Morelos 62550 (Mexico)]. e-mail: rvr@xanum.uam.mx

2008-07-01

This paper presents a general theoretical analysis of the problem of neutron motion in a nuclear reactor, where large variations on neutron cross sections normally preclude the use of the classical neutron diffusion equation. A volume-averaged neutron diffusion equation is derived which includes correction terms to diffusion and nuclear reaction effects. A method is presented to determine closure-relationships for the volume-averaged neutron diffusion equation (e.g., effective neutron diffusivity). In order to describe the distribution of neutrons in a highly heterogeneous configuration, it was necessary to extend the classical neutron diffusion equation. Thus, the volume averaged diffusion equation include two corrections factor: the first correction is related with the absorption process of the neutron and the second correction is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. (Author)

Three-dimensional h-adaptivity for the multigroup neutron diffusion equations

KAUST Repository

Wang, Yaqi

2009-04-01

Adaptive mesh refinement (AMR) has been shown to allow solving partial differential equations to significantly higher accuracy at reduced numerical cost. This paper presents a state-of-the-art AMR algorithm applied to the multigroup neutron diffusion equation for reactor applications. In order to follow the physics closely, energy group-dependent meshes are employed. We present a novel algorithm for assembling the terms coupling shape functions from different meshes and show how it can be made efficient by deriving all meshes from a common coarse mesh by hierarchic refinement. Our methods are formulated using conforming finite elements of any order, for any number of energy groups. The spatial error distribution is assessed with a generalization of an error estimator originally derived for the Poisson equation. Our implementation of this algorithm is based on the widely used Open Source adaptive finite element library deal.II and is made available as part of this library\\'s extensively documented tutorial. We illustrate our methods with results for 2-D and 3-D reactor simulations using 2 and 7 energy groups, and using conforming finite elements of polynomial degree up to 6. © 2008 Elsevier Ltd. All rights reserved.

A moving mesh finite difference method for equilibrium radiation diffusion equations

Energy Technology Data Exchange (ETDEWEB)

Yang, Xiaobo, E-mail: xwindyb@126.com [Department of Mathematics, College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221116 (China); Huang, Weizhang, E-mail: whuang@ku.edu [Department of Mathematics, University of Kansas, Lawrence, KS 66045 (United States); Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn [School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005 (China)

2015-10-01

An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation.

A moving mesh finite difference method for equilibrium radiation diffusion equations

International Nuclear Information System (INIS)

Yang, Xiaobo; Huang, Weizhang; Qiu, Jianxian

2015-01-01

An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed. Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multi-material, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation

Numerical Solutions for Convection-Diffusion Equation through Non-Polynomial Spline

Directory of Open Access Journals (Sweden)

Ravi Kanth A.S.V.

2016-01-01

Full Text Available In this paper, numerical solutions for convection-diffusion equation via non-polynomial splines are studied. We purpose an implicit method based on non-polynomial spline functions for solving the convection-diffusion equation. The method is proven to be unconditionally stable by using Von Neumann technique. Numerical results are illustrated to demonstrate the efficiency and stability of the purposed method.

An inverse diffusivity problem for the helium production–diffusion equation

International Nuclear Information System (INIS)

Bao, Gang; Xu, Xiang

2012-01-01

Thermochronology is a technique for the extraction of information about the thermal history of rocks. Such information is crucial for determining the depth below the surface at which rocks were located at a given time (Bao G et al 2011 Commun. Comput. Phys. 9 129). Mathematically, extracting the time–temperature path can be formulated as an inverse diffusivity problem for the helium production–diffusion equation which is the underlying process of thermochronology. In this paper, to reconstruct the diffusivity which depends on space only and accounts for the structural information of rocks, a local Hölder conditional stability is obtained by a Carleman estimate. A uniqueness result is also proven for extracting the thermal history, i.e. identifying the time-dependant part of the diffusion coefficient, provided that it is analytical with respect to time. Numerical examples are presented to illustrate the validity and effectiveness of the proposed regularization scheme. (paper)

Nonlinear analysis of a reaction-diffusion system: Amplitude equations

Energy Technology Data Exchange (ETDEWEB)

Zemskov, E. P., E-mail: zemskov@ccas.ru [Russian Academy of Sciences, Dorodnicyn Computing Center (Russian Federation)

2012-10-15

A reaction-diffusion system with a nonlinear diffusion term is considered. Based on nonlinear analysis, the amplitude equations are obtained in the cases of the Hopf and Turing instabilities in the system. Turing pattern-forming regions in the parameter space are determined for supercritical and subcritical instabilities in a two-component reaction-diffusion system.

Dynamical symmetries of semi-linear Schrodinger and diffusion equations

International Nuclear Information System (INIS)

Stoimenov, Stoimen; Henkel, Malte

2005-01-01

Conditional and Lie symmetries of semi-linear 1D Schrodinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrodinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf 3 ) C . We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of (conf 3 ) C are classified and the complete list of conditionally invariant semi-linear Schrodinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed

Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

KAUST Repository

Carrillo, J. A.; Desvillettes, L.; Fellner, K.

2009-01-01

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters. © Taylor & Francis Group, LLC.

Rigorous Derivation of a Nonlinear Diffusion Equation as Fast-Reaction Limit of a Continuous Coagulation-Fragmentation Model with Diffusion

KAUST Repository

Carrillo, J. A.

2009-10-30

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [5], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters. © Taylor & Francis Group, LLC.

Exponential attractors for a nonclassical diffusion equation

Directory of Open Access Journals (Sweden)

Qiaozhen Ma

2009-01-01

Full Text Available In this article, we prove the existence of exponential attractors for a nonclassical diffusion equation in ${H^{2}(Omega}cap{H}^{1}_{0}(Omega$ when the space dimension is less than 4.

Similarity Solutions for Multiterm Time-Fractional Diffusion Equation

OpenAIRE

Elsaid, A.; Abdel Latif, M. S.; Maneea, M.

2016-01-01

Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval (0,1] and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on ...

HEXAN - a hexagonal nodal code for solving the diffusion equation

International Nuclear Information System (INIS)

Makai, M.

1982-07-01

This report describes the theory of and provides a user's manual for the HEXAN program, which is a nodal program for the solution of the few-group diffusion equation in hexagonal geometry. Based upon symmetry considerations, the theory provides an analytical solution in a homogeneous node. WWER and HTGR test problem solutions are presented. The equivalence of the finite-difference scheme and the response matrix method is proven. The properties of a symmetric node's response matrix are investigated. (author)

Global dynamics of a nonlocal delayed reaction-diffusion equation on a half plane

Science.gov (United States)

Hu, Wenjie; Duan, Yueliang

2018-04-01

We consider a delayed reaction-diffusion equation with spatial nonlocality on a half plane that describes population dynamics of a two-stage species living in a semi-infinite environment. A Neumann boundary condition is imposed accounting for an isolated domain. To describe the global dynamics, we first establish some a priori estimate for nontrivial solutions after investigating asymptotic properties of the nonlocal delayed effect and the diffusion operator, which enables us to show the permanence of the equation with respect to the compact open topology. We then employ standard dynamical system arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated by the diffusive Nicholson's blowfly equation and the diffusive Mackey-Glass equation.

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

Science.gov (United States)

Indekeu, Joseph O.; Smets, Ruben

2017-08-01

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

International Nuclear Information System (INIS)

Indekeu, Joseph O; Smets, Ruben

2017-01-01

Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically. (paper)

Systematic homogenization and self-consistent flux and pin power reconstruction for nodal diffusion methods. 1: Diffusion equation-based theory

International Nuclear Information System (INIS)

Zhang, H.; Rizwan-uddin; Dorning, J.J.

1995-01-01

A diffusion equation-based systematic homogenization theory and a self-consistent dehomogenization theory for fuel assemblies have been developed for use with coarse-mesh nodal diffusion calculations of light water reactors. The theoretical development is based on a multiple-scales asymptotic expansion carried out through second order in a small parameter, the ratio of the average diffusion length to the reactor characteristic dimension. By starting from the neutron diffusion equation for a three-dimensional heterogeneous medium and introducing two spatial scales, the development systematically yields an assembly-homogenized global diffusion equation with self-consistent expressions for the assembly-homogenized diffusion tensor elements and cross sections and assembly-surface-flux discontinuity factors. The rector eigenvalue 1/k eff is shown to be obtained to the second order in the small parameter, and the heterogeneous diffusion theory flux is shown to be obtained to leading order in that parameter. The latter of these two results provides a natural procedure for the reconstruction of the local fluxes and the determination of pin powers, even though homogenized assemblies are used in the global nodal diffusion calculation

Utilization of Weibull equation to obtain soil-water diffusivity in horizontal infiltration

International Nuclear Information System (INIS)

Guerrini, I.A.

1982-06-01

Water movement was studied in horizontal infiltration experiments using laboratory columns of air-dry and homogeneous soil to obtain a simple and suitable equation for soil-water diffusivity. Many water content profiles for each one of the ten soil columns utilized were obtained through gamma-ray attenuation technique using a 137 Cs source. During the measurement of a particular water content profile, the soil column was held in the same position in order to measure changes in time and so to reduce the errors in water content determination. The Weibull equation utilized was excellent in fitting water content profiles experimental data. The use of an analytical function for ν, the Boltzmann variable, according to Weibull model, allowed to obtain a simple equation for soil water diffusivity. Comparisons among the equation here obtained for diffusivity and others solutions found in literature were made, and the unsuitability of a simple exponential variation of diffusivity with water content for the full range of the latter was shown. The necessity of admitting the time dependency for diffusivity was confirmed and also the possibility fixing that dependency on a well known value extended to generalized soil water infiltration studies was found. Finally, it was shown that the soil water diffusivity function given by the equation here proposed can be obtained just by the analysis of the wetting front advance as a function of time. (Author) [pt

Fractional Diffusion Limit for Collisional Kinetic Equations

KAUST Repository

Mellet, Antoine; Mischler, Sté phane; Mouhot, Clé ment

2010-01-01

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a

Semianalytic Solution of Space-Time Fractional Diffusion Equation

Directory of Open Access Journals (Sweden)

A. Elsaid

2016-01-01

Full Text Available We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM. Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.

Ionic diffusion through confined geometries: from Langevin equations to partial differential equations

International Nuclear Information System (INIS)

Nadler, Boaz; Schuss, Zeev; Singer, Amit; Eisenberg, R S

2004-01-01

Ionic diffusion through and near small domains is of considerable importance in molecular biophysics in applications such as permeation through protein channels and diffusion near the charged active sites of macromolecules. The motion of the ions in these settings depends on the specific nanoscale geometry and charge distribution in and near the domain, so standard continuum type approaches have obvious limitations. The standard machinery of equilibrium statistical mechanics includes microscopic details, but is also not applicable, because these systems are usually not in equilibrium due to concentration gradients and to the presence of an external applied potential, which drive a non-vanishing stationary current through the system. We present a stochastic molecular model for the diffusive motion of interacting particles in an external field of force and a derivation of effective partial differential equations and their boundary conditions that describe the stationary non-equilibrium system. The interactions can include electrostatic, Lennard-Jones and other pairwise forces. The analysis yields a new type of Poisson-Nernst-Planck equations, that involves conditional and unconditional charge densities and potentials. The conditional charge densities are the non-equilibrium analogues of the well studied pair correlation functions of equilibrium statistical physics. Our proposed theory is an extension of equilibrium statistical mechanics of simple fluids to stationary non-equilibrium problems. The proposed system of equations differs from the standard Poisson-Nernst-Planck system in two important aspects. First, the force term depends on conditional densities and thus on the finite size of ions, and second, it contains the dielectric boundary force on a discrete ion near dielectric interfaces. Recently, various authors have shown that both of these terms are important for diffusion through confined geometries in the context of ion channels

A Bloch-Torrey Equation for Diffusion in a Deforming Media

International Nuclear Information System (INIS)

Rohmer, Damien; Gullberg, Grant T.

2006-01-01

Diffusion Tensor Magnetic Resonance Imaging (DTMRI)technique enables the measurement of diffusion parameters and therefore, informs on the structure of the biological tissue. This technique is applied with success to the static organs such as brain. However, the diffusion measurement on the dynamically deformable organs such as the in-vivo heart is a complex problem that has however a great potential in the measurement of cardiac health. In order to understand the behavior of the Magnetic Resonance (MR)signal in a deforming media, the Bloch-Torrey equation that leads the MR behavior is expressed in general curvilinear coordinates. These coordinates enable to follow the heart geometry and deformations through time. The equation is finally discredited and presented in a numerical formulation using implicit methods, in order to get a stable scheme that can be applied to any smooth deformations. Diffusion process enables the link between the macroscopic behavior of molecules and the microscopic structure in which they evolve. The measurement of diffusion in biological tissues is therefore of major importance in understanding the complex underlying structure that cannot be studied directly. The Diffusion Tensor Magnetic Resonance Imaging(DTMRI) technique enables the measurement of diffusion parameters and therefore provides information on the structure of the biological tissue. This technique has been applied with success to static organs such as the brain. However, diffusion measurement of dynamically deformable organs such as the in-vivo heart remains a complex problem, which holds great potential in determining cardiac health. In order to understand the behavior of the magnetic resonance (MR) signal in a deforming media, the Bloch-Torrey equation that defines the MR behavior is expressed in general curvilinear coordinates. These coordinates enable us to follow the heart geometry and deformations through time. The equation is finally discredited and presented in a

Coarse-mesh discretized low-order quasi-diffusion equations for subregion averaged scalar fluxes

International Nuclear Information System (INIS)

Anistratov, D. Y.

2004-01-01

In this paper we develop homogenization procedure and discretization for the low-order quasi-diffusion equations on coarse grids for core-level reactor calculations. The system of discretized equations of the proposed method is formulated in terms of the subregion averaged group scalar fluxes. The coarse-mesh solution is consistent with a given fine-mesh discretization of the transport equation in the sense that it preserves a set of average values of the fine-mesh transport scalar flux over subregions of coarse-mesh cells as well as the surface currents, and eigenvalue. The developed method generates numerical solution that mimics the large-scale behavior of the transport solution within assemblies. (authors)

A family of analytical solutions of a nonlinear diffusion-convection equation

Science.gov (United States)

Hayek, Mohamed

2018-01-01

Despite its popularity in many engineering fields, the nonlinear diffusion-convection equation has no general analytical solutions. This work presents a family of closed-form analytical traveling wave solutions for the nonlinear diffusion-convection equation with power law nonlinearities. This kind of equations typically appears in nonlinear problems of flow and transport in porous media. The solutions that are addressed are simple and fully analytical. Three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient (increasing, decreasing or constant). It has shown that the structure of the traveling wave solution is strongly related to the diffusion term. The main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters. The proposed closed-form solutions are simple to use, do not require any numerical implementation, and may be implemented in a simple spreadsheet. The analytical expressions are also useful to mathematically analyze the structure and properties of the solutions.

Diffusion phenomenon for linear dissipative wave equations in an exterior domain

Science.gov (United States)

Ikehata, Ryo

Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.

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.

On the numerical solution of the neutron fractional diffusion equation

International Nuclear Information System (INIS)

Maleki Moghaddam, Nader; Afarideh, Hossein; Espinosa-Paredes, Gilberto

2014-01-01

Highlights: • The new version of neutron diffusion equation which established on the fractional derivatives is presented. • The Neutron Fractional Diffusion Equation (NFDE) is solved in the finite differences frame. • NFDE is solved using shifted Grünwald-Letnikov definition of fractional operators. • The results show that “K eff ” strongly depends on the order of fractional derivative. - Abstract: In order to core calculation in the nuclear reactors there is a new version of neutron diffusion equation which is established on the fractional partial derivatives, named Neutron Fractional Diffusion Equation (NFDE). In the NFDE model, neutron flux in each zone depends directly on the all previous zones (not only on the nearest neighbors). Under this circumstance, it can be said that the NFDE has the space history. We have developed a one-dimension code, NFDE-1D, which can simulate the reactor core using arbitrary exponent of differential operators. In this work a numerical solution of the NFDE is presented using shifted Grünwald-Letnikov definition of fractional derivative in finite differences frame. The model is validated with some numerical experiments where different orders of fractional derivative are considered (e.g. 0.999, 0.98, 0.96, and 0.94). The results show that the effective multiplication factor (K eff ) depends strongly on the order of fractional derivative

Analytical solution to the hybrid diffusion-transport equation

International Nuclear Information System (INIS)

Nanneh, M.M.; Williams, M.M.R.

1986-01-01

A special integral equation was derived in previous work using a hybrid diffusion-transport theory method for calculating the flux distribution in slab lattices. In this paper an analytical solution of this equation has been carried out on a finite reactor lattice. The analytical results of disadvantage factors are shown to be accurate in comparison with the numerical results and accurate transport theory calculations. (author)

New variable separation approach: application to nonlinear diffusion equations

International Nuclear Information System (INIS)

Zhang Shunli; Lou, S Y; Qu Changzheng

2003-01-01

The concept of the derivative-dependent functional separable solution (DDFSS), as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the DDFSS is obtained and some exact solutions to the resulting equations are described

Similarity Solutions for Multiterm Time-Fractional Diffusion Equation

Directory of Open Access Journals (Sweden)

A. Elsaid

2016-01-01

Full Text Available Similarity method is employed to solve multiterm time-fractional diffusion equation. The orders of the fractional derivatives belong to the interval (0,1] and are defined in the Caputo sense. We illustrate how the problem is reduced from a multiterm two-variable fractional partial differential equation to a multiterm ordinary fractional differential equation. Power series solution is obtained for the resulting ordinary problem and the convergence of the series solution is discussed. Based on the obtained results, we propose a definition for a multiterm error function with generalized coefficients.

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)

Operator Splitting Methods for Degenerate Convection-Diffusion Equations I: Convergence and Entropy Estimates

Energy Technology Data Exchange (ETDEWEB)

Holden, Helge; Karlsen, Kenneth H.; Lie, Knut-Andreas

1999-10-01

We present and analyze a numerical method for the solution of a class of scalar, multi-dimensional, nonlinear degenerate convection-diffusion equations. The method is based on operator splitting to separate the convective and the diffusive terms in the governing equation. The nonlinear, convective part is solved using front tracking and dimensional splitting, while the nonlinear diffusion equation is solved by a suitable difference scheme. We verify L{sup 1} compactness of the corresponding set of approximate solutions and derive precise entropy estimates. In particular, these results allow us to pass to the limit in our approximations and recover an entropy solution of the problem in question. The theory presented covers a large class of equations. Important subclasses are hyperbolic conservation laws, porous medium type equations, two-phase reservoir flow equations, and strongly degenerate equations coming from the recent theory of sedimentation-consolidation processes. A thorough numerical investigation of the method analyzed in this paper (and similar methods) is presented in a companion paper. (author)

Study on monostable and bistable reaction-diffusion equations by iteration of travelling wave maps

Science.gov (United States)

Yi, Taishan; Chen, Yuming

2017-12-01

In this paper, based on the iterative properties of travelling wave maps, we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction-diffusion equations. Precisely, for Dirichlet problems of monostable reaction-diffusion equations on the half line, by making links between travelling wave maps and integral operators associated with the Dirichlet diffusion kernel (the latter is NOT invariant under translation), we obtain some iteration properties of the Dirichlet diffusion and some a priori estimates on nontrivial solutions of Dirichlet problems under travelling wave transformation. We then provide the asymptotic behavior of nontrivial solutions in the space-time region for Dirichlet problems. These enable us to develop a unified method to obtain results on heterogeneous steady states, travelling waves, spreading speeds, and asymptotic spreading behavior for Dirichlet problem of monostable reaction-diffusion equations on R+ as well as of monostable/bistable reaction-diffusion equations on R.

The determination of an unknown boundary condition in a fractional diffusion equation

KAUST Repository

Rundell, William

2013-07-01

In this article we consider an inverse boundary problem, in which the unknown boundary function ∂u/∂v = f(u) is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. © 2013 Copyright Taylor and Francis Group, LLC.

A Nonlinear Diffusion Equation-Based Model for Ultrasound Speckle Noise Removal

Science.gov (United States)

Zhou, Zhenyu; Guo, Zhichang; Zhang, Dazhi; Wu, Boying

2018-04-01

Ultrasound images are contaminated by speckle noise, which brings difficulties in further image analysis and clinical diagnosis. In this paper, we address this problem in the view of nonlinear diffusion equation theories. We develop a nonlinear diffusion equation-based model by taking into account not only the gradient information of the image, but also the information of the gray levels of the image. By utilizing the region indicator as the variable exponent, we can adaptively control the diffusion type which alternates between the Perona-Malik diffusion and the Charbonnier diffusion according to the image gray levels. Furthermore, we analyze the proposed model with respect to the theoretical and numerical properties. Experiments show that the proposed method achieves much better speckle suppression and edge preservation when compared with the traditional despeckling methods, especially in the low gray level and low-contrast regions.

Development of 3D multi-group neutron diffusion code for hexagonal geometry

International Nuclear Information System (INIS)

Sun Wei; Wang Kan; Ni Dongyang; Li Qing

2013-01-01

Based on the theory of new flux expansion nodal method to solve the neutron diffusion equations, the intra-nodal fluence rate distribution was expanded in a series of analytic basic functions for each group. In order to improve the accuracy of calculation result, continuities of neutron fluence rate and current were utilized across the nodal surfaces. According to the boundary conditions, the iteration method was adopted to solve the diffusion equation, where inner iteration speedup method is Gauss-Seidel method and outer is Lyusternik-Wagner. A new speedup method (one-outer-iteration and multi-inner-iteration method) was proposed according to the characteristic that the convergence speed of multiplication factor is faster than that of neutron fluence rate and the update of inner iteration matrix is slow. Based on the proposed model, the code HANDF-D was developed and tested by 3D two-group vver440 benchmark, experiment 2 of HFETR, 3D four-group thermal reactor benchmark, and 3D seven-group fast reactor benchmark. The numerical results show that HANDF-D can predict accurately the multiplication factor and nodal powers. (authors)

Final report [on solving the multigroup diffusion equations

International Nuclear Information System (INIS)

Birkhoff, G.

1975-01-01

Progress achieved in the development of variational methods for solving the multigroup neutron diffusion equations is described. An appraisal is made of the extent to which improved variational methods could advantageously replace difference methods currently used

SCOTCH: a program for solution of the one-dimensional, two-group, space-time neutron diffusion equations with temperature feedback of multi-channel fluid dynamics for HTGR cores

International Nuclear Information System (INIS)

Ezaki, Masahiro; Mitake, Susumu; Ozawa, Tamotsu

1979-06-01

The SCOTCH program solves the one-dimensional (R or Z), two-group reactor kinetics equations with multi-channel temperature transients and fluid dynamics. Sub-program SCOTCH-RX simulates the space-time neutron diffusion in radial direction, and sub-program SCOTCH-AX simulates the same in axial direction. The program has about 8,000 steps of FORTRAN statement and requires about 102 kilo-words of computer memory. (author)

Liquefaction of Saturated Soil and the Diffusion Equation

Science.gov (United States)

Sawicki, Andrzej; Sławińska, Justyna

2015-06-01

The paper deals with the diffusion equation for pore water pressures with the source term, which is widely promoted in the marine engineering literature. It is shown that such an equation cannot be derived in a consistent way from the mass balance and the Darcy law. The shortcomings of the artificial source term are pointed out, including inconsistencies with experimental data. It is concluded that liquefaction and the preceding process of pore pressure generation and the weakening of the soil skeleton should be described by constitutive equations within the well-known framework of applied mechanics. Relevant references are provided

DWARF, 1-D Few-Group Neutron Diffusion with Thermal Feedback for Burnup and Xe Oscillation

International Nuclear Information System (INIS)

Anderson, E.C.; Putnam, G.E.

1975-01-01

1 - Description of problem or function: DWARF allows one-dimensional simulation of reactor burnup and xenon oscillation problems in slab, cylindrical, or spherical geometry using a few-group diffusion theory model. 2 - Method of solution: The few-group, neutron diffusion theory equations are reduced to a system of finite-difference equations that are solved for each group by the Gauss method at each time point. Fission neutron source iteration can be accelerated with Chebyshev extrapolation. A thermal feedback iterative loop is used to obtain consistent solutions for the distributions of reactor power, neutron flux, and fuel and coolant properties with the neutron group constants functions of the latter. Solutions for the new nuclide concentrations of a time-point are made with the flux assumed constant in the time interval. 3 - Restrictions on the complexity of the problem - Maxima of: 4 groups; 40 regions; 50 macroscopic materials (Only 10 are functions of the feedback variables); 50 nuclides per region; 250 mesh points

Resolution of the time dependent P{sub n} equations by a Godunov type scheme having the diffusion limit; Resolution des equations P{sub n} instationnaires par un schema de type Godunov, ayant la limite diffusion

Energy Technology Data Exchange (ETDEWEB)

Cargo, P.; Samba, G

2007-07-01

We consider the P{sub n} model to approximate the transport equation in one dimension of space. In a diffusive regime, the solution of this system is solution of a diffusion equation. We are looking for a numerical scheme having the diffusion limit property: in a diffusive regime, it gives the solution of the limiting diffusion equation on a mesh at the diffusion scale. The numerical scheme proposed is an extension of the Godunov type scheme proposed by L. Gosse to solve the P{sub 1} model without absorption term. Moreover, it has the well-balanced property: it preserves the steady solutions of the system. (authors)

From quantum stochastic differential equations to Gisin-Percival state diffusion

Science.gov (United States)

Parthasarathy, K. R.; Usha Devi, A. R.

2017-08-01

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ (L2(R+ ) ⊗(Cn⊕Cn ) ) and the Hilbert space L2(μ ) , where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B (t ) ,t ≥0 } , we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

International Nuclear Information System (INIS)

Guo, Ran; Du, Jiulin

2015-01-01

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

Energy Technology Data Exchange (ETDEWEB)

Guo, Ran; Du, Jiulin, E-mail: jiulindu@aliyun.com

2015-08-15

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution.

The numerical simulation of convection delayed dominated diffusion equation

Directory of Open Access Journals (Sweden)

Mohan Kumar P. Murali

2016-01-01

Full Text Available In this paper, we propose a fitted numerical method for solving convection delayed dominated diffusion equation. A fitting factor is introduced and the model equation is discretized by cubic spline method. The error analysis is analyzed for the consider problem. The numerical examples are solved using the present method and compared the result with the exact solution.

Simulation of the diffusion equation on a type-II quantum computer

International Nuclear Information System (INIS)

Berman, G.P.; Kamenev, D.I.; Ezhov, A.A.; Yepez, J.

2002-01-01

A lattice-gas algorithm for the one-dimensional diffusion equation is realized using radio frequency pulses in a one-dimensional spin system. The model is a large array of quantum two-qubit nodes interconnected by the nearest-neighbor classical communication channels. We present a quantum protocol for implementation of the quantum collision operator and a method for initialization and reinitialization of quantum states. Numerical simulations of the quantum-classical dynamics are in good agreement with the analytic solution for the diffusion equation

General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations

Science.gov (United States)

Lin, Guoxing

2018-05-01

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. A general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion { 0 integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. The theory and the CTRW simulation agree with each other. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.

Group foliation of finite difference equations

Science.gov (United States)

Thompson, Robert; Valiquette, Francis

2018-06-01

Using the theory of equivariant moving frames, a group foliation method for invariant finite difference equations is developed. This method is analogous to the group foliation of differential equations and uses the symmetry group of the equation to decompose the solution process into two steps, called resolving and reconstruction. Our constructions are performed algorithmically and symbolically by making use of discrete recurrence relations among joint invariants. Applications to invariant finite difference equations that approximate differential equations are given.

An interpolation between the wave and diffusion equations through the fractional evolution equations Dirac like

International Nuclear Information System (INIS)

Pierantozzi, T.; Vazquez, L.

2005-01-01

Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation, we analyze a possible interpolation between the Dirac and the diffusion equations in one space dimension. We study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the D'Alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations Dirac like. Such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator. For this system of fractional evolution equations, we also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case

Rarefied gas flows through a curved channel: Application of a diffusion-type equation

Science.gov (United States)

Aoki, Kazuo; Takata, Shigeru; Tatsumi, Eri; Yoshida, Hiroaki

2010-11-01

Rarefied gas flows through a curved two-dimensional channel, caused by a pressure or a temperature gradient, are investigated numerically by using a macroscopic equation of convection-diffusion type. The equation, which was derived systematically from the Bhatnagar-Gross-Krook model of the Boltzmann equation and diffuse-reflection boundary condition in a previous paper [K. Aoki et al., "A diffusion model for rarefied flows in curved channels," Multiscale Model. Simul. 6, 1281 (2008)], is valid irrespective of the degree of gas rarefaction when the channel width is much shorter than the scale of variations of physical quantities and curvature along the channel. Attention is also paid to a variant of the Knudsen compressor that can produce a pressure raise by the effect of the change of channel curvature and periodic temperature distributions without any help of moving parts. In the process of analysis, the macroscopic equation is (partially) extended to the case of the ellipsoidal-statistical model of the Boltzmann equation.

MODICO, 1-D Time-Dependent 1 Group, 2 Group Neutron Diffusion with Delayed Neutron Precursors

International Nuclear Information System (INIS)

Camiciola, P.; Cundari, D.; Montagnini, B.

1992-01-01

1 - Description of program or function: The program solves the 1-D time-dependent one and two group coarse-mesh neutron diffusion equations, coupled with the equations for the delayed-neutron precursor, in plane geometry. 2 - Method of solution: The program is based on a simple coarse-mesh cubic approximation formula for the spatial behaviour of the flux inside each interval. An implicit scheme (the time-integrated method) is used for the advancement of the solution. The resulting (block three-diagonal) matrix is inverted at each time step by Thomas' method. 3 - Restrictions on the complexity of the problem: Number of coarse- mesh intervals LE 80; number of material regions LE 10; number of delayed-neutron precursor groups LE 10. Typical mesh sizes range from 5 cm to 20 cm; typical step length (non-prompt critical transients) ranges from 0.005 to 0.1 seconds

EDEF: a program for solving the neutron diffusion equation using microcomputers

International Nuclear Information System (INIS)

Fernandes, A.; Maiorino, J.R.

1990-01-01

This work presents the development of a program to solve the two-group two-dimensional diffusion equation (with a buckling option to simulate axial leakage) applying the finite element method. It has been developed to microcomputers compatibles to the IBM-PC. Among the facilities of the program, we can mention the simplicity to represent two-dimensional complex domains, the input through a pre-processor and the output in which the fluxes are presented graphically. The program also calculates the multiplication factor, the peaking factor and the power distribution. (author) [pt

Application of Fokker-Planck equation in positron diffusion trapping model

International Nuclear Information System (INIS)

Bartosova, I.; Ballo, P.

2015-01-01

This paper is a theoretical prelude to future work involving positron diffusion in solids for the purpose of positron annihilation lifetime spectroscopy (PALS). PALS is a powerful tool used to study defects present in materials. However, the behavior of positrons in solids is a process hard to describe. Various models have been established to undertake this task. Our preliminary model is based on the Diffusion Trapping Model (DTM) described by partial differential Fokker-Planck equation and is solved via time discretization. Fokker-Planck equation describes the time evolution of the probability density function of velocity of a particle under the influence of various forces. (authors)

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

Science.gov (United States)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-01-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.

Existence and Stability of Traveling Waves for Degenerate Reaction-Diffusion Equation with Time Delay

Science.gov (United States)

Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue

2018-06-01

This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0. Furthermore, we prove the global existence and uniqueness of C^{α ,β }-solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1-space. The exponential convergence rate is also derived.

Dimensional reduction of a general advection–diffusion equation in 2D channels

Science.gov (United States)

Kalinay, Pavol; Slanina, František

2018-06-01

Diffusion of point-like particles in a two-dimensional channel of varying width is studied. The particles are driven by an arbitrary space dependent force. We construct a general recurrence procedure mapping the corresponding two-dimensional advection-diffusion equation onto the longitudinal coordinate x. Unlike the previous specific cases, the presented procedure enables us to find the one-dimensional description of the confined diffusion even for non-conservative (vortex) forces, e.g. caused by flowing solvent dragging the particles. We show that the result is again the generalized Fick–Jacobs equation. Despite of non existing scalar potential in the case of vortex forces, the effective one-dimensional scalar potential, as well as the corresponding quasi-equilibrium and the effective diffusion coefficient can be always found.

Mittag-Leffler functions as solutions of relaxation-oscillation and diffusion-wave fractional order equation

International Nuclear Information System (INIS)

Sandev, D. Trivche

2010-01-01

The fractional calculus basis, Mittag-Leffler functions, various relaxation-oscillation and diffusion-wave fractional order equation and systems of fractional order equations are considered in this thesis. To solve these fractional order equations analytical methods, such as the Laplace transform method and method of separation of variables are employed. Some applications of the fractional calculus are considered, particularly physical system with anomalous diffusive behavior. (Author)

Solutions for a diffusion equation with a backbone term

International Nuclear Information System (INIS)

Tateishi, A A; Lenzi, E K; Ribeiro, H V; Evangelista, L R; Mendes, R S; Da Silva, L R

2011-01-01

We investigate the diffusion equation ∂ t ρ=D y ∂ y 2 ρ+D x ∂ x 2 ρ+ D-bar x δ(y)∂ x μ ρ subjected to the boundary conditions ρ(±∞,y;t)=0 and ρ(x,±∞;t)=0, and the initial condition ρ(x,y;0)= ρ-hat (x,y). We obtain exact solutions in terms of the Green function approach and analyze the mean square displacement in the x and y directions. This analysis shows an anomalous spreading of the system which is characterized by different diffusive regimes connected to anomalous diffusion

An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations

KAUST Repository

Burrage, Kevin

2012-01-01

Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator. © 2012 Society for Industrial and Applied Mathematics.

Solution of time dependent atmospheric diffusion equation with a proposed diffusion coefficient

International Nuclear Information System (INIS)

Mayhoub, A.B.; Essa, KH.S.M.; Aly, SH.

2004-01-01

One-dimensional model for the dispersion of passive atmospheric contaminant (not included chemical reactions) in the atmospheric boundary layer is considered. On the basis of the gradient transfer theory (K-theory), the time dependent diffusion equation represents the dispersion of the pollutants is solved analytically. The solution depends on diffusion coefficient K', which is expressed in terms of the friction velocity 'u the vertical coordinate -L and the depth of the mixing layer 'h'. The solution is obtained to either the vertical coordinate 'z' is less or greater than the mixing height 'h'. The obtained solution may be applied to study the atmospheric dispersion of pollutants

Application of a Lie group admitted by a homogeneous equation for group classification of a corresponding inhomogeneous equation

Science.gov (United States)

Long, Feng-Shan; Karnbanjong, Adisak; Suriyawichitseranee, Amornrat; Grigoriev, Yurii N.; Meleshko, Sergey V.

2017-07-01

This paper proposes an algorithm for group classification of a nonhomogeneous equation using the group analysis provided for the corresponding homogeneous equation. The approach is illustrated by a partial differential equation, an integro-differential equation, and a delay partial differential equation.

Stability, accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation

International Nuclear Information System (INIS)

Zhou, Xiafeng; Guo, Jiong; Li, Fu

2015-01-01

Highlights: • NEMs are innovatively applied to solve convection diffusion equation. • Stability, accuracy and numerical diffusion for NEM are analyzed for the first time. • Stability and numerical diffusion depend on the NEM expansion order and its parity. • NEMs have higher accuracy than both second order upwind and QUICK scheme. • NEMs with different expansion orders are integrated into a unified discrete form. - Abstract: The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation. In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of

Stability, accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation

Energy Technology Data Exchange (ETDEWEB)

Zhou, Xiafeng, E-mail: zhou-xf11@mails.tsinghua.edu.cn; Guo, Jiong, E-mail: guojiong12@tsinghua.edu.cn; Li, Fu, E-mail: lifu@tsinghua.edu.cn

2015-12-15

Highlights: • NEMs are innovatively applied to solve convection diffusion equation. • Stability, accuracy and numerical diffusion for NEM are analyzed for the first time. • Stability and numerical diffusion depend on the NEM expansion order and its parity. • NEMs have higher accuracy than both second order upwind and QUICK scheme. • NEMs with different expansion orders are integrated into a unified discrete form. - Abstract: The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation. In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of

Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

Directory of Open Access Journals (Sweden)

Petr Stehlík

2015-01-01

Full Text Available We study reaction-diffusion equations with a general reaction function f on one-dimensional lattices with continuous or discrete time ux′  (or  Δtux=k(ux-1-2ux+ux+1+f(ux, x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.

Discrete formulation for two-dimensional multigroup neutron diffusion equations

Energy Technology Data Exchange (ETDEWEB)

Vosoughi, Naser E-mail: vosoughi@mehr.sharif.edu; Salehi, Ali A.; Shahriari, Majid

2003-02-01

The objective of this paper is to introduce a new numerical method for neutronic calculation in a reactor core. This method can produce the final finite form of the neutron diffusion equation by classifying the neutronic variables and using two kinds of cell complexes without starting from the conventional differential form of the neutron diffusion equation. The method with linear interpolation produces the same convergence as the linear continuous finite element method. The quadratic interpolation is proven; the convergence order depends on the shape of the dual cell. The maximum convergence order is achieved by choosing the dual cell based on two Gauss' points. The accuracy of the method was examined with a well-known IAEA two-dimensional benchmark problem. The numerical results demonstrate the effectiveness of the new method.

Discrete formulation for two-dimensional multigroup neutron diffusion equations

International Nuclear Information System (INIS)

Vosoughi, Naser; Salehi, Ali A.; Shahriari, Majid

2003-01-01

The objective of this paper is to introduce a new numerical method for neutronic calculation in a reactor core. This method can produce the final finite form of the neutron diffusion equation by classifying the neutronic variables and using two kinds of cell complexes without starting from the conventional differential form of the neutron diffusion equation. The method with linear interpolation produces the same convergence as the linear continuous finite element method. The quadratic interpolation is proven; the convergence order depends on the shape of the dual cell. The maximum convergence order is achieved by choosing the dual cell based on two Gauss' points. The accuracy of the method was examined with a well-known IAEA two-dimensional benchmark problem. The numerical results demonstrate the effectiveness of the new method

NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION

OpenAIRE

Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.

2013-01-01

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and technique...

Numerical method for solving the three-dimensional time-dependent neutron diffusion equation

International Nuclear Information System (INIS)

Khaled, S.M.; Szatmary, Z.

2005-01-01

A numerical time-implicit method has been developed for solving the coupled three-dimensional time-dependent multi-group neutron diffusion and delayed neutron precursor equations. The numerical stability of the implicit computation scheme and the convergence of the iterative associated processes have been evaluated. The computational scheme requires the solution of large linear systems at each time step. For this purpose, the point over-relaxation Gauss-Seidel method was chosen. A new scheme was introduced instead of the usual source iteration scheme. (author)

Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative

Directory of Open Access Journals (Sweden)

José Francisco Gómez Aguilar

2014-01-01

Full Text Available An alternative construction for the space-time fractional diffusion-advection equation for the sedimentation phenomena is presented. The order of the derivative is considered as 0<β, γ≤1 for the space and time domain, respectively. The fractional derivative of Caputo type is considered. In the spatial case we obtain the fractional solution for the underdamped, undamped, and overdamped case. In the temporal case we show that the concentration has amplitude which exhibits an algebraic decay at asymptotically large times and also shows numerical simulations where both derivatives are taken in simultaneous form. In order that the equation preserves the physical units of the system two auxiliary parameters σx and σt are introduced characterizing the existence of fractional space and time components, respectively. A physical relation between these parameters is reported and the solutions in space-time are given in terms of the Mittag-Leffler function depending on the parameters β and γ. The generalization of the fractional diffusion-advection equation in space-time exhibits anomalous behavior.

Advanced diffusion model in compacted bentonite based on modified Poisson-Boltzmann equations

International Nuclear Information System (INIS)

Yotsuji, K.; Tachi, Y.; Nishimaki, Y.

2012-01-01

Document available in extended abstract form only. Diffusion and sorption of radionuclides in compacted bentonite are the key processes in the safe geological disposal of radioactive waste. JAEA has developed the integrated sorption and diffusion (ISD) model for compacted bentonite by coupling the pore water chemistry, sorption and diffusion processes in consistent way. The diffusion model accounts consistently for cation excess and anion exclusion in narrow pores in compacted bentonite by the electric double layer (EDL) theory. The firstly developed ISD model could predict the diffusivity of the monovalent cation/anion in compacted bentonite as a function of dry density. This ISD model was modified by considering the visco-electric effect, and applied for diffusion data for various radionuclides measured under wide range of conditions (salinity, density, etc.). This modified ISD model can give better quantitative agreement with diffusion data for monovalent cation/anion, however, the model predictions still disagree with experimental data for multivalent cation and complex species. In this study we extract the additional key factors influencing diffusion model in narrow charged pores, and the effects of these factors were investigated to reach a better understanding of diffusion processes in compacted bentonite. We investigated here the dielectric saturation effect and the excluded volume effect into the present ISD model and numerically solved these modified Poisson-Boltzmann equations. In the vicinity of the negatively charged clay surfaces, it is necessary to evaluate concentration distribution of electrolytes considering the dielectric saturation effects. The Poisson-Boltzmann (P-B) equation coupled with the dielectric saturation effects was solved numerically by using Runge-Kutta and Shooting methods. Figure 1(a) shows the concentration distributions of Na + as numerical solutions of the modified and original P-B equations for 0.01 M pore water, 800 kg m -3

Analysis and application of diffusion equations involving a new fractional derivative without singular kernel

Directory of Open Access Journals (Sweden)

Lihong Zhang

2017-11-01

Full Text Available In this article, a family of nonlinear diffusion equations involving multi-term Caputo-Fabrizio time fractional derivative is investigated. Some maximum principles are obtained. We also demonstrate the application of the obtained results by deriving some estimation for solution to reaction-diffusion equations.

Analysis and Application of High Resolution Numerical Perturbation Algorithm for Convective-Diffusion Equation

International Nuclear Information System (INIS)

Gao Zhi; Shen Yi-Qing

2012-01-01

The high resolution numerical perturbation (NP) algorithm is analyzed and tested using various convective-diffusion equations. The NP algorithm is constructed by splitting the second order central difference schemes of both convective and diffusion terms of the convective-diffusion equation into upstream and downstream parts, then the perturbation reconstruction functions of the convective coefficient are determined using the power-series of grid interval and eliminating the truncated errors of the modified differential equation. The important nature, i.e. the upwind dominance nature, which is the basis to ensuring that the NP schemes are stable and essentially oscillation free, is firstly presented and verified. Various numerical cases show that the NP schemes are efficient, robust, and more accurate than the original second order central scheme

Asymptotic behavior of solutions of diffusion-like partial differential equations invariant to a family of affine groups

International Nuclear Information System (INIS)

Dresner, L.

1990-07-01

This report deals with the asymptotic behavior of certain solutions of partial differential equations in one dependent and two independent variables (call them c, z, and t, respectively). The partial differential equations are invariant to one-parameter families of one-parameter affine groups of the form: c' = λ α c, t' = λ β t, z' = λz, where λ is the group parameter that labels the individual transformations and α and β are parameters that label groups of the family. The parameters α and β are connected by a linear relation, Mα + Nβ = L, where M, N, and L are numbers determined by the structure of the partial differential equation. It is shown that when L/M and N/M are L/M t -N/M for large z or small t. Some practical applications of this result are discussed. 8 refs

Splitting Method for Solving the Coarse-Mesh Discretized Low-Order Quasi-Diffusion Equations

International Nuclear Information System (INIS)

Hiruta, Hikaru; Anistratov, Dmitriy Y.; Adams, Marvin L.

2005-01-01

In this paper, the development is presented of a splitting method that can efficiently solve coarse-mesh discretized low-order quasi-diffusion (LOQD) equations. The LOQD problem can reproduce exactly the transport scalar flux and current. To solve the LOQD equations efficiently, a splitting method is proposed. The presented method splits the LOQD problem into two parts: (a) the D problem that captures a significant part of the transport solution in the central parts of assemblies and can be reduced to a diffusion-type equation and (b) the Q problem that accounts for the complicated behavior of the transport solution near assembly boundaries. Independent coarse-mesh discretizations are applied: the D problem equations are approximated by means of a finite element method, whereas the Q problem equations are discretized using a finite volume method. Numerical results demonstrate the efficiency of the methodology presented. This methodology can be used to modify existing diffusion codes for full-core calculations (which already solve a version of the D problem) to account for transport effects

Solving the neutron diffusion equation on combinatorial geometry computational cells for reactor physics calculations

International Nuclear Information System (INIS)

Azmy, Y. Y.

2004-01-01

An approach is developed for solving the neutron diffusion equation on combinatorial geometry computational cells, that is computational cells composed by combinatorial operations involving simple-shaped component cells. The only constraint on the component cells from which the combinatorial cells are assembled is that they possess a legitimate discretization of the underlying diffusion equation. We use the Finite Difference (FD) approximation of the x, y-geometry diffusion equation in this work. Performing the same combinatorial operations involved in composing the combinatorial cell on these discrete-variable equations yields equations that employ new discrete variables defined only on the combinatorial cell's volume and faces. The only approximation involved in this process, beyond the truncation error committed in discretizing the diffusion equation over each component cell, is a consistent-order Legendre series expansion. Preliminary results for simple configurations establish the accuracy of the solution to the combinatorial geometry solution compared to straight FD as the system dimensions decrease. Furthermore numerical results validate the consistent Legendre-series expansion order by illustrating the second order accuracy of the combinatorial geometry solution, the same as standard FD. Nevertheless the magnitude of the error for the new approach is larger than FD's since it incorporates the additional truncated series approximation. (authors)

Entropy methods for diffusive partial differential equations

CERN Document Server

Jüngel, Ansgar

2016-01-01

This book presents a range of entropy methods for diffusive PDEs devised by many researchers in the course of the past few decades, which allow us to understand the qualitative behavior of solutions to diffusive equations (and Markov diffusion processes). Applications include the large-time asymptotics of solutions, the derivation of convex Sobolev inequalities, the existence and uniqueness of weak solutions, and the analysis of discrete and geometric structures of the PDEs. The purpose of the book is to provide readers an introduction to selected entropy methods that can be found in the research literature. In order to highlight the core concepts, the results are not stated in the widest generality and most of the arguments are only formal (in the sense that the functional setting is not specified or sufficient regularity is supposed). The text is also suitable for advanced master and PhD students and could serve as a textbook for special courses and seminars.

Explosive instabilities of reaction-diffusion equations including pinch effects

International Nuclear Information System (INIS)

Wilhelmsson, H.

1992-01-01

Particular solutions of reaction-diffusion equations for temperature are obtained for explosively unstable situations. As a result of the interplay between inertial, diffusion, pinch and source processes certain 'bell-shaped' distributions may grow explosively in time with preserved shape of the spatial distribution. The effect of the pinch, which requires a density inhomogeneity, is found to diminish the effect of diffusion, or inversely to support the inertial and source processes in creating the explosion. The results may be described in terms of elliptic integrals or. more simply, by means of expansions in the spatial coordinate. An application is the temperature evolution of a burning fusion plasma. (au) (18 refs.)

A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations

OpenAIRE

Boudin , Laurent; Grec , Bérénice; Salvarani , Francesco

2012-01-01

International audience; We consider the Maxwell-Stefan model of diffusion in a multicomponent gaseous mixture. After focusing on the main differences with the Fickian diffusion model, we study the equations governing a three-component gas mixture. We provide a qualitative and quantitative mathematical analysis of the model. The main properties of the standard explicit numerical scheme are also analyzed. We eventually include some numerical simulations pointing out the uphill diffusion phenome...

Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models

Directory of Open Access Journals (Sweden)

Narcisa Apreutesei

2014-05-01

Full Text Available In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.

Non probabilistic solution of uncertain neutron diffusion equation for imprecisely defined homogeneous bare reactor

International Nuclear Information System (INIS)

Chakraverty, S.; Nayak, S.

2013-01-01

Highlights: • Uncertain neutron diffusion equation of bare square homogeneous reactor is studied. • Proposed interval arithmetic is extended for fuzzy numbers. • The developed fuzzy arithmetic is used to handle uncertain parameters. • Governing differential equation is modelled by modified fuzzy finite element method. • Fuzzy critical eigenvalues and effective multiplication factors are investigated. - Abstract: The scattering of neutron collision inside a reactor depends upon geometry of the reactor, diffusion coefficient and absorption coefficient etc. In general these parameters are not crisp and hence we get uncertain neutron diffusion equation. In this paper we have investigated the above equation for a bare square homogeneous reactor. Here the uncertain governing differential equation is modelled by a modified fuzzy finite element method. Using modified fuzzy finite element method, obtained eigenvalues and effective multiplication factors are studied. Corresponding results are compared with the classical finite element method in special cases and various uncertain results have been discussed

New diffusion-sythetic acceleration methods for the SN equations with corner balance spatial differencing

International Nuclear Information System (INIS)

Wareing, T.A.

1993-01-01

New methods are presented for diffusion-synthetic accelerating the S N equations in slab and x-y geometries with the corner balance spatial differencing scheme. With the standard diffusion-synthetic acceleration method, the discretized diffusion problem is derived from the discretized S N problem to insure stability through consistent differencing. The major difference between our new methods and standard diffusion-synthetic acceleration is that the discretized diffusion problem is derived from a discretization of the P 1 equations, independently of the discretized S N problem. We present theoretical and numerical results to show that these new methods are unconditionally efficient in slab and x-y geometries with rectangular spatial meshes and isotropic scattering. (orig.)

NodHex3D: An application for solving the neutron diffusion equations in hexagonal-Z geometry and steady state

International Nuclear Information System (INIS)

Esquivel E, J.; Del Valle G, E.

2014-10-01

The system called NodHex3D is a graphical application that allows the solution of the neutron diffusion equation. The system considers fuel assemblies of hexagonal cross section. This application arose from the idea of expanding the development of neutron own codes, used primarily for academic purposes. The advantage associated with the use of NodHex3D, is that the kernel configuration and fuel batches is dynamically without affecting directly the base source code of the solution of the neutron diffusion equation. In addition to the kernel configuration to use, specify the values for the cross sections for each batch of fuel used, these values are: diffusion coefficient, removal cross section, absorption cross section, fission cross section and dispersion cross section. Important also, considering that the system is able to perform calculations for various energy groups. As evidence of the operation of NodHex3D, was proposed to model three-dimensional core of a nuclear reactor VVER-1000, based on the reference problem AER-FCM-101. The configuration of the reactor core consists of fuel assemblies (25 batches), composed of seven distinct materials, one of which reflector material, vacuum boundary conditions on the surface delimiting the reactor core. The diffusion equation for two energy groups solves, obtaining the value of the effective neutron multiplication factor. The obtained results are compared to those documented in the reference problem and by 3-DNT codes. (Author)

High order backward discretization of the neutron diffusion equation

Energy Technology Data Exchange (ETDEWEB)

Ginestar, D.; Bru, R.; Marin, J. [Universidad Politecnica de Valencia (Spain). Departamento de Matematica Aplicada; Verdu, G.; Munoz-Cobo, J.L. [Universidad Politecnica de Valencia (Spain). Departamento de Ingenieria Quimica y Nuclear; Vidal, V. [Universidad Politecnica de Valencia (Spain). Departamento de Sistemas Informaticos y Computacion

1997-11-21

Fast codes capable of dealing with three-dimensional geometries, are needed to be able to simulate spatially complicated transients in a nuclear reactor. We propose a new discretization technique for the time integration of the neutron diffusion equation, based on the backward difference formulas for systems of stiff ordinary differential equations. This method needs to solve a system of linear equations for each integration step, and for this purpose, we have developed an iterative block algorithm combined with a variational acceleration technique. We tested the algorithm with two benchmark problems, and compared the results with those provided by other codes, concluding that the performance and overall agreement are very good. (author).

A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation

Science.gov (United States)

Sun, HongGuang; Liu, Xiaoting; Zhang, Yong; Pang, Guofei; Garrard, Rhiannon

2017-09-01

Fractional-order diffusion equations (FDEs) extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media. Real-world diffusion can be multi-dimensional, requiring efficient numerical solvers that can handle long-term memory embedded in mass transport. To address this challenge, a semi-discrete Kansa method is developed to approximate the two-dimensional spatiotemporal FDE, where the Kansa approach first discretizes the FDE, then the Gauss-Jacobi quadrature rule solves the corresponding matrix, and finally the Mittag-Leffler function provides an analytical solution for the resultant time-fractional ordinary differential equation. Numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes. Applications further show that the numerical method can efficiently solve two-dimensional spatiotemporal FDE models with either a continuous or discrete mixing measure. Hence this study provides an efficient and fast computational method for modeling super-diffusive, sub-diffusive, and mixed diffusive processes in large, two-dimensional domains with irregular shapes.

Innovation diffusion equations on correlated scale-free networks

Energy Technology Data Exchange (ETDEWEB)

Bertotti, M.L., E-mail: marialetizia.bertotti@unibz.it [Free University of Bozen–Bolzano, Faculty of Science and Technology, Bolzano (Italy); Brunner, J., E-mail: johannes.brunner@tis.bz.it [TIS Innovation Park, Bolzano (Italy); Modanese, G., E-mail: giovanni.modanese@unibz.it [Free University of Bozen–Bolzano, Faculty of Science and Technology, Bolzano (Italy)

2016-07-29

Highlights: • The Bass diffusion model can be formulated on scale-free networks. • In the trickle-down version, the hubs adopt earlier and act as monitors. • We improve the equations in order to describe trickle-up diffusion. • Innovation is generated at the network periphery, and hubs can act as stiflers. • We compare diffusion times, in dependence on the scale-free exponent. - Abstract: We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the total adoption in a predictable way. In a major departure from the standard model, we model a trickle-up process by introducing heterogeneous publicity coefficients (which can also be negative for the hubs, thus turning them into stiflers) and a stochastic term which represents the erratic generation of innovation at the periphery of the network. The results confirm the robustness of the Bass model and expand considerably its range of applicability.

Innovation diffusion equations on correlated scale-free networks

International Nuclear Information System (INIS)

Bertotti, M.L.; Brunner, J.; Modanese, G.

2016-01-01

Highlights: • The Bass diffusion model can be formulated on scale-free networks. • In the trickle-down version, the hubs adopt earlier and act as monitors. • We improve the equations in order to describe trickle-up diffusion. • Innovation is generated at the network periphery, and hubs can act as stiflers. • We compare diffusion times, in dependence on the scale-free exponent. - Abstract: We introduce a heterogeneous network structure into the Bass diffusion model, in order to study the diffusion times of innovation or information in networks with a scale-free structure, typical of regions where diffusion is sensitive to geographic and logistic influences (like for instance Alpine regions). We consider both the diffusion peak times of the total population and of the link classes. In the familiar trickle-down processes the adoption curve of the hubs is found to anticipate the total adoption in a predictable way. In a major departure from the standard model, we model a trickle-up process by introducing heterogeneous publicity coefficients (which can also be negative for the hubs, thus turning them into stiflers) and a stochastic term which represents the erratic generation of innovation at the periphery of the network. The results confirm the robustness of the Bass model and expand considerably its range of applicability.

Wielandt method applied to the diffusion equations discretized by finite element nodal methods

International Nuclear Information System (INIS)

Mugica R, A.; Valle G, E. del

2003-01-01

Nowadays the numerical methods of solution to the diffusion equation by means of algorithms and computer programs result so extensive due to the great number of routines and calculations that should carry out, this rebounds directly in the execution times of this programs, being obtained results in relatively long times. This work shows the application of an acceleration method of the convergence of the classic method of those powers that it reduces notably the number of necessary iterations for to obtain reliable results, what means that the compute times they see reduced in great measure. This method is known in the literature like Wielandt method and it has incorporated to a computer program that is based on the discretization of the neutron diffusion equations in plate geometry and stationary state by polynomial nodal methods. In this work the neutron diffusion equations are described for several energy groups and their discretization by means of those called physical nodal methods, being illustrated in particular the quadratic case. It is described a model problem widely described in the literature which is solved for the physical nodal grade schemes 1, 2, 3 and 4 in three different ways: to) with the classic method of the powers, b) method of the powers with the Wielandt acceleration and c) method of the powers with the Wielandt modified acceleration. The results for the model problem as well as for two additional problems known as benchmark problems are reported. Such acceleration method can also be implemented to problems of different geometry to the proposal in this work, besides being possible to extend their application to problems in 2 or 3 dimensions. (Author)

Darboux transformations for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix

International Nuclear Information System (INIS)

Schulze-Halberg, Axel

2012-01-01

We construct a Darboux transformation for (1+2)-dimensional Fokker-Planck equations with constant diffusion matrix. Our transformation is based on the two-dimensional supersymmetry formalism for the Schrödinger equation. The transformed Fokker-Planck equation and its solutions are obtained in explicit form.

NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION.

Science.gov (United States)

Liu, F; Meerschaert, M M; McGough, R J; Zhuang, P; Liu, Q

2013-03-01

In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

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.

Nonlinear reaction-diffusion equations with delay: some theorems, test problems, exact and numerical solutions

Science.gov (United States)

Polyanin, A. D.; Sorokin, V. G.

2017-12-01

The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.

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

Attractor of reaction-diffusion equations in Banach spaces

Directory of Open Access Journals (Sweden)

José Valero

2001-04-01

Full Text Available In this paper we prove first some abstract theorems on existence of global attractors for differential inclusions generated by w-dissipative operators. Then these results are applied to reaction-diffusion equations in which the Babach space Lp is used as phase space. Finally, new results concerning the fractal dimension of the global attractor in the space L2 are obtained.

A perturbational h4 exponential finite difference scheme for the convective diffusion equation

International Nuclear Information System (INIS)

Chen, G.Q.; Gao, Z.; Yang, Z.F.

1993-01-01

A perturbational h 4 compact exponential finite difference scheme with diagonally dominant coefficient matrix and upwind effect is developed for the convective diffusion equation. Perturbations of second order are exerted on the convective coefficients and source term of an h 2 exponential finite difference scheme proposed in this paper based on a transformation to eliminate the upwind effect of the convective diffusion equation. Four numerical examples including one- to three-dimensional model equations of fluid flow and a problem of natural convective heat transfer are given to illustrate the excellent behavior of the present exponential schemes. Besides, the h 4 accuracy of the perturbational scheme is verified using double precision arithmetic

General solution of the aerosol dynamic equation: growth and diffusion processes

International Nuclear Information System (INIS)

Elgarayhi, A.; Elhanbaly, A.

2004-01-01

The dispersion of aerosol particles in a fluid media is studied considering the main mechanism for condensation and diffusion. This has been done when the technique of Lie is used for solving the aerosol dynamic equation. This method is very useful in sense that it reduces the partial differential equation to some ordinary differential equations. So, different classes of similarity solutions have been obtained. The quantity of well-defined physical interest is the mean particle volume has been calculated

Construction and analysis of lattice Boltzmann methods applied to a 1D convection-diffusion equation

International Nuclear Information System (INIS)

Dellacherie, Stephane

2014-01-01

To solve the 1D (linear) convection-diffusion equation, we construct and we analyze two LBM schemes built on the D1Q2 lattice. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in L∞ of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in L∞ under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results. (authors)

A Bloch-Torrey Equation for Diffusion in a Deforming Media

Energy Technology Data Exchange (ETDEWEB)

Rohmer, Damien; Gullberg, Grant T.

2006-12-29

Diffusion Tensor Magnetic Resonance Imaging (DTMRI)technique enables the measurement of diffusion parameters and therefore,informs on the structure of the biological tissue. This technique isapplied with success to the static organs such as brain. However, thediffusion measurement on the dynamically deformable organs such as thein-vivo heart is a complex problem that has however a great potential inthe measurement of cardiac health. In order to understand the behavior ofthe Magnetic Resonance (MR)signal in a deforming media, the Bloch-Torreyequation that leads the MR behavior is expressed in general curvilinearcoordinates. These coordinates enable to follow the heart geometry anddeformations through time. The equation is finally discretized andpresented in a numerical formulation using implicit methods, in order toget a stable scheme that can be applied to any smooth deformations.Diffusion process enables the link between the macroscopic behavior ofmolecules and themicroscopic structure in which they evolve. Themeasurement of diffusion in biological tissues is therefore of majorimportance in understanding the complex underlying structure that cannotbe studied directly. The Diffusion Tensor Magnetic ResonanceImaging(DTMRI) technique enables the measurement of diffusion parametersand therefore provides information on the structure of the biologicaltissue. This technique has been applied with success to static organssuch as the brain. However, diffusion measurement of dynamicallydeformable organs such as the in-vivo heart remains a complex problem,which holds great potential in determining cardiac health. In order tounderstand the behavior of the magnetic resonance (MR) signal in adeforming media, the Bloch-Torrey equation that defines the MR behavioris expressed in general curvilinear coordinates. These coordinates enableus to follow the heart geometry and deformations through time. Theequation is finally discretized and presented in a numerical formulationusing

Diffusion-accelerated solution of the 2-D x-y Sn equations with linear-bilinear nodal differencing

International Nuclear Information System (INIS)

Wareing, T.A.; Walters, W.F.; Morel, J.E.

1994-01-01

Recently a new diffusion-synthetic acceleration scheme was developed for solving the 2-D S n Equations in x-y geometry with bilinear-discontinuous finite element spatial discretization using a bilinear-discontinuous diffusion differencing scheme for the diffusion acceleration equations. This method differs from previous methods in that it is conditional efficient for problems with isotropic or nearly isotropic scattering. We have used the same bilinear-discontinuous diffusion scheme, and associated solution technique, to accelerate the x-y geometry S n equations with linear-bilinear nodal spatial differencing. We find that this leads to an unconditionally efficient solution method for problems with isotropic or nearly isotropic scattering. computational results are given which demonstrate this property

Spectral and evolutionary analysis of advection-diffusion equations and the shear flow paradigm

International Nuclear Information System (INIS)

Thyagaraja, A.; Loureiro, N.; Knight, P.J.

2002-01-01

Advection-diffusion equations occur in a wide variety of fields in many contexts of active and passive transport in fluids and plasmas. The effects of sheared advective flows in the presence of irreversible processes such as diffusion and viscosity are of considerable current interest in tokamak and astrophysical contexts, where they are thought to play a key role in both transport and the dynamical structures characteristic of electromagnetic plasma turbulence. In this paper we investigate the spectral and evolutionary properties of relatively simple, linear, advection-diffusion equations. We apply analytical approaches based on standard Green's function methods to obtain insight into the nature of the spectra when the advective and diffusive effects occur separately and in combination. In particular, the physically interesting limit of small (but finite) diffusion is studied in detail. The analytical work is extended and supplemented by numerical techniques involving a direct solution of the eigenvalue problem as well as evolutionary studies of the initial value problem using a parallel code, CADENCE. The three approaches are complementary and entirely consistent with each other when appropriate comparison is made. They reveal different aspects of the properties of the advection-diffusion equation, such as the ability of sheared flows to generate a direct cascade to high wave numbers transverse to the advection and the consequent enhancement of even small amounts of diffusivity. The invariance properties of the spectra in the low diffusivity limit and the ability of highly sheared, jet-like flows to 'confine' transport to low shear regions are demonstrated. The implications of these properties in a wider context are discussed and set in perspective. (author)

Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds

KAUST Repository

Desvillettes, Laurent; Fellner, Klemens

2008-01-01

In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in L1 to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global L∞ bound via interpolation of a polynomially growing H1 bound with the almost exponential L1 convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.

A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation

Directory of Open Access Journals (Sweden)

Yunying Zheng

2011-01-01

Full Text Available The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also included which are in line with the theoretical analysis.

Compositeness condition in the renormalization group equation

International Nuclear Information System (INIS)

Bando, Masako; Kugo, Taichiro; Maekawa, Nobuhiro; Sasakura, Naoki; Watabiki, Yoshiyuki; Suehiro, Kazuhiko

1990-01-01

The problems in imposing compositeness conditions as boundary conditions in renormalization group equations are discussed. It is pointed out that one has to use the renormalization group equation directly in cutoff theory. In some cases, however, it can be approximated by the renormalization group equation in continuum theory if the mass dependent renormalization scheme is adopted. (orig.)

Nonlocal Symmetries to Systems of Nonlinear Diffusion Equations

International Nuclear Information System (INIS)

Qu Changzheng; Kang Jing

2008-01-01

In this paper, we study potential symmetries to certain systems of nonlinear diffusion equations. Those systems have physical applications in soil science, mathematical biology, and invariant curve flows in R 3 . Lie point symmetries of the potential system, which cannot be projected to vector fields of the given dependent and independent variables, yield potential symmetries. The class of the system that admits potential symmetries is expanded.

On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion

International Nuclear Information System (INIS)

Iyiola, O.S.; Tasbozan, O.; Kurt, A.; Çenesiz, Y.

2017-01-01

In this paper, we consider the system of conformable time-fractional Robertson equations with one-dimensional diffusion having widely varying diffusion coefficients. Due to the mismatched nature of the initial and boundary conditions associated with Robertson equation, there are spurious oscillations appearing in many computational algorithms. Our goal is to obtain an approximate solutions of this system of equations using the q-homotopy analysis method (q-HAM) and examine the widely varying diffusion coefficients and the fractional order of the derivative.

Singular solution of the Feller diffusion equation via a spectral decomposition

Science.gov (United States)

Gan, Xinjun; Waxman, David

2015-01-01

Feller studied a branching process and found that the distribution for this process approximately obeys a diffusion equation [W. Feller, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley and Los Angeles, 1951), pp. 227-246]. This diffusion equation and its generalizations play an important role in many scientific problems, including, physics, biology, finance, and probability theory. We work under the assumption that the fundamental solution represents a probability density and should account for all of the probability in the problem. Thus, under the circumstances where the random process can be irreversibly absorbed at the boundary, this should lead to the presence of a Dirac delta function in the fundamental solution at the boundary. However, such a feature is not present in the standard approach (Laplace transformation). Here we require that the total integrated probability is conserved. This yields a fundamental solution which, when appropriate, contains a term proportional to a Dirac delta function at the boundary. We determine the fundamental solution directly from the diffusion equation via spectral decomposition. We obtain exact expressions for the eigenfunctions, and when the fundamental solution contains a Dirac delta function at the boundary, every eigenfunction of the forward diffusion operator contains a delta function. We show how these combine to produce a weight of the delta function at the boundary which ensures the total integrated probability is conserved. The solution we present covers cases where parameters are time dependent, thereby greatly extending its applicability.

Solution of two-dimensional neutron diffusion equation for triangular region by finite Fourier transformation

International Nuclear Information System (INIS)

Kobayashi, Keisuke; Ishibashi, Hideo

1978-01-01

A two-dimensional neutron diffusion equation for a triangular region is shown to be solved by the finite Fourier transformation. An application of the Fourier transformation to the diffusion equation for triangular region yields equations whose unknowns are the expansion coefficients of the neutron flux and current in Fourier series or Legendre polynomials expansions only at the region boundary. Some numerical calculations have revealed that the present technique gives accurate results. It is shown also that the solution using the expansion in Legendre polynomials converges with relatively few terms even if the solution in Fourier series exhibits the Gibbs' phenomenon. (auth.)

FEM for time-fractional diffusion equations, novel optimal error analyses

OpenAIRE

Mustapha, Kassem

2016-01-01

A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with respect to both the convergence order of the approximate solution and the regularity of the initial data. By using novel energy arguments, for each fixed time $t$, optimal error bounds in the spatial $L^2$- and $H^1$-norms are derived for both cases: smooth...

CYLFUX, Fast Reactor Reactivity Transients Simulation in LWR by 2-D 2 Group Diffusion

International Nuclear Information System (INIS)

Schmidt, A.

1973-01-01

1 - Nature of physical problem solved: A 2-dimensional calculation of the 2-group, space-dependent neutron diffusion equations is performed in r-z geometry using an arbitrary number of groups of delayed neutron precursors. The program is designed to simulate fast reactivity excursions in light water reactors taking into account Doppler feedback via adiabatic heatup of fuel. Axial motions of control rods may be considered including scram action on option. 2 - Method of solution: The differential equations are solved at each time step by an explicit finite difference method using two time levels. The stationary distributions are obtained by using the same algorithm. 3 - Restrictions on the complexity of the problem: No restriction to the number of space points and delayed neutron energy groups besides the computer size

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

Science.gov (United States)

Zhang, Linghai

2017-10-01

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 traveling wave front as well as the existence and instability of a standing pulse solution if 0 traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.

Diffusion with space memory modelled with distributed order space fractional differential equations

Directory of Open Access Journals (Sweden)

M. Caputo

2003-06-01

Full Text Available Distributed order fractional differential equations (Caputo, 1995, 2001; Bagley and Torvik, 2000a,b were fi rst used in the time domain; they are here considered in the space domain and introduced in the constitutive equation of diffusion. The solution of the classic problems are obtained, with closed form formulae. In general, the Green functions act as low pass fi lters in the frequency domain. The major difference with the case when a single space fractional derivative is present in the constitutive equations of diffusion (Caputo and Plastino, 2002 is that the solutions found here are potentially more fl exible to represent more complex media (Caputo, 2001a. The difference between the space memory medium and that with the time memory is that the former is more fl exible to represent local phenomena while the latter is more fl exible to represent variations in space. Concerning the boundary value problem, the difference with the solution of the classic diffusion medium, in the case when a constant boundary pressure is assigned and in the medium the pressure is initially nil, is that one also needs to assign the fi rst order space derivative at the boundary.

Performance of a parallel algorithm for solving the neutron diffusion equation on the hypercube

International Nuclear Information System (INIS)

Kirk, B.L.; Azmy, Y.Y.

1989-01-01

The one-group, steady state neutron diffusion equation in two- dimensional Cartesian geometry is solved using the nodal method technique. By decoupling sets of equations representing the neutron current continuity along the length of rows and columns of computational cells a new iterative algorithm is derived that is more suitable to solving large practical problems. This algorithm is highly parallelizable and is implemented on the Intel iPSC/2 hypercube in three versions which differ essentially in the total size of communicated data. Even though speedup was achieved, the efficiency is very low when many processors are used leading to the conclusion that the hypercube is not as well suited for this algorithm as shared memory machines. 10 refs., 1 fig., 3 tabs

The Nodal Polynomial Expansion method to solve the multigroup diffusion equations

International Nuclear Information System (INIS)

Ribeiro, R.D.M.

1983-03-01

The methodology of the solutions of the multigroup diffusion equations and uses the Nodal Polynomial Expansion Method is covered. The EPON code was developed based upon the above mentioned method for stationary state, rectangular geometry, one-dimensional or two-dimensional and for one or two energy groups. Then, one can study some effects such as the influence of the baffle on the thermal flux by calculating the flux and power distribution in nuclear reactors. Furthermore, a comparative study with other programs which use Finite Difference (CITATION and PDQ5) and Finite Element (CHD and FEMB) Methods was undertaken. As a result, the coherence, feasibility, speed and accuracy of the methodology used were demonstrated. (Author) [pt

Differential equations and finite groups

NARCIS (Netherlands)

Put, Marius van der; Ulmer, Felix

2000-01-01

The classical solution of the Riemann-Hilbert problem attaches to a given representation of the fundamental group a regular singular linear differential equation. We present a method to compute this differential equation in the case of a representation with finite image. The approach uses Galois

Monte Carlo Finite Volume Element Methods for the Convection-Diffusion Equation with a Random Diffusion Coefficient

Directory of Open Access Journals (Sweden)

Qian Zhang

2014-01-01

Full Text Available The paper presents a framework for the construction of Monte Carlo finite volume element method (MCFVEM for the convection-diffusion equation with a random diffusion coefficient, which is described as a random field. We first approximate the continuous stochastic field by a finite number of random variables via the Karhunen-Loève expansion and transform the initial stochastic problem into a deterministic one with a parameter in high dimensions. Then we generate independent identically distributed approximations of the solution by sampling the coefficient of the equation and employing finite volume element variational formulation. Finally the Monte Carlo (MC method is used to compute corresponding sample averages. Statistic error is estimated analytically and experimentally. A quasi-Monte Carlo (QMC technique with Sobol sequences is also used to accelerate convergence, and experiments indicate that it can improve the efficiency of the Monte Carlo method.

A mixed finite element method for nonlinear diffusion equations

KAUST Repository

Burger, Martin; Carrillo, José ; Wolfram, Marie-Therese

2010-01-01

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.

Three-group albedo method applied to the diffusion phenomenon with up-scattering of neutrons

International Nuclear Information System (INIS)

Terra, Andre M. Barge Pontes Torres; Silva, Jorge A. Valle da; Cabral, Ronaldo G.

2007-01-01

The main objective of this research is to develop a three-group neutron Albedo algorithm considering the up-scattering of neutrons in order to analyse the diffusion phenomenon in nonmultiplying media. The neutron Albedo method is an analytical method that does not try to solve describing explicit equations for the neutron fluxes. Thus the neutron Albedo methodology is very different from the conventional methodology, as the neutron diffusion theory model. Graphite is analyzed as a model case. One major application is in the determination of the nonleakage probabilities with more understandable results in physical terms than conventional radiation transport method calculations. (author)

Fifth-order amplitude equation for traveling waves in isothermal double diffusive convection

International Nuclear Information System (INIS)

Mendoza, S.; Becerril, R.

2009-01-01

Third-order amplitude equations for isothermal double diffusive convection are known to hold the tricritical condition all along the oscillatory branch, predicting that stable traveling waves exist Only at the onset of the instability. In order to properly describe stable traveling waves, we perform a fifth-order calculation and present explicitly the corresponding amplitude equation.

Mixed and mixed-hybrid elements for the diffusion equation

International Nuclear Information System (INIS)

Coulomb, F.; Fedon-Magnaud, C.

1987-04-01

To solve the diffusion equation, one often uses a Lagrangian finite element method. We want to introduce the mixed elements which allow a simultaneous approximation of the same order for the flux and its gradient. Though the linear systems are not positive definite, it is possible to make them so by eliminating some of the unknowns

The arbitrary order mimetic finite difference method for a diffusion equation with a non-symmetric diffusion tensor

Science.gov (United States)

Gyrya, V.; Lipnikov, K.

2017-11-01

We present the arbitrary order mimetic finite difference (MFD) discretization for the diffusion equation with non-symmetric tensorial diffusion coefficient in a mixed formulation on general polygonal meshes. The diffusion tensor is assumed to be positive definite. The asymmetry of the diffusion tensor requires changes to the standard MFD construction. We present new approach for the construction that guarantees positive definiteness of the non-symmetric mass matrix in the space of discrete velocities. The numerically observed convergence rate for the scalar quantity matches the predicted one in the case of the lowest order mimetic scheme. For higher orders schemes, we observed super-convergence by one order for the scalar variable which is consistent with the previously published result for a symmetric diffusion tensor. The new scheme was also tested on a time-dependent problem modeling the Hall effect in the resistive magnetohydrodynamics.

Stochastic interpretation of the advection-diffusion equation and its relevance to bed load transport

Science.gov (United States)

Ancey, C.; Bohorquez, P.; Heyman, J.

2015-12-01

The advection-diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it, with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in rivers. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection-diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection-diffusion equation for particle activity. Yet while this derivation is formally valid in the continuum limit, it runs into difficulty in practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due

Simulation of a parallel processor on a serial processor: The neutron diffusion equation

International Nuclear Information System (INIS)

Honeck, H.C.

1981-01-01

Parallel processors could provide the nuclear industry with very high computing power at a very moderate cost. Will we be able to make effective use of this power. This paper explores the use of a very simple parallel processor for solving the neutron diffusion equation to predict power distributions in a nuclear reactor. We first describe a simple parallel processor and estimate its theoretical performance based on the current hardware technology. Next, we show how the parallel processor could be used to solve the neutron diffusion equation. We then present the results of some simulations of a parallel processor run on a serial processor and measure some of the expected inefficiencies. Finally we extrapolate the results to estimate how actual design codes would perform. We find that the standard numerical methods for solving the neutron diffusion equation are still applicable when used on a parallel processor. However, some simple modifications to these methods will be necessary if we are to achieve the full power of these new computers. (orig.) [de

Calculation of the power factor using the neutron diffusion hybrid equation

International Nuclear Information System (INIS)

Costa da Silva, Adilson; Carvalho da Silva, Fernando; Senra Martinez, Aquilino

2013-01-01

Highlights: ► A neutron diffusion hybrid equation with an external neutron source was used. ► Nodal expansion method to obtain the neutron flux was used. ► Nuclear power factors in each fuel element in the reactor core were calculated. ► The results obtained were very accurate. -- Abstract: In this paper, we used a neutron diffusion hybrid equation with an external neutron source to calculate nuclear power factors in each fuel element in the reactor core. We used the nodal expansion method to obtain the neutron flux for a given control rods bank position. The results were compared with results obtained for eigenvalue problem near criticality condition and fixed source problem during the start-up of the reactor, where external neutron sources are extremely important for the stabilization of external neutron detectors.

Application of finite Fourier transformation for the solution of the diffusion equation

International Nuclear Information System (INIS)

Kobayashi, Keisuke

1991-01-01

The application of the finite Fourier transformation to the solution of the neutron diffusion equation in one dimension, two dimensional x-y and triangular geometries is discussed. It can be shown that the equation obtained by the Nodal Green's function method in Cartesian coordinates can be derived as a special case of the finite Fourier transformation method. (author)

Solution of diffusion equation in deformable spheroids

Energy Technology Data Exchange (ETDEWEB)

Ayyoubzadeh, Seyed Mohsen [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Safari, Mohammad Javad, E-mail: iFluka@gmail.com [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of); Vosoughi, Naser [Department of Energy Engineering, Sharif University of Technology, Tehran (Iran, Islamic Republic of)

2011-05-15

Research highlights: > Developing an explicit solution for the diffusion equation in spheroidal geometry. > Proving an orthogonality relation for spheroidal eigenfunctions. > Developing a relation for the extrapolation distance in spheroidal geometry. > Considering the sphere and slab as limiting cases for a spheroid. > Cross-validation of the analytical solution with Monte Carlo simulations. - Abstract: The time-dependent diffusion of neutrons in a spheroid as a function of the focal distance has been studied. The solution is based on an orthogonal basis and an extrapolation distanced related boundary condition for the spheroidal geometry. It has been shown that spheres and disks are two limiting cases for the spheroids, for which there is a smooth transition for the systems properties between these two limits. Furthermore, it is demonstrated that a slight deformation from a sphere does not affect the fundamental mode properties, to the first order. The calculations for both multiplying and non-multiplying media have been undertaken, showing good agreement with direct Monte Carlo simulations.

Second order time evolution of the multigroup diffusion and P1 equations for radiation transport

International Nuclear Information System (INIS)

Olson, Gordon L.

2011-01-01

Highlights: → An existing multigroup transport algorithm is extended to be second-order in time. → A new algorithm is presented that does not require a grey acceleration solution. → The two algorithms are tested with 2D, multi-material problems. → The two algorithms have comparable computational requirements. - Abstract: An existing solution method for solving the multigroup radiation equations, linear multifrequency-grey acceleration, is here extended to be second order in time. This method works for simple diffusion and for flux-limited diffusion, with or without material conduction. A new method is developed that does not require the solution of an averaged grey transport equation. It is effective solving both the diffusion and P 1 forms of the transport equation. Two dimensional, multi-material test problems are used to compare the solution methods.

Support Operators Method for the Diffusion Equation in Multiple Materials

Energy Technology Data Exchange (ETDEWEB)

Winters, Andrew R. [Los Alamos National Laboratory; Shashkov, Mikhail J. [Los Alamos National Laboratory

2012-08-14

A second-order finite difference scheme for the solution of the diffusion equation on non-uniform meshes is implemented. The method allows the heat conductivity to be discontinuous. The algorithm is formulated on a one dimensional mesh and is derived using the support operators method. A key component of the derivation is that the discrete analog of the flux operator is constructed to be the negative adjoint of the discrete divergence, in an inner product that is a discrete analog of the continuum inner product. The resultant discrete operators in the fully discretized diffusion equation are symmetric and positive definite. The algorithm is generalized to operate on meshes with cells which have mixed material properties. A mechanism to recover intermediate temperature values in mixed cells using a limited linear reconstruction is introduced. The implementation of the algorithm is verified and the linear reconstruction mechanism is compared to previous results for obtaining new material temperatures.

Density-Dependent Conformable Space-time Fractional Diffusion-Reaction Equation and Its Exact Solutions

Science.gov (United States)

Hosseini, Kamyar; Mayeli, Peyman; Bekir, Ahmet; Guner, Ozkan

2018-01-01

In this article, a special type of fractional differential equations (FDEs) named the density-dependent conformable fractional diffusion-reaction (DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the \\exp (-φ (\\varepsilon )) -expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.

HEXAGA-II-120, -60, -30 two-dimensional multi-group neutron diffusion programmes for a uniform triangular mesh with arbitrary group scattering

International Nuclear Information System (INIS)

Woznicki, Z.

1979-06-01

This report presents the AGA two-sweep iterative methods belonging to the family of factorization techniques in their practical application in the HEXAGA-II two-dimensional programme to obtain the numerical solution to the multi-group, time-independent, (real and/or adjoint) neutron diffusion equations for a fine uniform triangular mesh. An arbitrary group scattering model is permitted. The report written for the users provides the description of input and output. The use of HEXAGA-II is illustrated by two sample reactor problems. (orig.) [de

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.

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.)

Natural gas diffusion model and diffusion computation in well Cai25 Bashan Group oil and gas reservoir

Institute of Scientific and Technical Information of China (English)

无

2001-01-01

Natural gas diffusion through the cap rock is mainly by means ofdissolving in water, so its concentration can be replaced by solubility, which varies with temperature, pressure and salinity in strata. Under certain geological conditions the maximal solubility is definite, so the diffusion com-putation can be handled approximately by stable state equation. Furthermore, on the basis of the restoration of the paleo-buried history, the diffusion is calculated with the dynamic method, and the result is very close to the real diffusion value in the geological history.

Numerical solution of the equation of neutrons transport on plane geometry by analytical schemes using acceleration by synthetic diffusion

International Nuclear Information System (INIS)

Alonso-Vargas, G.

1991-01-01

A computer program has been developed which uses a technique of synthetic acceleration by diffusion by analytical schemes. Both in the diffusion equation as in that of transport, analytical schemes were used which allowed a substantial time saving in the number of iterations required by source iteration method to obtain the K e ff. The program developed ASD (Synthetic Diffusion Acceleration) by diffusion was written in FORTRAN and can be executed on a personal computer with a hard disc and mathematical O-processor. The program is unlimited as to the number of regions and energy groups. The results obtained by the ASD program for K e ff is nearly completely concordant with those of obtained utilizing the ANISN-PC code for different analytical type problems in this work. The ASD program allowed obtention of an approximate solution of the neutron transport equation with a relatively low number of internal reiterations with good precision. One of its applications would be in the direct determinations of axial distribution neutronic flow in a fuel assembly as well as in the obtention of the effective multiplication factor. (Author)

Generalization of the Nernst-Einstein equation for self-diffusion in high-defect-concentration solids

International Nuclear Information System (INIS)

McKee, R.A.

1981-01-01

It is shown that the Nernst-Einstein equation can be generalized for a high defect concentration solid to relate the mobility or conductivity to the self-diffusion coefficient. This relationship is derived assuming that the diffusing particles interact strongly and that the mobility is concentration-dependent. It is derived for interstitial disordered structures, but it is perfectly general to any mechanism of self diffusion as long as diffusion in a pure system is considered

Applicability of the Fokker-Planck equation to the description of diffusion effects on nucleation

Science.gov (United States)

Sorokin, M. V.; Dubinko, V. I.; Borodin, V. A.

2017-01-01

The nucleation of islands in a supersaturated solution of surface adatoms is considered taking into account the possibility of diffusion profile formation in the island vicinity. It is shown that the treatment of diffusion-controlled cluster growth in terms of the Fokker-Planck equation is justified only provided certain restrictions are satisfied. First of all, the standard requirement that diffusion profiles of adatoms quickly adjust themselves to the actual island sizes (adiabatic principle) can be realized only for sufficiently high island concentration. The adiabatic principle is essential for the probabilities of adatom attachment to and detachment from island edges to be independent of the adatom diffusion profile establishment kinetics, justifying the island nucleation treatment as the Markovian stochastic process. Second, it is shown that the commonly used definition of the "diffusion" coefficient in the Fokker-Planck equation in terms of adatom attachment and detachment rates is justified only provided the attachment and detachment are statistically independent, which is generally not the case for the diffusion-limited growth of islands. We suggest a particular way to define the attachment and detachment rates that allows us to satisfy this requirement as well. When applied to the problem of surface island nucleation, our treatment predicts the steady-state nucleation barrier, which coincides with the conventional thermodynamic expression, even though no thermodynamic equilibrium is assumed and the adatom diffusion is treated explicitly. The effect of adatom diffusional profiles on the nucleation rate preexponential factor is also discussed. Monte Carlo simulation is employed to analyze the applicability domain of the Fokker-Planck equation and the diffusion effect beyond it. It is demonstrated that a diffusional cloud is slowing down the nucleation process for a given monomer interaction with the nucleus edge.

Bifurcation of positive solutions to scalar reaction-diffusion equations with nonlinear boundary condition

Science.gov (United States)

Liu, Ping; Shi, Junping

2018-01-01

The bifurcation of non-trivial steady state solutions of a scalar reaction-diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.

A meshless method for solving two-dimensional variable-order time fractional advection-diffusion equation

Science.gov (United States)

Tayebi, A.; Shekari, Y.; Heydari, M. H.

2017-07-01

Several physical phenomena such as transformation of pollutants, energy, particles and many others can be described by the well-known convection-diffusion equation which is a combination of the diffusion and advection equations. In this paper, this equation is generalized with the concept of variable-order fractional derivatives. The generalized equation is called variable-order time fractional advection-diffusion equation (V-OTFA-DE). An accurate and robust meshless method based on the moving least squares (MLS) approximation and the finite difference scheme is proposed for its numerical solution on two-dimensional (2-D) arbitrary domains. In the time domain, the finite difference technique with a θ-weighted scheme and in the space domain, the MLS approximation are employed to obtain appropriate semi-discrete solutions. Since the newly developed method is a meshless approach, it does not require any background mesh structure to obtain semi-discrete solutions of the problem under consideration, and the numerical solutions are constructed entirely based on a set of scattered nodes. The proposed method is validated in solving three different examples including two benchmark problems and an applied problem of pollutant distribution in the atmosphere. In all such cases, the obtained results show that the proposed method is very accurate and robust. Moreover, a remarkable property so-called positive scheme for the proposed method is observed in solving concentration transport phenomena.

Moving-boundary problems for the time-fractional diffusion equation

Directory of Open Access Journals (Sweden)

Sabrina D. Roscani

2017-02-01

Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.

Determination of neutron buildup factor using analytical solution of one-dimensional neutron diffusion equation in cylindrical geometry

Energy Technology Data Exchange (ETDEWEB)

Fernandes, Julio Cesar L.; Vilhena, Marco Tullio, E-mail: julio.lombaldo@ufrgs.b, E-mail: vilhena@pq.cnpq.b [Universidade Federal do Rio Grande do Sul (DMPA/UFRGS), Porto Alegre, RS (Brazil). Dept. de Matematica Pura e Aplicada. Programa de Pos Graduacao em Matematica Aplicada; Borges, Volnei; Bodmann, Bardo Ernest, E-mail: bardo.bodmann@ufrgs.b, E-mail: borges@ufrgs.b [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

2011-07-01

The principal idea of this work, consist on formulate an analytical method to solved problems for diffusion of neutrons with isotropic scattering in one-dimensional cylindrical geometry. In this area were develop many works that study the same problem in different system of coordinates as well as cartesian system, nevertheless using numerical methods to solve the shielding problem. In view of good results in this works, we starting with the idea that we can represent a source in the origin of the cylindrical system by a Delta Dirac distribution, we describe the physical modeling and solved the neutron diffusion equation inside of cylinder of radius R. For the case of transport equation, the formulation of discrete ordinates S{sub N} consists in discretize the angular variables in N directions and in using a quadrature angular set for approximate the sources of scattering, where the Diffusion equation consist on S{sub 2} approximated transport equation in discrete ordinates. We solved the neutron diffusion equation with an analytical form by the finite Hankel transform. Was presented also the build-up factor for the case that we have neutron flux inside the cylinder. (author)

Determination of neutron buildup factor using analytical solution of one-dimensional neutron diffusion equation in cylindrical geometry

International Nuclear Information System (INIS)

Fernandes, Julio Cesar L.; Vilhena, Marco Tullio; Borges, Volnei; Bodmann, Bardo Ernest

2011-01-01

The principal idea of this work, consist on formulate an analytical method to solved problems for diffusion of neutrons with isotropic scattering in one-dimensional cylindrical geometry. In this area were develop many works that study the same problem in different system of coordinates as well as cartesian system, nevertheless using numerical methods to solve the shielding problem. In view of good results in this works, we starting with the idea that we can represent a source in the origin of the cylindrical system by a Delta Dirac distribution, we describe the physical modeling and solved the neutron diffusion equation inside of cylinder of radius R. For the case of transport equation, the formulation of discrete ordinates S N consists in discretize the angular variables in N directions and in using a quadrature angular set for approximate the sources of scattering, where the Diffusion equation consist on S 2 approximated transport equation in discrete ordinates. We solved the neutron diffusion equation with an analytical form by the finite Hankel transform. Was presented also the build-up factor for the case that we have neutron flux inside the cylinder. (author)

On the renormalization group equations of quantum electrodynamics

International Nuclear Information System (INIS)

Hirayama, Minoru

1980-01-01

The renormalization group equations of quantum electrodynamics are discussed. The solution of the Gell-Mann-Low equation is presented in a convenient form. The interrelation between the Nishijima-Tomozawa equation and the Gell-Mann-Low equation is clarified. The reciprocal effective charge, so to speak, turns out to play an important role to discuss renormalization group equations. Arguments are given that the reciprocal effective charge vanishes as the renormalization momentum tends to infinity. (author)

A finite difference method for space fractional differential equations with variable diffusivity coefficient

KAUST Repository

Mustapha, K.

2017-06-03

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\\\\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

A finite difference method for space fractional differential equations with variable diffusivity coefficient

KAUST Repository

Mustapha, K.; Furati, K.; Knio, Omar; Maitre, O. Le

2017-01-01

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

On the solutions of fractional reaction-diffusion equations

Directory of Open Access Journals (Sweden)

Jagdev Singh

2013-05-01

Full Text Available In this paper, we obtain the solution of a fractional reaction-diffusion equation associated with the generalized Riemann-Liouville fractional derivative as the time derivative and Riesz-Feller fractional derivative as the space-derivative. The results are derived by the application of the Laplace and Fourier transforms in compact and elegant form in terms of Mittag-Leffler function and H-function. The results obtained here are of general nature and include the results investigated earlier by many authors.

An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations

KAUST Repository

Burrage, Kevin; Hale, Nicholas; Kay, David

2012-01-01

Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time

Instability of traveling waves of the convective-diffusive Cahn-Hilliard equation

International Nuclear Information System (INIS)

Gao Hongjun; Liu Changchun

2004-01-01

In this paper we study the instability of the traveling waves of the convective-diffusive Cahn-Hilliard equation. We prove that it is nonlinearly unstable under H 2 perturbations, for some traveling wave solution that is asymptotic to a constant as x→∞

Galerkin method for solving diffusion equations

International Nuclear Information System (INIS)

Tsapelkin, E.S.

1975-01-01

A programme for the solution of the three-dimensional two-group multizone neutron diffusion problem in (x, y, z)-geometry is described. The programme XYZ-5 gives the currents of both groups, the effective neutron multiplication coefficient and several integral properties of the reactor. The solution was found with the Galerkin method using speciallly constructed and chosen coordinate functions. The programme is written in ALGOL-60 and consists of 5 parts. Its text is given

A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations

Science.gov (United States)

Chen, Hao; Lv, Wen; Zhang, Tongtong

2018-05-01

We study preconditioned iterative methods for the linear system arising in the numerical discretization of a two-dimensional space-fractional diffusion equation. Our approach is based on a formulation of the discrete problem that is shown to be the sum of two Kronecker products. By making use of an alternating Kronecker product splitting iteration technique we establish a class of fixed-point iteration methods. Theoretical analysis shows that the new method converges to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real. The basic iteration is accelerated by a Krylov subspace method like GMRES. The corresponding preconditioner is in a form of a Kronecker product structure and requires at each iteration the solution of a set of discrete one-dimensional fractional diffusion equations. We use structure preserving approximations to the discrete one-dimensional fractional diffusion operators in the action of the preconditioning matrix. Numerical examples are presented to illustrate the effectiveness of this approach.

Enhanced finite difference scheme for the neutron diffusion equation using the importance function

International Nuclear Information System (INIS)

Vagheian, Mehran; Vosoughi, Naser; Gharib, Morteza

2016-01-01

Highlights: • An enhanced finite difference scheme for the neutron diffusion equation is proposed. • A seven-step algorithm is considered based on the importance function. • Mesh points are distributed through entire reactor core with respect to the importance function. • The results all proved that the proposed algorithm is highly efficient. - Abstract: Mesh point positions in Finite Difference Method (FDM) of discretization for the neutron diffusion equation can remarkably affect the averaged neutron fluxes as well as the effective multiplication factor. In this study, by aid of improving the mesh point positions, an enhanced finite difference scheme for the neutron diffusion equation is proposed based on the neutron importance function. In order to determine the neutron importance function, the adjoint (backward) neutron diffusion calculations are performed in the same procedure as for the forward calculations. Considering the neutron importance function, the mesh points can be improved through the entire reactor core. Accordingly, in regions with greater neutron importance, density of mesh elements is higher than that in regions with less importance. The forward calculations are then performed for both of the uniform and improved non-uniform mesh point distributions and the results (the neutron fluxes along with the corresponding eigenvalues) for the two cases are compared with each other. The results are benchmarked against the reference values (with fine meshes) for Kang and Rod Bundle BWR benchmark problems. These benchmark cases revealed that the improved non-uniform mesh point distribution is highly efficient.

A Hennart nodal method for the diffusion equation

International Nuclear Information System (INIS)

Lesaint, P.; Noceir, S.; Verwaerde, D.

1995-01-01

A modification of the Hennart nodal method for neutron diffusion problems is presented. The final system of equations obtained by this method is not positive definite. However, a flux elimination technique leads to a simple positive definite system, which can be solved by the traditional iterative methods. Calculations of a two-dimensional International Atomic Energy Agency benchmark problem are performed and compared with results of the original Hennart nodal method and some finite element methods. The high computational efficiency of this modified nodal method is clearly demonstrated

A boundary integral equation for boundary element applications in multigroup neutron diffusion theory

International Nuclear Information System (INIS)

Ozgener, B.

1998-01-01

A boundary integral equation (BIE) is developed for the application of the boundary element method to the multigroup neutron diffusion equations. The developed BIE contains no explicit scattering term; the scattering effects are taken into account by redefining the unknowns. Boundary elements of the linear and constant variety are utilised for validation of the developed boundary integral formulation

Solution of differential equations by application of transformation groups

Science.gov (United States)

Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.

1968-01-01

Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.

Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates

Directory of Open Access Journals (Sweden)

Ya-Juan Hao

2013-01-01

Full Text Available The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.

Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

Directory of Open Access Journals (Sweden)

Ai-Min Yang

2013-01-01

Full Text Available We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

On an adaptive time stepping strategy for solving nonlinear diffusion equations

International Nuclear Information System (INIS)

Chen, K.; Baines, M.J.; Sweby, P.K.

1993-01-01

A new time step selection procedure is proposed for solving non- linear diffusion equations. It has been implemented in the ASWR finite element code of Lorenz and Svoboda [10] for 2D semiconductor process modelling diffusion equations. The strategy is based on equi-distributing the local truncation errors of the numerical scheme. The use of B-splines for interpolation (as well as for the trial space) results in a banded and diagonally dominant matrix. The approximate inverse of such a matrix can be provided to a high degree of accuracy by another banded matrix, which in turn can be used to work out the approximate finite difference scheme corresponding to the ASWR finite element method, and further to calculate estimates of the local truncation errors of the numerical scheme. Numerical experiments on six full simulation problems arising in semiconductor process modelling have been carried out. Results show that our proposed strategy is more efficient and better conserves the total mass. 18 refs., 6 figs., 2 tabs

HEXAGA-III-120, -30. Three dimensional multi-group neutron diffusion programmes for a uniform triangular mesh with arbitrary group scattering

International Nuclear Information System (INIS)

Woznicki, Z.I.

1983-07-01

This report presents the HEXAGA-III-programme solving multi-group time-independent real and/or adjoint neutron diffusion equations for three-dimensional-triangular-z-geometry. The method of solution is based on the AGA two-sweep iterative method belonging to the family of factorization techniques. An arbitrary neutron scattering model is permitted. The report written for users provides the description of the programme input and output and the use of HEXAGA-III is illustrated by a sample reactor problem. (orig.) [de

New diffusion-like solutions of one-speed transport equations in spherical geometry

International Nuclear Information System (INIS)

Sahni, D.C.

1988-01-01

Stationary, one-speed, spherically symmetric transport equations are considered in a conservative medium. Closed-form expressions are obtained for the angular flux ψ(r, μ) that yield a total flux varying as 1/r by using Sonine transforms. Properties of this solution are studied and it is shown that the solution can not be identified as a diffusion mode solution of the transport equation. Limitations of the Sonine transform technique are noted. (author)

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

Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

OpenAIRE

Cortazar, C.; Elgueta, M.; Quiros, F.; Wolanski, N.

2011-01-01

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, $u_t=J*u-u:=Lu$, in an exterior domain, $\\Omega$, which excludes one or several holes, and with zero Dirichlet data on $\\mathbb{R}^N\\setminus\\Omega$. When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves...

Solution of the atmospheric diffusion equation with a realistic diffusion coefficient and time dependent mixing height

International Nuclear Information System (INIS)

Mayhoub, A.B.; Etman, S.M.

1997-01-01

One dimensional model for the dispersion of a passive atmospheric contaminant (neglecting chemical reactions) in the atmospheric boundary layer is introduced. The differential equation representing the dispersion of pollutants is solved on the basis of gradient-transfer theory (K- theory). The present approach deals with a more appropriate and realistic profile for the diffusion coefficient K, which is expressed in terms of the friction velocity U, the vertical coordinate z and the depth of the mixing layer h, which is taken time dependent. After some mathematical simplification, the equation analytic obtained solution can be easily applied to case study concerning atmospheric dispersion of pollutants

Improved diffusion coefficients generated from Monte Carlo codes

International Nuclear Information System (INIS)

Herman, B. R.; Forget, B.; Smith, K.; Aviles, B. N.

2013-01-01

Monte Carlo codes are becoming more widely used for reactor analysis. Some of these applications involve the generation of diffusion theory parameters including macroscopic cross sections and diffusion coefficients. Two approximations used to generate diffusion coefficients are assessed using the Monte Carlo code MC21. The first is the method of homogenization; whether to weight either fine-group transport cross sections or fine-group diffusion coefficients when collapsing to few-group diffusion coefficients. The second is a fundamental approximation made to the energy-dependent P1 equations to derive the energy-dependent diffusion equations. Standard Monte Carlo codes usually generate a flux-weighted transport cross section with no correction to the diffusion approximation. Results indicate that this causes noticeable tilting in reconstructed pin powers in simple test lattices with L2 norm error of 3.6%. This error is reduced significantly to 0.27% when weighting fine-group diffusion coefficients by the flux and applying a correction to the diffusion approximation. Noticeable tilting in reconstructed fluxes and pin powers was reduced when applying these corrections. (authors)

Modeling single-file diffusion with step fractional Brownian motion and a generalized fractional Langevin equation

International Nuclear Information System (INIS)

Lim, S C; Teo, L P

2009-01-01

Single-file diffusion behaves as normal diffusion at small time and as subdiffusion at large time. These properties can be described in terms of fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We introduce a new stochastic process called Riemann–Liouville step fractional Brownian motion which can be regarded as a special case of multifractional Brownian motion with a step function type of Hurst exponent tailored for single-file diffusion. Such a step fractional Brownian motion can be obtained as a solution of the fractional Langevin equation with zero damping. Various kinds of fractional Langevin equations and their generalizations are then considered in order to decide whether their solutions provide the correct description of the long and short time behaviors of single-file diffusion. The cases where the dissipative memory kernel is a Dirac delta function, a power-law function and a combination of these functions are studied in detail. In addition to the case where the short time behavior of single-file diffusion behaves as normal diffusion, we also consider the possibility of a process that begins as ballistic motion

Molecular dynamics on diffusive time scales from the phase-field-crystal equation.

Science.gov (United States)

Chan, Pak Yuen; Goldenfeld, Nigel; Dantzig, Jon

2009-03-01

We extend the phase-field-crystal model to accommodate exact atomic configurations and vacancies by requiring the order parameter to be non-negative. The resulting theory dictates the number of atoms and describes the motion of each of them. By solving the dynamical equation of the model, which is a partial differential equation, we are essentially performing molecular dynamics simulations on diffusive time scales. To illustrate this approach, we calculate the two-point correlation function of a fluid.

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

Energy Technology Data Exchange (ETDEWEB)

Bailey, T S; Adams, M L [Texas A M Univ., Dept. of Nuclear Engineering, College Station, TX (United States); Yang, B; Zika, M R [Lawrence Livermore National Lab., Livermore, CA (United States)

2005-07-01

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2-dimensional) or polyhedral (3-dimensional) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids. (authors)

Generalized Analytical Treatment Of The Source Strength In The Solution Of The Diffusion Equation

International Nuclear Information System (INIS)

Essa, Kh.S.M.; EI-Otaify, M.S.

2007-01-01

The source release strength (which is an integral part of the mathematical formulation of the diffusion equation) together with the boundary conditions leads to three different forms of the diffusion equation. The obtained forms have been solved analytically under different boundary conditions, by using transformation of axis, cosine, and Fourier transformation. Three equivalent alternative mathematical formulations of the problem have been obtained. The estimated solution of the concentrations at the ground source has been used for comparison with observed concentrations data for SF 6 tracer experiments in low wind and unstable conditions at lIT Delhi sports ground. A good agreement between estimated and observed concentrations is found

Hybrid simplified spherical harmonics with diffusion equation for light propagation in tissues

International Nuclear Information System (INIS)

Chen, Xueli; Sun, Fangfang; Yang, Defu; Ren, Shenghan; Liang, Jimin; Zhang, Qian

2015-01-01

Aiming at the limitations of the simplified spherical harmonics approximation (SP N ) and diffusion equation (DE) in describing the light propagation in tissues, a hybrid simplified spherical harmonics with diffusion equation (HSDE) based diffuse light transport model is proposed. In the HSDE model, the living body is first segmented into several major organs, and then the organs are divided into high scattering tissues and other tissues. DE and SP N are employed to describe the light propagation in these two kinds of tissues respectively, which are finally coupled using the established boundary coupling condition. The HSDE model makes full use of the advantages of SP N and DE, and abandons their disadvantages, so that it can provide a perfect balance between accuracy and computation time. Using the finite element method, the HSDE is solved for light flux density map on body surface. The accuracy and efficiency of the HSDE are validated with both regular geometries and digital mouse model based simulations. Corresponding results reveal that a comparable accuracy and much less computation time are achieved compared with the SP N model as well as a much better accuracy compared with the DE one. (paper)

Hybrid simplified spherical harmonics with diffusion equation for light propagation in tissues.

Science.gov (United States)

Chen, Xueli; Sun, Fangfang; Yang, Defu; Ren, Shenghan; Zhang, Qian; Liang, Jimin

2015-08-21

Aiming at the limitations of the simplified spherical harmonics approximation (SPN) and diffusion equation (DE) in describing the light propagation in tissues, a hybrid simplified spherical harmonics with diffusion equation (HSDE) based diffuse light transport model is proposed. In the HSDE model, the living body is first segmented into several major organs, and then the organs are divided into high scattering tissues and other tissues. DE and SPN are employed to describe the light propagation in these two kinds of tissues respectively, which are finally coupled using the established boundary coupling condition. The HSDE model makes full use of the advantages of SPN and DE, and abandons their disadvantages, so that it can provide a perfect balance between accuracy and computation time. Using the finite element method, the HSDE is solved for light flux density map on body surface. The accuracy and efficiency of the HSDE are validated with both regular geometries and digital mouse model based simulations. Corresponding results reveal that a comparable accuracy and much less computation time are achieved compared with the SPN model as well as a much better accuracy compared with the DE one.

Nonperturbative renormalization group study of the stochastic Navier-Stokes equation.

Science.gov (United States)

Mejía-Monasterio, Carlos; Muratore-Ginanneschi, Paolo

2012-07-01

We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4-2ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant.

A asymptotic numerical method for the steady-state convection diffusion equation

International Nuclear Information System (INIS)

Wu Qiguang

1988-01-01

In this paper, A asymptotic numerical method for the steady-state Convection diffusion equation is proposed, which need not take very fine mesh size in the neighbourhood of the boundary layer. Numerical computation for model problem show that we can obtain the numerical solution in the boundary layer with moderate step size

Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation

Science.gov (United States)

Agarwal, P.; El-Sayed, A. A.

2018-06-01

In this paper, a new numerical technique for solving the fractional order diffusion equation is introduced. This technique basically depends on the Non-Standard finite difference method (NSFD) and Chebyshev collocation method, where the fractional derivatives are described in terms of the Caputo sense. The Chebyshev collocation method with the (NSFD) method is used to convert the problem into a system of algebraic equations. These equations solved numerically using Newton's iteration method. The applicability, reliability, and efficiency of the presented technique are demonstrated through some given numerical examples.

Reduced equations of motion for quantum systems driven by diffusive Markov processes.

Science.gov (United States)

Sarovar, Mohan; Grace, Matthew D

2012-09-28

The expansion of a stochastic Liouville equation for the coupled evolution of a quantum system and an Ornstein-Uhlenbeck process into a hierarchy of coupled differential equations is a useful technique that simplifies the simulation of stochastically driven quantum systems. We expand the applicability of this technique by completely characterizing the class of diffusive Markov processes for which a useful hierarchy of equations can be derived. The expansion of this technique enables the examination of quantum systems driven by non-Gaussian stochastic processes with bounded range. We present an application of this extended technique by simulating Stark-tuned Förster resonance transfer in Rydberg atoms with nonperturbative position fluctuations.

Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction–diffusion equation

KAUST Repository

Luchko, Yuri

2013-05-30

In this paper, we consider a reaction-diffusion problem with an unknown nonlinear source function that has to be determined from overposed data. The underlying model is in the form of a time-fractional reaction-diffusion equation and the work generalizes some known results for the inverse problems posed for PDEs of parabolic type. For the inverse problem under consideration, a uniqueness result is proved and a numerical algorithm with some theoretical qualification is presented in the one-dimensional case. The key both to the uniqueness result and to the numerical algorithm relies on the maximum principle which has recently been shown to hold for the fractional diffusion equation. In order to show the effectiveness of the proposed method, results of numerical simulations are presented. © 2013 IOP Publishing Ltd.

Analytical approximate solutions of the time-domain diffusion equation in layered slabs.

Science.gov (United States)

Martelli, Fabrizio; Sassaroli, Angelo; Yamada, Yukio; Zaccanti, Giovanni

2002-01-01

Time-domain analytical solutions of the diffusion equation for photon migration through highly scattering two- and three-layered slabs have been obtained. The effect of the refractive-index mismatch with the external medium is taken into account, and approximate boundary conditions at the interface between the diffusive layers have been considered. A Monte Carlo code for photon migration through a layered slab has also been developed. Comparisons with the results of Monte Carlo simulations showed that the analytical solutions correctly describe the mean path length followed by photons inside each diffusive layer and the shape of the temporal profile of received photons, while discrepancies are observed for the continuous-wave reflectance or transmittance.

Analysis of nonlinear parabolic equations modeling plasma diffusion across a magnetic field

International Nuclear Information System (INIS)

Hyman, J.M.; Rosenau, P.

1984-01-01

We analyse the evolutionary behavior of the solution of a pair of coupled quasilinear parabolic equations modeling the diffusion of heat and mass of a magnetically confined plasma. The solutions's behavior, due to the nonlinear diffusion coefficients, exhibits many new phenomena. In short time, the solution converges into a highly organized symmetric pattern that is almost completely independent of initial data. The asymptotic dynamics then become very simple and take place in a finite dimensional space. These conclusions are backed by extensive numerical experimentation

Domain decomposition method for solving the neutron diffusion equation

International Nuclear Information System (INIS)

Coulomb, F.

1989-03-01

The aim of this work is to study methods for solving the neutron diffusion equation; we are interested in methods based on a classical finite element discretization and well suited for use on parallel computers. Domain decomposition methods seem to answer this preoccupation. This study deals with a decomposition of the domain. A theoretical study is carried out for Lagrange finite elements and some examples are given; in the case of mixed dual finite elements, the study is based on examples [fr

New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis.

Science.gov (United States)

Ingo, Carson; Magin, Richard L; Parrish, Todd B

2014-11-01

Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag-Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.

New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis

Directory of Open Access Journals (Sweden)

Carson Ingo

2014-11-01

Full Text Available Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.

Pseudodynamic systems approach based on a quadratic approximation of update equations for diffuse optical tomography.

Science.gov (United States)

Biswas, Samir Kumar; Kanhirodan, Rajan; Vasu, Ram Mohan; Roy, Debasish

2011-08-01

We explore a pseudodynamic form of the quadratic parameter update equation for diffuse optical tomographic reconstruction from noisy data. A few explicit and implicit strategies for obtaining the parameter updates via a semianalytical integration of the pseudodynamic equations are proposed. Despite the ill-posedness of the inverse problem associated with diffuse optical tomography, adoption of the quadratic update scheme combined with the pseudotime integration appears not only to yield higher convergence, but also a muted sensitivity to the regularization parameters, which include the pseudotime step size for integration. These observations are validated through reconstructions with both numerically generated and experimentally acquired data.

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

Energy Technology Data Exchange (ETDEWEB)

Bailey, T.S.; Adams, M.L. [Texas A M Univ., Dept. of Nuclear Engineering, College Station, TX (United States); Yang, B.; Zika, M.R. [Lawrence Livermore National Lab., Livermore, CA (United States)

2005-07-01

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses piecewise linear weight and basis functions in the finite element approximation, and it can be applied on arbitrary polygonal (2-dimensional) or polyhedral (3-dimensional) grids. We show that this new PWL method gives solutions comparable to those from Palmer's finite-volume method. However, since the PWL method produces a symmetric positive definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids. (authors)

Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates

Directory of Open Access Journals (Sweden)

Povstenko YZ

2011-01-01

Full Text Available Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.

Bifurcation Analysis of Gene Propagation Model Governed by Reaction-Diffusion Equations

Directory of Open Access Journals (Sweden)

Guichen Lu

2016-01-01

Full Text Available We present a theoretical analysis of the attractor bifurcation for gene propagation model governed by reaction-diffusion equations. We investigate the dynamical transition problems of the model under the homogeneous boundary conditions. By using the dynamical transition theory, we give a complete characterization of the bifurcated objects in terms of the biological parameters of the problem.

Multigrid solution of the convection-diffusion equation with high-Reynolds number

Energy Technology Data Exchange (ETDEWEB)

Zhang, Jun [George Washington Univ., Washington, DC (United States)

1996-12-31

A fourth-order compact finite difference scheme is employed with the multigrid technique to solve the variable coefficient convection-diffusion equation with high-Reynolds number. Scaled inter-grid transfer operators and potential on vectorization and parallelization are discussed. The high-order multigrid method is unconditionally stable and produces solution of 4th-order accuracy. Numerical experiments are included.

Solution of 3-dimensional diffusion equation by finite Fourier transformation

International Nuclear Information System (INIS)

Krishnani, P.D.

1978-01-01

Three dimensional diffusion equation in Cartesian co-ordinates is solved by using the finite Fourier transformation. This method is different from the usual Fourier transformation method in the sense that the solutions are obtained without performing the inverse Fourier transformation. The advantage has been taken of the fact that the flux is finite and integrable in the finite region. By applying this condition, a two-dimensional integral equation, involving flux and its normal derivative at the boundary, is obtained. By solving this equation with given boundary conditions, all of the boundary values are determined. In order to calculate the flux inside the region, flux is expanded into three-dimensional Fourier series. The Fourier coefficients of the flux in the region are calculated from the boundary values. The advantage of this method is that the integrated flux is obtained without knowing the fluxes inside the region as in the case of finite difference method. (author)

Global Regularity and Time Decay for the 2D Magnetohydrodynamic Equations with Fractional Dissipation and Partial Magnetic Diffusion

Science.gov (United States)

Dong, Bo-Qing; Jia, Yan; Li, Jingna; Wu, Jiahong

2018-05-01

This paper focuses on a system of the 2D magnetohydrodynamic (MHD) equations with the kinematic dissipation given by the fractional operator (-Δ )^α and the magnetic diffusion by partial Laplacian. We are able to show that this system with any α >0 always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion. The results presented here are the sharpest on the global regularity problem for the 2D MHD equations with only partial magnetic diffusion.

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.

HEXAGA-II. A two-dimensional multi-group neutron diffusion programme for a uniform triangular mesh with arbitrary group scattering for the IBM/370-168 computer

International Nuclear Information System (INIS)

Woznicki, Z.

1976-05-01

This report presents the AGA two-sweep iterative methods belonging to the family of factorization techniques in their practical application in the HEXAGA-II two-dimensional programme to obtain the numerical solution to the multi-group, time-independent, (real and/or adjoint) neutron diffusion equations for a fine uniform triangular mesh. An arbitrary group scattering model is permitted. The report written for the users provides the description of input and output. The use of HEXAGA-II is illustrated by two sample reactor problems. (orig.) [de

Chaotic dynamics and diffusion in a piecewise linear equation

International Nuclear Information System (INIS)

Shahrear, Pabel; Glass, Leon; Edwards, Rod

2015-01-01

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems

Chaotic dynamics and diffusion in a piecewise linear equation

Science.gov (United States)

Shahrear, Pabel; Glass, Leon; Edwards, Rod

2015-03-01

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system

Directory of Open Access Journals (Sweden)

Md. Nur Alam

2016-06-01

Full Text Available In this article, we apply the exp(-Φ(ξ-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.

Development of Multigrid Methods for diffusion, Advection, and the incompressible Navier-Stokes Equations

Energy Technology Data Exchange (ETDEWEB)

Gjesdal, Thor

1997-12-31

This thesis discusses the development and application of efficient numerical methods for the simulation of fluid flows, in particular the flow of incompressible fluids. The emphasis is on practical aspects of algorithm development and on application of the methods either to linear scalar model equations or to the non-linear incompressible Navier-Stokes equations. The first part deals with cell centred multigrid methods and linear correction scheme and presents papers on (1) generalization of the method to arbitrary sized grids for diffusion problems, (2) low order method for advection-diffusion problems, (3) attempt to extend the basic method to advection-diffusion problems, (4) Fourier smoothing analysis of multicolour relaxation schemes, and (5) analysis of high-order discretizations for advection terms. The second part discusses a multigrid based on pressure correction methods, non-linear full approximation scheme, and papers on (1) systematic comparison of the performance of different pressure correction smoothers and some other algorithmic variants, low to moderate Reynolds numbers, and (2) systematic study of implementation strategies for high order advection schemes, high-Re flow. An appendix contains Fortran 90 data structures for multigrid development. 160 refs., 26 figs., 22 tabs.

How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems

OpenAIRE

Cortazar, C.; Elgueta, M.; Rossi, J. D.; Wolanski, N.

2006-01-01

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.

An efficient explicit numerical scheme for diffusion-type equations with a highly inhomogeneous and highly anisotropic diffusion tensor

International Nuclear Information System (INIS)

Larroche, O.

2007-01-01

A locally split-step explicit (LSSE) algorithm was developed for efficiently solving a multi-dimensional advection-diffusion type equation involving a highly inhomogeneous and highly anisotropic diffusion tensor, which makes the problem very ill-conditioned for standard implicit methods involving the iterative solution of large linear systems. The need for such an optimized algorithm arises, in particular, in the frame of thermonuclear fusion applications, for the purpose of simulating fast charged-particle slowing-down with an ion Fokker-Planck code. The LSSE algorithm is presented in this paper along with the results of a model slowing-down problem to which it has been applied

Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion

Science.gov (United States)

Ślęzak, Jakub; Metzler, Ralf; Magdziarz, Marcin

2018-02-01

Recent advances in single particle tracking and supercomputing techniques demonstrate the emergence of normal or anomalous, viscoelastic diffusion in conjunction with non-Gaussian distributions in soft, biological, and active matter systems. We here formulate a stochastic model based on a generalised Langevin equation in which non-Gaussian shapes of the probability density function and normal or anomalous diffusion have a common origin, namely a random parametrisation of the stochastic force. We perform a detailed analysis demonstrating how various types of parameter distributions for the memory kernel result in exponential, power law, or power-log law tails of the memory functions. The studied system is also shown to exhibit a further unusual property: the velocity has a Gaussian one point probability density but non-Gaussian joint distributions. This behaviour is reflected in the relaxation from a Gaussian to a non-Gaussian distribution observed for the position variable. We show that our theoretical results are in excellent agreement with stochastic simulations.

Parallel computing for homogeneous diffusion and transport equations in neutronics; Calcul parallele pour les equations de diffusion et de transport homogenes en neutronique

Energy Technology Data Exchange (ETDEWEB)

Pinchedez, K

1999-06-01

Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)

A nonlinear Fokker-Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics

Science.gov (United States)

Marin, D.; Ribeiro, M. A.; Ribeiro, H. V.; Lenzi, E. K.

2018-07-01

We investigate the solutions for a set of coupled nonlinear Fokker-Planck equations coupled by the diffusion coefficient in presence of external forces. The coupling by the diffusion coefficient implies that the diffusion of each species is influenced by the other and vice versa due to this term, which represents an interaction among them. The solutions for the stationary case are given in terms of the Tsallis distributions, when arbitrary external forces are considered. We also use the Tsallis distributions to obtain a time dependent solution for a linear external force. The results obtained from this analysis show a rich class of behavior related to anomalous diffusion, which can be characterized by compact or long-tailed distributions.

Initial-boundary value problems for multi-term time-fractional diffusion equations with x-dependent coefficients

OpenAIRE

Li, Zhiyuan; Huang, Xinchi; Yamamoto, Masahiro

2018-01-01

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion L...

assessment of concentration of air pollutants using analytical and numerical solution of the atmospheric diffusion equation

International Nuclear Information System (INIS)

Esmail, S.F.H.

2011-01-01

The mathematical formulation of numerous physical problems a results in differential equations actually partial or ordinary differential equations.In our study we are interested in solutions of partial differential equations.The aim of this work is to calculate the concentrations of the pollution, by solving the atmospheric diffusion equation(ADE) using different mathematical methods of solution. It is difficult to solve the general form of ADE analytically, so we use some assumptions to get its solution.The solutions of it depend on the eddy diffusivity profiles(k) and the wind speed u. We use some physical assumptions to simplify its formula and solve it. In the present work, we solve the ADE analytically in three dimensions using Green's function method, Laplace transform method, normal mode method and these separation of variables method. Also, we use ADM as a numerical method. Finally, comparisons are made with the results predicted by the previous methods and the observed data.

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

Strong Maximum Principle for Multi-Term Time-Fractional Diffusion Equations and its Application to an Inverse Source Problem

OpenAIRE

Liu, Yikan

2015-01-01

In this paper, we establish a strong maximum principle for fractional diffusion equations with multiple Caputo derivatives in time, and investigate a related inverse problem of practical importance. Exploiting the solution properties and the involved multinomial Mittag-Leffler functions, we improve the weak maximum principle for the multi-term time-fractional diffusion equation to a stronger one, which is parallel to that for its single-term counterpart as expected. As a direct application, w...

Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation

International Nuclear Information System (INIS)

Li, Gongsheng; Zhang, Dali; Jia, Xianzheng; Yamamoto, Masahiro

2013-01-01

This paper deals with an inverse problem of simultaneously identifying the space-dependent diffusion coefficient and the fractional order in the 1D time-fractional diffusion equation with smooth initial functions by using boundary measurements. The uniqueness results for the inverse problem are proved on the basis of the inverse eigenvalue problem, and the Lipschitz continuity of the solution operator is established. A modified optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function is put forward for the discretization of the minimization problem. Numerical inversions are performed for the diffusion coefficient taking on different functional forms and the additional data having random noise. Several factors which have important influences on the realization of the algorithm are discussed, including the approximate space of the diffusion coefficient, the regularization parameter and the initial iteration. The inversion solutions are good approximations to the exact solutions with stability and adaptivity demonstrating that the optimal perturbation algorithm with the sigmoid-type regularization parameter is efficient for the simultaneous inversion. (paper)

Studies on the numerical solution of three-dimensional stationary diffusion equations using the finite element method

International Nuclear Information System (INIS)

Franke, H.P.

1976-05-01

The finite element method is applied to the solution of the stationary 3D group diffusion equations. For this, a programme system with the name of FEM3D is established which also includes a module for semi-automatic mesh generation. Tetrahedral finite elements are used. The neutron fluxes are described by complete first- or second-order Lagrangian polynomials. General homogeneous boundary conditions are allowed. The studies show that realistic three-dimensional problems can be solved at less expense by iterative methods, in particular so when especially adapted matrix handling and storage schemes are used efficiently. (orig./RW) [de

Prediction of the neutrons subcritical multiplication using the diffusion hybrid equation with external neutron sources

Energy Technology Data Exchange (ETDEWEB)

Costa da Silva, Adilson; Carvalho da Silva, Fernando [COPPE/UFRJ, Programa de Engenharia Nuclear, Caixa Postal 68509, 21941-914, Rio de Janeiro (Brazil); Senra Martinez, Aquilino, E-mail: aquilino@lmp.ufrj.br [COPPE/UFRJ, Programa de Engenharia Nuclear, Caixa Postal 68509, 21941-914, Rio de Janeiro (Brazil)

2011-07-15

Highlights: > We proposed a new neutron diffusion hybrid equation with external neutron source. > A coarse mesh finite difference method for the adjoint flux and reactivity calculation was developed. > 1/M curve to predict the criticality condition is used. - Abstract: We used the neutron diffusion hybrid equation, in cartesian geometry with external neutron sources to predict the subcritical multiplication of neutrons in a pressurized water reactor, using a 1/M curve to predict the criticality condition. A Coarse Mesh Finite Difference Method was developed for the adjoint flux calculation and to obtain the reactivity values of the reactor. The results obtained were compared with benchmark values in order to validate the methodology presented in this paper.

Prediction of the neutrons subcritical multiplication using the diffusion hybrid equation with external neutron sources

International Nuclear Information System (INIS)

Costa da Silva, Adilson; Carvalho da Silva, Fernando; Senra Martinez, Aquilino

2011-01-01

Highlights: → We proposed a new neutron diffusion hybrid equation with external neutron source. → A coarse mesh finite difference method for the adjoint flux and reactivity calculation was developed. → 1/M curve to predict the criticality condition is used. - Abstract: We used the neutron diffusion hybrid equation, in cartesian geometry with external neutron sources to predict the subcritical multiplication of neutrons in a pressurized water reactor, using a 1/M curve to predict the criticality condition. A Coarse Mesh Finite Difference Method was developed for the adjoint flux calculation and to obtain the reactivity values of the reactor. The results obtained were compared with benchmark values in order to validate the methodology presented in this paper.

Smoothed particle hydrodynamics model for Landau-Lifshitz-Navier-Stokes and advection-diffusion equations.

Science.gov (United States)

Kordilla, Jannes; Pan, Wenxiao; Tartakovsky, Alexandre

2014-12-14

We propose a novel smoothed particle hydrodynamics (SPH) discretization of the fully coupled Landau-Lifshitz-Navier-Stokes (LLNS) and stochastic advection-diffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and the self-diffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations is found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for coupled LLNS and advection-diffusion equations, we simulate the interface between two miscible fluids. We study formation of the so-called "giant fluctuations" of the front between light and heavy fluids with and without gravity, where the light fluid lies on the top of the heavy fluid. We find that the power spectra of the simulated concentration field are in good agreement with the experiments and analytical solutions. In the absence of gravity, the power spectra decay as the power -4 of the wavenumber-except for small wavenumbers that diverge from this power law behavior due to the effect of finite domain size. Gravity suppresses the fluctuations, resulting in much weaker dependence of the power spectra on the wavenumber. Finally, the model is used to study the effect of thermal fluctuation on the Rayleigh-Taylor instability, an unstable dynamics of the front between a heavy fluid overlaying a light fluid. The front dynamics is shown to agree well with the analytical solutions.

Covariant Derivatives and the Renormalization Group Equation

Science.gov (United States)

Dolan, Brian P.

The renormalization group equation for N-point correlation functions can be interpreted in a geometrical manner as an equation for Lie transport of amplitudes in the space of couplings. The vector field generating the diffeomorphism has components given by the β functions of the theory. It is argued that this simple picture requires modification whenever any one of the points at which the amplitude is evaluated becomes close to any other. This modification necessitates the introduction of a connection on the space of couplings and new terms appear in the renormalization group equation involving covariant derivatives of the β function and the curvature associated with the connection. It is shown how the connection is related to the operator product expansion coefficients, but there remains an arbitrariness in its definition.

Solution of the multigroup diffusion equation for two-dimensional triangular regions by finite Fourier transformation

International Nuclear Information System (INIS)

Takeshi, Y.; Keisuke, K.

1983-01-01

The multigroup neutron diffusion equation for two-dimensional triangular geometry is solved by the finite Fourier transformation method. Using the zero-th-order equation of the integral equation derived by this method, simple algebraic expressions for the flux are derived and solved by the alternating direction implicit method. In sample calculations for a benchmark problem of a fast breeder reactor, it is shown that the present method gives good results with fewer mesh points than the usual finite difference method

Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

Science.gov (United States)

Hasnain, Shahid; Saqib, Muhammad; Mashat, Daoud Suleiman

2017-07-01

This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit) to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

Fourth order Douglas implicit scheme for solving three dimension reaction diffusion equation with non-linear source term

Directory of Open Access Journals (Sweden)

Shahid Hasnain

2017-07-01

Full Text Available This research paper represents a numerical approximation to non-linear three dimension reaction diffusion equation with non-linear source term from population genetics. Since various initial and boundary value problems exist in three dimension reaction diffusion phenomena, which are studied numerically by different numerical methods, here we use finite difference schemes (Alternating Direction Implicit and Fourth Order Douglas Implicit to approximate the solution. Accuracy is studied in term of L2, L∞ and relative error norms by random selected grids along time levels for comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Numerical results showed that Fourth Order Douglas Implicit scheme is very efficient and reliable for solving 3-D non-linear reaction diffusion equation.

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.

Basic studies for the solution of the criticality equation: two groups of energy and one dimension

International Nuclear Information System (INIS)

Britto Aghina, L.O. de.

1994-12-01

This work collects six basic studies for the numerical solution of the criticality equation for thermal reactors. Use is made of the diffusion theory for two groups of energy and one dimension, applicable to bare reactors, bare equivalent, infinite bare equivalent and reflected reactors. These studies were written in Mathcad 4.0/WIN programming, a practical form for use by the researchers and operators working with the Argonaut Reactor at the Instituto de Engenharia Nuclear (IEN). (author). 11 refs, 20 figs, 8 tabs

Introductory Applications of Partial Differential Equations With Emphasis on Wave Propagation and Diffusion

CERN Document Server

Lamb, George L

1995-01-01

INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS. With Emphasis on Wave Propagation and Diffusion. This is the ideal text for students and professionals who have some familiarity with partial differential equations, and who now wish to consolidate and expand their knowledge. Unlike most other texts on this topic, it interweaves prior knowledge of mathematics and physics, especially heat conduction and wave motion, into a presentation that demonstrates their interdependence. The result is a superb teaching text that reinforces the reader's understanding of both mathematics and physic

Generalized Callan-Symanzik equations and the Renormalization Group

International Nuclear Information System (INIS)

MacDowell, S.W.

1975-01-01

A set of generalized Callan-Symanzik equations derived by Symanzik, relating Green's functions with arbitrary number of mass insertions, is shown be equivalent to the new Renormalization Group equation proposed by S. Weinberg

Eternal solutions to a singular diffusion equation with critical gradient absorption

International Nuclear Information System (INIS)

Iagar, Razvan Gabriel; Laurençot, Philippe

2013-01-01

The existence of non-negative radially symmetric eternal solutions of exponential self-similar type u(t, x) = e −pβt/(2−p) f β (|x|e −βt ; β) is investigated for the singular diffusion equation with critical gradient absorption ∂ t u−Δ p u+|∇u| p/2 =0  in (0,∞)×R N , where 2N/(N + 1) < p < 2. Such solutions are shown to exist only if the parameter β ranges in a bounded interval (0, β * ], which is in sharp contrast to well-known singular diffusion equations, such as ∂ t φ − Δ p φ = 0 when p = 2N/(N + 1), N ⩾ 1, or the porous medium equation ∂ t φ − Δφ m  = 0 when m = (N − 2)/N, N ⩾ 3. Moreover, the profile f(r; β) decays to zero as r → ∞ in a faster way for β = β * than for β ∈ (0, β * ) but the algebraic leading order is the same in both cases. In fact, for large r, f(r; β * ) decays as r −p/(2−p) while f(r; β) behaves as (log r) 2/(2−p) r −p/(2−p) when β ∈ (0, β * ). (paper)

On symmetry groups of a 2D nonlinear diffusion equation with source

Indian Academy of Sciences (India)

offers a powerful tool for simplifying the form of partial differential equations ... to its involutive form, then apply the classical method to the reduced PDE but with an ..... The author is grateful for the financial support offered by the Romanian ...

Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

Science.gov (United States)

Cortázar, Carmen; Elgueta, Manuel; Quirós, Fernando; Wolanski, Noemí

2012-08-01

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, u t = J* u- u := Lu, in an exterior domain, Ω, which excludes one or several holes, and with zero Dirichlet data on {R^NsetminusΩ} . When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behaviors can be presented in a unified way through a suitable global approximation.

Construction of Difference Equations Using Lie Groups

International Nuclear Information System (INIS)

Axford, R.A.

1998-01-01

The theory of prolongations of the generators of groups of point transformations to the grid point values of dependent variables and grid spacings is developed and applied to the construction of group invariant numerical algorithms. The concepts of invariant difference operators and generalized discrete sources are introduced for the discretization of systems of inhomogeneous differential equations and shown to produce exact difference equations. Invariant numerical flux functions are constructed from the general solutions of first order partial differential equations that come out of the evaluation of the Lie derivatives of conservation forms of difference schemes. It is demonstrated that invariant numerical flux functions with invariant flux or slope limiters can be determined to yield high resolution difference schemes. The introduction of an invariant flux or slope limiter can be done so as not to break the symmetry properties of a numerical flux-function

Discussing the precipitation behavior of {sigma} phase using diffusion equation and thermodynamic simulation in dissimilar stainless steels

Energy Technology Data Exchange (ETDEWEB)

Hsieh, Chih-Chun [Department of Materials Science and Engineering, National Chung Hsing University, 250 Kuo-Kuang Rd., Taichung 402, Taiwan (China); Wu, Weite, E-mail: wwu@dragon.nchu.edu.t [Department of Materials Science and Engineering, National Chung Hsing University, 250 Kuo-Kuang Rd., Taichung 402, Taiwan (China)

2010-09-17

Research highlights: This article concentrates the phase transformation in {delta} {yields} {sigma} in dissimilar stainless steels using the Vitek equation and thermodynamics simulation during the multi-pass welding. The phase transformation in {delta} {yields} {sigma} is very important to the properties of stainless steel composites. In this study, the diffusion behavior of Cr, Ni and Si in the {delta}, {sigma}, and {gamma} phases were discussed using the DSC analysis and diffusion equation calculation. This method has a novelty for discussing the phase transformation in {delta} {yields} {sigma} in the dissimilar stainless steel. We hope that we can give a scientific contribution for the phase transformation of the dissimilar stainless steels during the multi-pass welding. - Abstract: This study performed a precipitation examination of the {sigma} phase using the Vitek diffusion equation and thermodynamic simulation in dissimilar stainless steels during multi-pass welding. The results of the experiment demonstrate that the diffusion rates (D{sub Cr}{sup {delta}} and D{sub Ni}{sup {delta}}) of Cr and Ni are higher in {delta}-ferrite than (D{sub Cr}{sup {gamma}} and D{sub Ni}{sup {gamma}}) in the {gamma} phase and that they facilitate the precipitation of {sigma} phase in the third pass fusion zone. When the diffusion activation energy of Cr in {delta}-ferrite is equal to that of Ni in {delta}-ferrite (Q{sub dCr}{sup {delta}}=Q{sub dNi}{sup {delta}}), phase transformation of the {delta} {yields} {sigma} can be occurred.

Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator

Science.gov (United States)

Owolabi, Kolade M.

2018-03-01

In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain (-∞, ∞), which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.

Nodal methods with non linear feedback for the three dimensional resolution of the diffusion's multigroup equations

International Nuclear Information System (INIS)

Ferri, A.A.

1986-01-01

Nodal methods applied in order to calculate the power distribution in a nuclear reactor core are presented. These methods have received special attention, because they yield accurate results in short computing times. Present nodal schemes contain several unknowns per node and per group. In the methods presented here, non linear feedback of the coupling coefficients has been applied to reduce this number to only one unknown per node and per group. The resulting algorithm is a 7- points formula, and the iterative process has proved stable in the response matrix scheme. The intranodal flux shape is determined by partial integration of the diffusion equations over two of the coordinates, leading to a set of three coupled one-dimensional equations. These can be solved by using a polynomial approximation or by integration (analytic solution). The tranverse net leakage is responsible for the coupling between the spatial directions, and two alternative methods are presented to evaluate its shape: direct parabolic approximation and local model expansion. Numerical results, which include the IAEA two-dimensional benchmark problem illustrate the efficiency of the developed methods. (M.E.L.) [es

Steady state solution of the Fokker-Planck equation combined with unidirectional quasilinear diffusion under detailed balance conditions

International Nuclear Information System (INIS)

Hizanidis, K.

1984-04-01

The relativistic collisional Fokker-Planck equation combined with an externally imposed unidirectional quasilinear (rf) diffusion is solved for arbitrary values of rf diffusion coefficient under conditions of detailed balance of the staionary joint distribution involved. The detailed balance condition imposes a restriction on the functional form of the quasilinear diffusion coefficient which might be associated with the existence of a saturated spectrum of fluctuation in a quasilinearly rf-driven plasma

Assessment of Effective Factor of Hydrogen Diffusion Equation Using FE Analysis

International Nuclear Information System (INIS)

Kim, Nak Hyun; Oh, Chang Sik; Kim, Yun Jae

2010-01-01

The coupled model with hydrogen transport and elasto-plasticity behavior was introduced. In this paper, the effective factor of the hydrogen diffusion equation has been described. To assess the effective factor, finite element (FE) analyses including hydrogen transport and mechanical loading for boundary layer specimens with low-strength steel properties are carried out. The results of the FE analyses are compared with those from previous studies conducted by Taha and Sofronis (2001)

Modified two-fluid model for the two-group interfacial area transport equation

International Nuclear Information System (INIS)

Sun Xiaodong; Ishii, Mamoru; Kelly, Joseph M.

2003-01-01

This paper presents a modified two-fluid model that is ready to be applied in the approach of the two-group interfacial area transport equation. The two-group interfacial area transport equation was developed to provide a mechanistic constitutive relation for the interfacial area concentration in the two-fluid model. In the two-group transport equation, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 while cap/slug/churn-turbulent bubbles as Group 2. Therefore, this transport equation can be employed in the flow regimes spanning from bubbly, cap bubbly, slug to churn-turbulent flows. However, the introduction of the two groups of bubbles requires two gas velocity fields. Yet it is not practical to solve two momentum equations for the gas phase alone. In the current modified two-fluid model, a simplified approach is proposed. The momentum equation for the averaged velocity of both Group-1 and Group-2 bubbles is retained. By doing so, the velocity difference between Group-1 and Group-2 bubbles needs to be determined. This may be made either based on simplified momentum equations for both Group-1 and Group-2 bubbles or by a modified drift-flux model

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

Directory of Open Access Journals (Sweden)

Roman Cherniha

2018-04-01

Full Text Available This review is devoted to search for Lie and Q-conditional (nonclassical symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show that the result depends essentially on the type of equivalence transformations used for the classification. Moreover, a complete description of Q-conditional symmetries for PDEs from the class in question is also presented. It is shown that all the well-known results for reaction-diffusion equations with exponential nonlinearities follow as particular cases from the results derived for this class of reaction-diffusion-convection equations. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found by means of different techniques. Finally, an application of the exact solutions for solving boundary-value problems arising in population dynamics is presented.

Fractional cable equation for general geometry: A model of axons with swellings and anomalous diffusion

Science.gov (United States)

López-Sánchez, Erick J.; Romero, Juan M.; Yépez-Martínez, Huitzilin

2017-09-01

Different experimental studies have reported anomalous diffusion in brain tissues and notably this anomalous diffusion is expressed through fractional derivatives. Axons are important to understand neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease, and Parkinson's disease. Indeed, abnormal accumulation of proteins and organelles in axons is a hallmark of these diseases. The diffusion in the axons can become anomalous as a result of this abnormality. In this case the voltage propagation in axons is affected. Another hallmark of different neurodegenerative diseases is given by discrete swellings along the axon. In order to model the voltage propagation in axons with anomalous diffusion and swellings, in this paper we propose a fractional cable equation for a general geometry. This generalized equation depends on fractional parameters and geometric quantities such as the curvature and torsion of the cable. For a cable with a constant radius we show that the voltage decreases when the fractional effect increases. In cables with swellings we find that when the fractional effect or the swelling radius increases, the voltage decreases. Similar behavior is obtained when the number of swellings and the fractional effect increase. Moreover, we find that when the radius swelling (or the number of swellings) and the fractional effect increase at the same time, the voltage dramatically decreases.

Do group-specific equations provide the best estimates of stature?

Science.gov (United States)

Albanese, John; Osley, Stephanie E; Tuck, Andrew

2016-04-01

An estimate of stature can be used by a forensic anthropologist with the preliminary identification of an unknown individual when human skeletal remains are recovered. Fordisc is a computer application that can be used to estimate stature; like many other methods it requires the user to assign an unknown individual to a specific group defined by sex, race/ancestry, and century of birth before an equation is applied. The assumption is that a group-specific equation controls for group differences and should provide the best results most often. In this paper we assess the utility and benefits of using group-specific equations to estimate stature using Fordisc. Using the maximum length of the humerus and the maximum length of the femur from individuals with documented stature, we address the question: Do sex-, race/ancestry- and century-specific stature equations provide the best results when estimating stature? The data for our sample of 19th Century White males (n=28) were entered into Fordisc and stature was estimated using 22 different equation options for a total of 616 trials: 19th and 20th Century Black males, 19th and 20th Century Black females, 19th and 20th Century White females, 19th and 20th Century White males, 19th and 20th Century any, and 20th Century Hispanic males. The equations were assessed for utility in any one case (how many times the estimated range bracketed the documented stature) and in aggregate using 1-way ANOVA and other approaches. This group-specific equation that should have provided the best results was outperformed by several other equations for both the femur and humerus. These results suggest that group-specific equations do not provide better results for estimating stature while at the same time are more difficult to apply because an unknown must be allocated to a given group before stature can be estimated. Copyright © 2016 Elsevier Ireland Ltd. All rights reserved.

An iterative algorithm for solving the multidimensional neutron diffusion nodal method equations on parallel computers

International Nuclear Information System (INIS)

Kirk, B.L.; Azmy, Y.Y.

1992-01-01

In this paper the one-group, steady-state neutron diffusion equation in two-dimensional Cartesian geometry is solved using the nodal integral method. The discrete variable equations comprise loosely coupled sets of equations representing the nodal balance of neutrons, as well as neutron current continuity along rows or columns of computational cells. An iterative algorithm that is more suitable for solving large problems concurrently is derived based on the decomposition of the spatial domain and is accelerated using successive overrelaxation. This algorithm is very well suited for parallel computers, especially since the spatial domain decomposition occurs naturally, so that the number of iterations required for convergence does not depend on the number of processors participating in the calculation. Implementation of the authors' algorithm on the Intel iPSC/2 hypercube and Sequent Balance 8000 parallel computer is presented, and measured speedup and efficiency for test problems are reported. The results suggest that the efficiency of the hypercube quickly deteriorates when many processors are used, while the Sequent Balance retains very high efficiency for a comparable number of participating processors. This leads to the conjecture that message-passing parallel computers are not as well suited for this algorithm as shared-memory machines

Zeta Functions, Renormalization Group Equations, and the Effective Action

International Nuclear Information System (INIS)

Hochberg, D.; Perez-Mercader, J.; Molina-Paris, C.; Visser, M.

1998-01-01

We demonstrate how to extract all the one-loop renormalization group equations for arbitrary quantum field theories from knowledge of an appropriate Seeley-DeWitt coefficient. By formally solving the renormalization group equations to one loop, we renormalization group improve the classical action and use this to derive the leading logarithms in the one-loop effective action for arbitrary quantum field theories. copyright 1998 The American Physical Society

Parallel computing for homogeneous diffusion and transport equations in neutronics

International Nuclear Information System (INIS)

Pinchedez, K.

1999-06-01

Parallel computing meets the ever-increasing requirements for neutronic computer code speed and accuracy. In this work, two different approaches have been considered. We first parallelized the sequential algorithm used by the neutronics code CRONOS developed at the French Atomic Energy Commission. The algorithm computes the dominant eigenvalue associated with PN simplified transport equations by a mixed finite element method. Several parallel algorithms have been developed on distributed memory machines. The performances of the parallel algorithms have been studied experimentally by implementation on a T3D Cray and theoretically by complexity models. A comparison of various parallel algorithms has confirmed the chosen implementations. We next applied a domain sub-division technique to the two-group diffusion Eigen problem. In the modal synthesis-based method, the global spectrum is determined from the partial spectra associated with sub-domains. Then the Eigen problem is expanded on a family composed, on the one hand, from eigenfunctions associated with the sub-domains and, on the other hand, from functions corresponding to the contribution from the interface between the sub-domains. For a 2-D homogeneous core, this modal method has been validated and its accuracy has been measured. (author)

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.

Group-theoretical interpretation of the Korteweg-de Vries type equations

International Nuclear Information System (INIS)

Berezin, F.A.; Perelomov, A.M.

1978-01-01

The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schroedinger equation (with nonlocal potential) plays the same role as the one-dimensional Schroedinger equation does in the theory of the Korteweg-de Vries equation

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

Energy Technology Data Exchange (ETDEWEB)

Bailey, Teresa S. [Texas A and M University, Department of Nuclear Engineering, College Station, TX 77843-3133 (United States)], E-mail: baileyte@tamu.edu; Adams, Marvin L. [Texas A and M University, Department of Nuclear Engineering, College Station, TX 77843-3133 (United States)], E-mail: mladams@tamu.edu; Yang, Brian [Lawrence Livermore National Laboratory, Livermore, CA 94551 (United States); Zika, Michael R. [Lawrence Livermore National Laboratory, Livermore, CA 94551 (United States)], E-mail: zika@llnl.gov

2008-04-01

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer's vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer's. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

International Nuclear Information System (INIS)

Bailey, Teresa S.; Adams, Marvin L.; Yang, Brian; Zika, Michael R.

2008-01-01

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer's vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer's. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer's method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids

Quarkonia from charmonium and renormalization group equations

International Nuclear Information System (INIS)

Ditsas, P.; McDougall, N.A.; Moorhouse, R.G.

1978-01-01

A prediction of the upsilon and strangeonium spectra is made from the charmonium spectrum by solving the Salpeter equation using an identical potential to that used in charmonium. Effective quark masses and coupling parameters αsub(s) are functions of the inter-quark distance according to the renormalization group equations. The use of the Fermi-Breit Hamiltonian for obtaining the charmonium hyperfine splitting is criticized. (Auth.)

Discontinuous finite element solution of the radiation diffusion equation on arbitrary polygonal meshes and locally adapted quadrilateral grids

International Nuclear Information System (INIS)

Ragusa, Jean C.

2015-01-01

In this paper, we propose a piece-wise linear discontinuous (PWLD) finite element discretization of the diffusion equation for arbitrary polygonal meshes. It is based on the standard diffusion form and uses the symmetric interior penalty technique, which yields a symmetric positive definite linear system matrix. A preconditioned conjugate gradient algorithm is employed to solve the linear system. Piece-wise linear approximations also allow a straightforward implementation of local mesh adaptation by allowing unrefined cells to be interpreted as polygons with an increased number of vertices. Several test cases, taken from the literature on the discretization of the radiation diffusion equation, are presented: random, sinusoidal, Shestakov, and Z meshes are used. The last numerical example demonstrates the application of the PWLD discretization to adaptive mesh refinement

Space-time versus world-sheet renormalization group equation in string theory

International Nuclear Information System (INIS)

Brustein, R.; Roland, K.

1991-05-01

We discuss the relation between space-time renormalization group equation for closed string field theory and world-sheet renormalization group equation for first-quantized strings. Restricting our attention to massless states we argue that there is a one-to-one correspondence between the fixed point solutions of the two renormalization group equations. In particular, we show how to extract the Fischler-Susskind mechanism from the string field theory equation in the case of the bosonic string. (orig.)

Fast resolution of the neutron diffusion equation through public domain Ode codes

Energy Technology Data Exchange (ETDEWEB)

Garcia, V.M.; Vidal, V.; Garayoa, J. [Universidad Politecnica de Valencia, Departamento de Sistemas Informaticos, Valencia (Spain); Verdu, G. [Universidad Politecnica de Valencia, Departamento de Ingenieria Quimica y Nuclear, Valencia (Spain); Gomez, R. [I.E.S. de Tavernes Blanques, Valencia (Spain)

2003-07-01

The time-dependent neutron diffusion equation is a partial differential equation with source terms. The resolution method usually includes discretizing the spatial domain, obtaining a large system of linear, stiff ordinary differential equations (ODEs), whose resolution is computationally very expensive. Some standard techniques use a fixed time step to solve the ODE system. This can result in errors (if the time step is too large) or in long computing times (if the time step is too little). To speed up the resolution method, two well-known public domain codes have been selected: DASPK and FCVODE that are powerful codes for the resolution of large systems of stiff ODEs. These codes can estimate the error after each time step, and, depending on this estimation can decide which is the new time step and, possibly, which is the integration method to be used in the next step. With these mechanisms, it is possible to keep the overall error below the chosen tolerances, and, when the system behaves smoothly, to take large time steps increasing the execution speed. In this paper we address the use of the public domain codes DASPK and FCVODE for the resolution of the time-dependent neutron diffusion equation. The efficiency of these codes depends largely on the preconditioning of the big systems of linear equations that must be solved. Several pre-conditioners have been programmed and tested; it was found that the multigrid method is the best of the pre-conditioners tested. Also, it has been found that DASPK has performed better than FCVODE, being more robust for our problem.We can conclude that the use of specialized codes for solving large systems of ODEs can reduce drastically the computational work needed for the solution; and combining them with appropriate pre-conditioners, the reduction can be still more important. It has other crucial advantages, since it allows the user to specify the allowed error, which cannot be done in fixed step implementations; this, of course

Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation

Science.gov (United States)

Abuasad, Salah; Hashim, Ishak

2018-04-01

In this paper, we present the homotopy decomposition method with a modified definition of beta fractional derivative for the first time to find exact solution of one-dimensional time-fractional diffusion equation. In this method, the solution takes the form of a convergent series with easily computable terms. The exact solution obtained by the proposed method is compared with the exact solution obtained by using fractional variational homotopy perturbation iteration method via a modified Riemann-Liouville derivative.

An analytic algorithm for the space-time fractional reaction-diffusion equation

Directory of Open Access Journals (Sweden)

M. G. Brikaa

2015-11-01

Full Text Available In this paper, we solve the space-time fractional reaction-diffusion equation by the fractional homotopy analysis method. Solutions of different examples of the reaction term will be computed and investigated. The approximation solutions of the studied models will be put in the form of convergent series to be easily computed and simulated. Comparison with the approximation solution of the classical case of the studied modeled with their approximation errors will also be studied.

The nature and role of advection in advection-diffusion equations used for modelling bed load transport

Science.gov (United States)

Ancey, Christophe; Bohorquez, Patricio; Heyman, Joris

2016-04-01

The advection-diffusion equation arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Stochastic models can also be used to derive this equation, with the significant advantage that they provide information on the statistical properties of particle activity. Stochastic models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. We develop an approach based on birth-death Markov processes, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received little attention. We show that particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due to velocity fluctuations), with the important consequence that local measurements depend on both the intrinsic properties of particle displacement and the dimensions of the measurement system.

HEXAB-3D, 3-D Few-Group Diffusion for Hexagonal Core Geometry

International Nuclear Information System (INIS)

Apostolov, T.G.; Ivanov, K.N.; Manolova, M.A.

2002-01-01

1 - Description of program or function: A three-dimensional few-group calculational model based on diffusion theory has been developed for calculating the basic neutron physical characteristics of power reactors which have a hexagonal core configuration with a heterogeneous region structure in axial direction. There are two versions of the model: - HEXAB-III-30 - the solution range in horizontal plane is 30 - sector of reactor core - HEXAB-III-360 - the solution range in horizontal plane is full core. 2 - Method of solution: In the HEXAB-3D code the nine-point mesh-centered finite-difference approximation of neutron diffusion equation is applied. The standard inner-outer iterative strategy is used. Inner iterations are solved using two different incomplete factorization techniques: AGA two-sweep iterative method and modified AGA two-sweep iterative method both accelerated by the double successive over-relaxation procedure. The power method, combined with two- or three-term Chebishev polynomial acceleration for outer iterations is applied in the code. To improve the accuracy of the calculated integral and local reactor parameters without significantly increasing computer time and storage an effective approach has been developed. It decreases errors due to the use of coarse mesh by means of correcting the coefficients of finite- difference scheme. 3 - Restrictions on the complexity of the problem: Maximum of 10 energy groups, 30 horizontal layers and 100 material compositions

Hybrid nodal methods in the solution of the diffusion equations in X Y geometry

International Nuclear Information System (INIS)

Hernandez M, N.; Alonso V, G.; Valle G, E. del

2003-01-01

In 1979, Hennart and collaborators applied several schemes of classic finite element in the numerical solution of the diffusion equations in X Y geometry and stationary state. Almost two decades then, in 1996, himself and other collaborators carried out a similar work but using nodal schemes type finite element. Continuing in this last direction, in this work a group it is described a set of several Hybrid Nodal schemes denominated (NH) as well as their application to solve the diffusion equations in multigroup in stationary state and X Y geometry. The term hybrid nodal it means that such schemes interpolate not only Legendre moments of face and of cell but also the values of the scalar flow of neutrons in the four corners of each cell or element of the spatial discretization of the domain of interest. All the schemes here considered are polynomials like they were it their predecessors. Particularly, its have developed and applied eight different hybrid nodal schemes that its are very nearby related with those developed by Hennart and collaborators in the past. It is treated of schemes in those that nevertheless that decreases the number of interpolation parameters it is conserved the accurate in relation to the bi-quadratic and bi-cubic schemes. Of these eight, three were described and applied in a previous work. It is the bi-lineal classic scheme as well as the hybrid nodal schemes, bi-quadratic and bi-cubic for that here only are described the other 5 hybrid nodal schemes although they are provided numerical results for several test problems with all them. (Author)

Langevin equation with multiplicative white noise: Transformation of diffusion processes into the Wiener process in different prescriptions

International Nuclear Information System (INIS)

Kwok, Sau Fa

2012-01-01

A Langevin equation with multiplicative white noise and its corresponding Fokker–Planck equation are considered in this work. From the Fokker–Planck equation a transformation into the Wiener process is provided for different orders of prescription in discretization rule for the stochastic integrals. A few applications are also discussed. - Highlights: ► Fokker–Planck equation corresponding to the Langevin equation with mul- tiplicative white noise is presented. ► Transformation of diffusion processes into the Wiener process in different prescriptions is provided. ► The prescription parameter is associated with the growth rate for a Gompertz-type model.

Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain

Energy Technology Data Exchange (ETDEWEB)

Tumelero, Fernanda; Bodmann, Bardo E. J.; Vilhena, Marco T. [Universidade Federal do Rio Grande do Sul (PROMEC/UFRGS), Porto Alegre, RS (Brazil). Programa de Pos Graduacao em Engenharia Mecanica; Lapa, Celso M.F., E-mail: fernanda.tumelero@yahoo.com.br, E-mail: bardo.bodmann@ufrgs.br, E-mail: mtmbvilhena@gmail.com, E-mail: lapa@ien.gov.br [Instituto de Engenharia Nuclear (IEN/CNEN-RJ), Rio de Janeiro, RJ (Brazil)

2017-07-01

In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the rst recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution. (author)

Fourier spectral methods for fractional-in-space reaction-diffusion equations

KAUST Repository

Bueno-Orovio, Alfonso

2014-04-01

© 2014, Springer Science+Business Media Dordrecht. Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of ℝ. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

Analytical modeling for fractional multi-dimensional diffusion equations by using Laplace transform

Directory of Open Access Journals (Sweden)

Devendra Kumar

2015-01-01

Full Text Available In this paper, we propose a simple numerical algorithm for solving multi-dimensional diffusion equations of fractional order which describes density dynamics in a material undergoing diffusion by using homotopy analysis transform method. The fractional derivative is described in the Caputo sense. This homotopy analysis transform method is an innovative adjustment in Laplace transform method and makes the calculation much simpler. The technique is not limited to the small parameter, such as in the classical perturbation method. The scheme gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive.

From baking a cake to solving the diffusion equation

Science.gov (United States)

Olszewski, Edward A.

2006-06-01

We explain how modifying a cake recipe by changing either the dimensions of the cake or the amount of cake batter alters the baking time. We restrict our consideration to the génoise and obtain a semiempirical relation for the baking time as a function of oven temperature, initial temperature of the cake batter, and dimensions of the unbaked cake. The relation, which is based on the diffusion equation, has three parameters whose values are estimated from data obtained by baking cakes in cylindrical pans of various diameters. The relation takes into account the evaporation of moisture at the top surface of the cake, which is the dominant factor affecting the baking time of a cake.

Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method

Science.gov (United States)

Jain, Sonal

2018-01-01

In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.

Fractional diffusion equation with distributed-order material derivative. Stochastic foundations

International Nuclear Information System (INIS)

Magdziarz, M; Teuerle, M

2017-01-01

In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Lévy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type. (paper)

A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

Directory of Open Access Journals (Sweden)

Inci Cilingir Sungu

2015-01-01

Full Text Available A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation

Science.gov (United States)

Liang, Yingjie; Chen, Wen; Magin, Richard L.

2016-07-01

Analytical solutions to the fractional diffusion equation are often obtained by using Laplace and Fourier transforms, which conveniently encode the order of the time and the space derivatives (α and β) as non-integer powers of the conjugate transform variables (s, and k) for the spectral and the spatial frequencies, respectively. This study presents a new solution to the fractional diffusion equation obtained using the Laplace transform and expressed as a Fox's H-function. This result clearly illustrates the kinetics of the underlying stochastic process in terms of the Laplace spectral frequency and entropy. The spectral entropy is numerically calculated by using the direct integration method and the adaptive Gauss-Kronrod quadrature algorithm. Here, the properties of spectral entropy are investigated for the cases of sub-diffusion and super-diffusion. We find that the overall spectral entropy decreases with the increasing α and β, and that the normal or Gaussian case with α = 1 and β = 2, has the lowest spectral entropy (i.e., less information is needed to describe the state of a Gaussian process). In addition, as the neighborhood over which the entropy is calculated increases, the spectral entropy decreases, which implies a spatial averaging or coarse graining of the material properties. Consequently, the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion. Future studies should be designed to examine changes of spectral entropy in physical, chemical and biological systems undergoing phase changes, chemical reactions and tissue regeneration.

Two-Phase Fluid Simulation Using a Diffuse Interface Model with Peng--Robinson Equation of State

KAUST Repository

Qiao, Zhonghua; Sun, Shuyu

2014-01-01

In this paper, two-phase fluid systems are simulated using a diffusive interface model with the Peng-Robinson equation of state (EOS), a widely used realistic EOS for hydrocarbon fluid in the petroleum industry. We first utilize the gradient theory

Assembly Discontinuity Factors for the Neutron Diffusion Equation discretized with the Finite Volume Method. Application to BWR

International Nuclear Information System (INIS)

Bernal, A.; Roman, J.E.; Miró, R.; Verdú, G.

2016-01-01

Highlights: • A method is proposed to solve the eigenvalue problem of the Neutron Diffusion Equation in BWR. • The Neutron Diffusion Equation is discretized with the Finite Volume Method. • The currents are calculated by using a polynomial expansion of the neutron flux. • The current continuity and boundary conditions are defined implicitly to reduce the size of the matrices. • Different structured and unstructured meshes were used to discretize the BWR. - Abstract: The neutron flux spatial distribution in Boiling Water Reactors (BWRs) can be calculated by means of the Neutron Diffusion Equation (NDE), which is a space- and time-dependent differential equation. In steady state conditions, the time derivative terms are zero and this equation is rewritten as an eigenvalue problem. In addition, the spatial partial derivatives terms are transformed into algebraic terms by discretizing the geometry and using numerical methods. As regards the geometrical discretization, BWRs are complex systems containing different components of different geometries and materials, but they are usually modelled as parallelepiped nodes each one containing only one homogenized material to simplify the solution of the NDE. There are several techniques to correct the homogenization in the node, but the most commonly used in BWRs is that based on Assembly Discontinuity Factors (ADFs). As regards numerical methods, the Finite Volume Method (FVM) is feasible and suitable to be applied to the NDE. In this paper, a FVM based on a polynomial expansion method has been used to obtain the matrices of the eigenvalue problem, assuring the accomplishment of the ADFs for a BWR. This eigenvalue problem has been solved by means of the SLEPc library.

Higher-order Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique

KAUST Repository

El-Beltagy, Mohamed A.

2014-01-06

Using Wiener-Hermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The Wiener-Hermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic Wiener-Hermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.

Higher-order Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique

KAUST Repository

El-Beltagy, Mohamed A.; Al-Mulla, Noah

2014-01-01

Using Wiener-Hermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The Wiener-Hermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic Wiener-Hermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.

Mixed-hybrid finite element method for the transport equation and diffusion approximation of transport problems

International Nuclear Information System (INIS)

Cartier, J.

2006-04-01

This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)

An inherently parallel method for solving discretized diffusion equations

International Nuclear Information System (INIS)

Eccleston, B.R.; Palmer, T.S.

1999-01-01

A Monte Carlo approach to solving linear systems of equations is being investigated in the context of the solution of discretized diffusion equations. While the technique was originally devised decades ago, changes in computer architectures (namely, massively parallel machines) have driven the authors to revisit this technique. There are a number of potential advantages to this approach: (1) Analog Monte Carlo techniques are inherently parallel; this is not necessarily true to today's more advanced linear equation solvers (multigrid, conjugate gradient, etc.); (2) Some forms of this technique are adaptive in that they allow the user to specify locations in the problem where resolution is of particular importance and to concentrate the work at those locations; and (3) These techniques permit the solution of very large systems of equations in that matrix elements need not be stored. The user could trade calculational speed for storage if elements of the matrix are calculated on the fly. The goal of this study is to compare the parallel performance of Monte Carlo linear solvers to that of a more traditional parallelized linear solver. The authors observe the linear speedup that they expect from the Monte Carlo algorithm, given that there is no domain decomposition to cause significant communication overhead. Overall, PETSc outperforms the Monte Carlo solver for the test problem. The PETSc parallel performance improves with larger numbers of unknowns for a given number of processors. Parallel performance of the Monte Carlo technique is independent of the size of the matrix and the number of processes. They are investigating modifications to the scheme to accommodate matrix problems with positive off-diagonal elements. They are also currently coding an on-the-fly version of the algorithm to investigate the solution of very large linear systems

Using Directional Diffusion Coefficients for Nonlinear Diffusion Acceleration of the First Order SN Equations in Near-Void Regions

Energy Technology Data Exchange (ETDEWEB)

Schunert, Sebastian; Hammer, Hans; Lou, Jijie; Wang, Yaqi; Ortensi, Javier; Gleicher, Frederick; Baker, Benjamin; DeHart, Mark; Martineau, Richard

2016-11-01

The common definition of the diffusion coeffcient as the inverse of three times the transport cross section is not compat- ible with voids. Morel introduced a non-local tensor diffusion coeffcient that remains finite in voids[1]. It can be obtained by solving an auxiliary transport problem without scattering or fission. Larsen and Trahan successfully applied this diffusion coeffcient for enhancing the accuracy of diffusion solutions of very high temperature reactor (VHTR) problems that feature large, optically thin channels in the z-direction [2]. It is demonstrated that a significant reduction of error can be achieved in particular in the optically thin region. Along the same line of thought, non-local diffusion tensors are applied modeling the TREAT reactor confirming the findings of Larsen and Trahan [3]. Previous work of the authors have introduced a flexible Nonlinear-Diffusion Acceleration (NDA) method for the first order S N equations discretized with the discontinuous finite element method (DFEM), [4], [5], [6]. This NDA method uses a scalar diffusion coeffcient in the low-order system that is obtained as the flux weighted average of the inverse transport cross section. Hence, it su?ers from very large and potentially unbounded diffusion coeffcients in the low order problem. However, it was noted that the choice of the diffusion coeffcient does not influence consistency of the method at convergence and hence the di?usion coeffcient is essentially a free parameter. The choice of the di?usion coeffcient does, however, affect the convergence behavior of the nonlinear di?usion iterations. Within this work we use Morel’s non-local di?usion coef- ficient in the aforementioned NDA formulation in lieu of the flux weighted inverse of three times the transport cross section. The goal of this paper is to demonstrate that significant en- hancement of the spectral properties of NDA can be achieved in near void regions. For testing the spectral properties of the NDA

Recursive solutions for multi-group neutron kinetics diffusion equations in homogeneous three-dimensional rectangular domains with time dependent perturbations

Energy Technology Data Exchange (ETDEWEB)

Petersen, Claudio Z. [Universidade Federal de Pelotas, Capao do Leao (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Bodmann, Bardo E.J.; Vilhena, Marco T. [Universidade Federal do Rio Grande do Sul, Porto Alegre, RS (Brazil). Programa de Pos-graduacao em Engenharia Mecanica; Barros, Ricardo C. [Universidade do Estado do Rio de Janeiro, Nova Friburgo, RJ (Brazil). Inst. Politecnico

2014-12-15

In the present work we solve in analytical representation the three dimensional neutron kinetic diffusion problem in rectangular Cartesian geometry for homogeneous and bounded domains for any number of energy groups and precursor concentrations. The solution in analytical representation is constructed using a hierarchical procedure, i.e. the original problem is reduced to a problem previously solved by the authors making use of a combination of the spectral method and a recursive decomposition approach. Time dependent absorption cross sections of the thermal energy group are considered with step, ramp and Chebyshev polynomial variations. For these three cases, we present numerical results and discuss convergence properties and compare our results to those available in the literature.

Numerical simulation of diffuse double layer around microporous electrodes based on the Poisson–Boltzmann equation

International Nuclear Information System (INIS)

Kitazumi, Yuki; Shirai, Osamu; Yamamoto, Masahiro; Kano, Kenji

2013-01-01

Graphical abstract: - Highlights: • Diffuse double layers overlap with each other in the micropore. • The overlapping of the diffuse double layer affects the double layer capacitance. • The electric field becomes weak in the micropore. • The electroneutrality is unsatisfactory in the micropore. - Abstract: The structure of the diffuse double layer around a nm-sized micropore on porous electrodes has been studied by numerical simulation using the Poisson–Boltzmann equation. The double layer capacitance of the microporous electrode strongly depends on the electrode potential, the electrolyte concentration, and the size of the micropore. The potential and the electrolyte concentration dependence of the capacitance is different from that of the planner electrode based on the Gouy's theory. The overlapping of the diffuse double layer becomes conspicuous in the micropore. The overlapped diffuse double layer provides the mild electric field. The intensified electric field exists at the rim of the orifice of the micropore because of the expansion of the diffuse double layers. The characteristic features of microporous electrodes are caused by the heterogeneity of the electric field around the micropores

Multi-group diffusion perturbation calculation code. PERKY (2002)

Energy Technology Data Exchange (ETDEWEB)

Iijima, Susumu; Okajima, Shigeaki [Japan Atomic Energy Research Inst., Tokai, Ibaraki (Japan). Tokai Research Establishment

2002-12-01

Perturbation calculation code based on the diffusion theory ''PERKY'' is designed for nuclear characteristic analyses of fast reactor. The code calculates reactivity worth on the multi-group diffusion perturbation theory in two or three dimensional core model and kinetics parameters such as effective delayed neutron fraction, prompt neutron lifetime and absolute reactivity scale factor ({rho}{sub 0} {delta}k/k) for FCA experiments. (author)

Single molecule diffusion and the solution of the spherically symmetric residence time equation.

Science.gov (United States)

Agmon, Noam

2011-06-16

The residence time of a single dye molecule diffusing within a laser spot is propotional to the total number of photons emitted by it. With this application in mind, we solve the spherically symmetric "residence time equation" (RTE) to obtain the solution for the Laplace transform of the mean residence time (MRT) within a d-dimensional ball, as a function of the initial location of the particle and the observation time. The solutions for initial conditions of potential experimental interest, starting in the center, on the surface or uniformly within the ball, are explicitly presented. Special cases for dimensions 1, 2, and 3 are obtained, which can be Laplace inverted analytically for d = 1 and 3. In addition, the analytic short- and long-time asymptotic behaviors of the MRT are derived and compared with the exact solutions for d = 1, 2, and 3. As a demonstration of the simplification afforded by the RTE, the Appendix obtains the residence time distribution by solving the Feynman-Kac equation, from which the MRT is obtained by differentiation. Single-molecule diffusion experiments could be devised to test the results for the MRT presented in this work. © 2011 American Chemical Society

Nodal integral method for the neutron diffusion equation in cylindrical geometry

International Nuclear Information System (INIS)

Azmy, Y.Y.

1987-01-01

The nodal methodology is based on retaining a higher a higher degree of analyticity in the process of deriving the discrete-variable equations compared to conventional numerical methods. As a result, extensive numerical testing of nodal methods developed for a wide variety of partial differential equations and comparison of the results to conventional methods have established the superior accuracy of nodal methods on coarse meshes. Moreover, these tests have shown that nodal methods are more computationally efficient than finite difference and finite-element methods in the sense that they require shorter CPU times to achieve comparable accuracy in the solutions. However, nodal formalisms and the final discrete-variable equations they produce are, in general, more complicated than their conventional counterparts. This, together with anticipated difficulties in applying the transverse-averaging procedure in curvilinear coordinates, has limited the applications of nodal methods, so far, to Cartesian geometry, and with additional approximations to hexagonal geometry. In this paper the authors report recent progress in deriving and numerically implementing a nodal integral method (NIM) for solving the neutron diffusion equation in cylindrical r-z geometry. Also, presented are comparisons of numerical solutions to two test problems with those obtained by the Exterminator-2 code, which indicate the superior accuracy of the nodal integral method solutions on much coarser meshes

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

The PLUS family: A set of computer programs to evaluate analytical solutions of the diffusion equation and thermoelasticity

International Nuclear Information System (INIS)

Montan, D.N.

1987-02-01

This report is intended to describe, document and provide instructions for the use of new versions of a set of computer programs commonly referred to as the PLUS family. These programs were originally designed to numerically evaluate simple analytical solutions of the diffusion equation. The new versions include linear thermo-elastic effects from thermal fields calculated by the diffusion equation. After the older versions of the PLUS family were documented a year ago, it was realized that the techniques employed in the programs were well suited to the addition of linear thermo-elastic phenomena. This has been implemented and this report describes the additions. 3 refs., 14 figs

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.

The Transport Equation in Optically Thick Media: Discussion of IMC and its Diffusion Limit

Energy Technology Data Exchange (ETDEWEB)

Szoke, A. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks, E. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2016-07-12

We discuss the limits of validity of the Implicit Monte Carlo (IMC) method for the transport of thermally emitted radiation. The weakened coupling between the radiation and material energy of the IMC method causes defects in handling problems with strong transients. We introduce an approach to asymptotic analysis for the transport equation that emphasizes the fact that the radiation and material temperatures are always different in time-dependent problems, and we use it to show that IMC does not produce the correct diffusion limit. As this is a defect of IMC in the continuous equations, no improvement to its discretization can remedy it.

A comparison of certain variational solutions of neutron diffusion equation

International Nuclear Information System (INIS)

Altiparmakov, D.V.; Milgram, M.S.

1987-01-01

Using the R-function theory and the variational method of Kantorovich, an approximate solution of the neutron diffusion equation is constructed for a homogeneous spatial domain of arbitrary shape. Calculations have been carried out by five different types of trial functions for certain characteristic domains of polygonal shape (square, triangle, hexagon, rhombus nad L-shaped domain). In the case of non-convex polygons, the consequence of the R-function solution is very poor and a separate treatment of singularity seems to be necessary. Compared to the R-function solution, the singular function development is mathematically more complicated but superior in respect to convergence rate. (author)

Multigrid solution of diffusion equations on distributed memory multiprocessor systems

International Nuclear Information System (INIS)

Finnemann, H.

1988-01-01

The subject is the solution of partial differential equations for simulation of the reactor core on high-performance computers. The parallelization and implementation of nodal multigrid diffusion algorithms on array and ring configurations of the DIRMU multiprocessor system is outlined. The particular iteration scheme employed in the nodal expansion method appears similarly efficient in serial and parallel environments. The combination of modern multi-level techniques with innovative hardware (vector-multiprocessor systems) provides powerful tools needed for real time simulation of physical systems. The parallel efficiencies range from 70 to 90%. The same performance is estimated for large problems on large multiprocessor systems being designed at present. (orig.) [de

Preconditioned iterative methods for space-time fractional advection-diffusion equations

Science.gov (United States)

Zhao, Zhi; Jin, Xiao-Qing; Lin, Matthew M.

2016-08-01

In this paper, we propose practical numerical methods for solving a class of initial-boundary value problems of space-time fractional advection-diffusion equations. First, we propose an implicit method based on two-sided Grünwald formulae and discuss its stability and consistency. Then, we develop the preconditioned generalized minimal residual (preconditioned GMRES) method and preconditioned conjugate gradient normal residual (preconditioned CGNR) method with easily constructed preconditioners. Importantly, because resulting systems are Toeplitz-like, fast Fourier transform can be applied to significantly reduce the computational cost. We perform numerical experiments to demonstrate the efficiency of our preconditioners, even in cases with variable coefficients.

Enriched reproducing kernel particle method for fractional advection-diffusion equation

Science.gov (United States)

Ying, Yuping; Lian, Yanping; Tang, Shaoqiang; Liu, Wing Kam

2018-06-01

The reproducing kernel particle method (RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection-diffusion equation (FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

Solution of the diffusion equation in the GPT theory by the Laplace transform technique

International Nuclear Information System (INIS)

Lemos, R.S.M.; Vilhena, M.T.; Segatto, C.F.; Silva, M.T.

2003-01-01

In this work we present a analytical solution to the auxiliary and importance functions attained from the solution of a multigroup diffusion problem in a multilayered slab by the Laplace Transform technique. We also obtain the the transcendental equation for the effective multiplication factor, resulting from the application of the boundary and interface conditions. (author)

Method of independently operating a group of stages within a diffusion cascade

International Nuclear Information System (INIS)

Benedict, M.; Allen, J.F.; Levey, H.B.

1976-01-01

A method of operating a group of the diffusion stages of a productive diffusion cascade with counter-current flow is described. The group consists of a top and a bottom stage which isolates the group from the cascade. The diffused gas produced in the top stage is circulated to the feed of the bottom stage, while at the same time undiffused gas from the bottom stage is circulated to the feed of the top stage whereby major changes in inventory distribution within the group of stages are prevented

Local multiplicative Schwarz algorithms for convection-diffusion equations

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Cai, Xiao-Chuan; Sarkis, Marcus

1995-01-01

We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diffusion equations discretized by finite element or finite difference methods. The preconditioners consist of two components, namely, the usual two-level additive Schwarz preconditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the flow. We prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. We show by numerical examples that the new algorithms are less sensitive to the direction of the flow than either the classical multiplicative Schwarz algorithms, and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for fluid flow applications than the classical additive or multiplicative Schwarz algorithms.

A variable timestep generalized Runge-Kutta method for the numerical integration of the space-time diffusion equations

International Nuclear Information System (INIS)

Aviles, B.N.; Sutton, T.M.; Kelly, D.J. III.

1991-09-01

A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block-factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized form of the time-dependent neutron diffusion equations. Preliminary numerical evaluation using a one-dimensional finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method. 12 refs., 5 figs., 4 tabs

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

KAUST Repository

Cañizo, J.A.

2010-03-01

We present a new a priori estimate for discrete coagulation-fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction-diffusion equations. In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case. © 2009 Elsevier Masson SAS. All rights reserved.

Improvement of the symbolic Monte-Carlo method for the transport equation: P1 extension and coupling with diffusion

International Nuclear Information System (INIS)

Clouet, J.F.; Samba, G.

2005-01-01

We use asymptotic analysis to study the diffusion limit of the Symbolic Implicit Monte-Carlo (SIMC) method for the transport equation. For standard SIMC with piecewise constant basis functions, we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation. Nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution. This improvement allows the calculation in opaque medium on a mesh resolving the diffusion scale much larger than the transport scale. Anyway, the huge number of particles which is necessary to get a correct answer makes this computation time consuming. Thus, we have derived from this asymptotic study an hybrid method coupling deterministic calculation in the opaque medium and Monte-Carlo calculation in the transparent medium. This method gives exactly the same results as the previous one but at a much lower price. We present numerical examples which illustrate the analysis. (authors)

The validity of quantum-classical multi-channel diffusion equations describing interlevel transitions in the condensed phase. The adiabatic representation

CERN Document Server

Basilevsky, M V

2002-01-01

We develop an approach for derivation of quantum-classical relaxation equations for a two-channel problem. The treatment is based on the adiabatic channel wavefunctions and the system-bath coupling is modelled as a bilinear interaction in momentum representation. In the quantum-classical limit we obtain Liouville equations with the relaxation operator containing diffusion terms diagonal in Liouvillian space and the off-diagonal part which is responsible for thermal interlevel transitions. The high-frequency interlevel quantum beats are fully taken into account in this relaxation term. In the framework of the present formulation and as a consequence of the momentum-dependent interaction the Smoluchovsky diffusion limit can be reached without invoking Fokker-Planck equations as an intermediate step. The inherent property of equations so obtained is that the partial rates of interlevel transitions obey the principle of detailed balance. This result could not be gained in earlier treatments of the two-level diffu...

Lie Group Classifications and Non-differentiable Solutions for Time-Fractional Burgers Equation

International Nuclear Information System (INIS)

Wu Guocheng

2011-01-01

Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)

Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations.

Science.gov (United States)

Yang, Xuguang; Shi, Baochang; Chai, Zhenhua

2014-07-01

In this paper, two modified lattice Boltzmann Bhatnagar-Gross-Krook (LBGK) models for incompressible Navier-Stokes equations and convection-diffusion equations are proposed via the addition of correction terms in the evolution equations. Utilizing this modification, the value of the dimensionless relaxation time in the LBGK model can be kept in a proper range, and thus the stability of the LBGK model can be improved. Although some gradient operators are included in the correction terms, they can be computed efficiently using local computational schemes such that the present LBGK models still retain the intrinsic parallelism characteristic of the lattice Boltzmann method. Numerical studies of the steady Poiseuille flow and unsteady Womersley flow show that the modified LBGK model has a second-order convergence rate in space, and the compressibility effect in the common LBGK model can be eliminated. In addition, to test the stability of the present models, we also performed some simulations of the natural convection in a square cavity, and we found that the results agree well with those reported in the previous work, even at a very high Rayleigh number (Ra = 10(12)).

Solution of the multilayer multigroup neutron diffusion equation in cartesian geometry by fictitious borders power method

Energy Technology Data Exchange (ETDEWEB)

Zanette, Rodrigo; Petersen, Caudio Zen [Univ. Federal de Pelotas, Capao do Leao (Brazil). Programa de Pos Graduacao em Modelagem Matematica; Schramm, Marcello [Univ. Federal de Pelotas (Brazil). Centro de Engenharias; Zabadal, Jorge Rodolfo [Univ. Federal do Rio Grande do Sul, Tramandai (Brazil)

2017-05-15

In this paper a solution for the one-dimensional steady state Multilayer Multigroup Neutron Diffusion Equation in cartesian geometry by Fictitious Borders Power Method and a perturbative analysis of this solution is presented. For each new iteration of the power method, the neutron flux is reconstructed by polynomial interpolation, so that it always remains in a standard form. However when the domain is long, an almost singular matrix arises in the interpolation process. To eliminate this singularity the domain segmented in R regions, called fictitious regions. The last step is to solve the neutron diffusion equation for each fictitious region in analytical form locally. The results are compared with results present in the literature. In order to analyze the sensitivity of the solution, a perturbation in the nuclear parameters is inserted to determine how a perturbation interferes in numerical results of the solution.

A Solution of the Convective-Diffusion Equation for Solute Mass Transfer inside a Capillary Membrane Bioreactor

Directory of Open Access Journals (Sweden)

B. Godongwana

2010-01-01

Full Text Available This paper presents an analytical model of substrate mass transfer through the lumen of a membrane bioreactor. The model is a solution of the convective-diffusion equation in two dimensions using a regular perturbation technique. The analysis accounts for radial-convective flow as well as axial diffusion of the substrate specie. The model is applicable to the different modes of operation of membrane bioreactor (MBR systems (e.g., dead-end, open-shell, or closed-shell mode, as well as the vertical or horizontal orientation. The first-order limit of the Michaelis-Menten equation for substrate consumption was used to test the developed model against available analytical results. The results obtained from the application of this model, along with a biofilm growth kinetic model, will be useful in the derivation of an efficiency expression for enzyme production in an MBR.

Linearized pseudo-Einstein equations on the Heisenberg group

Science.gov (United States)

Barletta, Elisabetta; Dragomir, Sorin; Jacobowitz, Howard

2017-02-01

We study the pseudo-Einstein equation R11bar = 0 on the Heisenberg group H1 = C × R. We consider first order perturbations θɛ =θ0 + ɛ θ and linearize the pseudo-Einstein equation about θ0 (the canonical Tanaka-Webster flat contact form on H1 thought of as a strictly pseudoconvex CR manifold). If θ =e2uθ0 the linearized pseudo-Einstein equation is Δb u - 4 | Lu|2 = 0 where Δb is the sublaplacian of (H1 ,θ0) and L bar is the Lewy operator. We solve the linearized pseudo-Einstein equation on a bounded domain Ω ⊂H1 by applying subelliptic theory i.e. existence and regularity results for weak subelliptic harmonic maps. We determine a solution u to the linearized pseudo-Einstein equation, possessing Heisenberg spherical symmetry, and such that u(x) → - ∞ as | x | → + ∞.

On progress of the solution of the stationary 2-dimensional neutron diffusion equation: a polynomial approximation method with error analysis

International Nuclear Information System (INIS)

Ceolin, C.; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T.

2015-01-01

Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)

On progress of the solution of the stationary 2-dimensional neutron diffusion equation: a polynomial approximation method with error analysis

Energy Technology Data Exchange (ETDEWEB)

Ceolin, C., E-mail: celina.ceolin@gmail.com [Universidade Federal de Santa Maria (UFSM), Frederico Westphalen, RS (Brazil). Centro de Educacao Superior Norte; Schramm, M.; Bodmann, B.E.J.; Vilhena, M.T., E-mail: celina.ceolin@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica

2015-07-01

Recently the stationary neutron diffusion equation in heterogeneous rectangular geometry was solved by the expansion of the scalar fluxes in polynomials in terms of the spatial variables (x; y), considering the two-group energy model. The focus of the present discussion consists in the study of an error analysis of the aforementioned solution. More specifically we show how the spatial subdomain segmentation is related to the degree of the polynomial and the Lipschitz constant. This relation allows to solve the 2-D neutron diffusion problem for second degree polynomials in each subdomain. This solution is exact at the knots where the Lipschitz cone is centered. Moreover, the solution has an analytical representation in each subdomain with supremum and infimum functions that shows the convergence of the solution. We illustrate the analysis with a selection of numerical case studies. (author)

An incomplete assembly with thresholding algorithm for systems of reaction-diffusion equations in three space dimensions IAT for reaction-diffusion systems

International Nuclear Information System (INIS)

Moore, Peter K.

2003-01-01

Solving systems of reaction-diffusion equations in three space dimensions can be prohibitively expensive both in terms of storage and CPU time. Herein, I present a new incomplete assembly procedure that is designed to reduce storage requirements. Incomplete assembly is analogous to incomplete factorization in that only a fixed number of nonzero entries are stored per row and a drop tolerance is used to discard small values. The algorithm is incorporated in a finite element method-of-lines code and tested on a set of reaction-diffusion systems. The effect of incomplete assembly on CPU time and storage and on the performance of the temporal integrator DASPK, algebraic solver GMRES and preconditioner ILUT is studied

Boundedness for a system of reaction-diffusion equations with more general Arrhenius term. Pt. 1

International Nuclear Information System (INIS)

Okoya, S.S.

1992-11-01

In this paper, we consider an extended model of a coupled nonlinear reaction-diffusion equation with Neumann-Neumann boundary conditions. We obtain upper linear growth bound for one of the components. We also find the corresponding bound for the case of Dirichlet-Dirichlet boundary conditions. (author). 12 refs

Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport

International Nuclear Information System (INIS)

Litvinenko, Yuri E.; Effenberger, Frederic

2014-01-01

Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we analytically solve a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.

Solution of the Multigroup-Diffusion equation by the response matrix method

International Nuclear Information System (INIS)

Oliveira, C.R.E.

1980-10-01

A preliminary analysis of the response matrix method is made, considering its application to the solution of the multigroup diffusion equations. The one-dimensional formulation is presented and used to test some flux expansions, seeking the application of the method to the two-dimensional problem. This formulation also solves the equations that arise from the integro-differential synthesis algorithm. The slow convergence of the power method, used to solve the eigenvalue problem, and its acceleration by means of the Chebyshev polynomial method, are also studied. An algorithm for the estimation of the dominance ratio is presented, based on the residues of two successive iteration vectors. This ratio, which is not known a priori, is fundamental for the efficiency of the method. Some numerical problems are solved, testing the 1D formulation of the response matrix method, its application to the synthesis algorithm and also, at the same time, the algorithm to accelerate the source problem. (Author) [pt

Hysteresis and Phase Transitions in a Lattice Regularization of an Ill-Posed Forward-Backward Diffusion Equation

Science.gov (United States)

Helmers, Michael; Herrmann, Michael

2018-03-01

We consider a lattice regularization for an ill-posed diffusion equation with a trilinear constitutive law and study the dynamics of phase interfaces in the parabolic scaling limit. Our main result guarantees for a certain class of single-interface initial data that the lattice solutions satisfy asymptotically a free boundary problem with a hysteretic Stefan condition. The key challenge in the proof is to control the microscopic fluctuations that are inevitably produced by the backward diffusion when a particle passes the spinodal region.

Fourth-order numerical solutions of diffusion equation by using SOR method with Crank-Nicolson approach

Science.gov (United States)

Muhiddin, F. A.; Sulaiman, J.

2017-09-01

The aim of this paper is to investigate the effectiveness of the Successive Over-Relaxation (SOR) iterative method by using the fourth-order Crank-Nicolson (CN) discretization scheme to derive a five-point Crank-Nicolson approximation equation in order to solve diffusion equation. From this approximation equation, clearly, it can be shown that corresponding system of five-point approximation equations can be generated and then solved iteratively. In order to access the performance results of the proposed iterative method with the fourth-order CN scheme, another point iterative method which is Gauss-Seidel (GS), also presented as a reference method. Finally the numerical results obtained from the use of the fourth-order CN discretization scheme, it can be pointed out that the SOR iterative method is superior in terms of number of iterations, execution time, and maximum absolute error.

Nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

Nonlinear differential equations

International Nuclear Information System (INIS)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics

Parabolic equations in biology growth, reaction, movement and diffusion

CERN Document Server

Perthame, Benoît

2015-01-01

This book presents several fundamental questions in mathematical biology such as Turing instability, pattern formation, reaction-diffusion systems, invasion waves and Fokker-Planck equations. These are classical modeling tools for mathematical biology with applications to ecology and population dynamics, the neurosciences, enzymatic reactions, chemotaxis, invasion waves etc. The book presents these aspects from a mathematical perspective, with the aim of identifying those qualitative properties of the models that are relevant for biological applications. To do so, it uncovers the mechanisms at work behind Turing instability, pattern formation and invasion waves. This involves several mathematical tools, such as stability and instability analysis, blow-up in finite time, asymptotic methods and relative entropy properties. Given the content presented, the book is well suited as a textbook for master-level coursework.

Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

KAUST Repository

Peng, Qiujin; Qiao, Zhonghua; Sun, Shuyu

2017-01-01

In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L-infinity convergence of these two schemes are proved. Numerical results demonstrate the good approximation of the fourth order equation and confirm reliability of these two schemes.

Stability and Convergence Analysis of Second-Order Schemes for a Diffuse Interface Model with Peng-Robinson Equation of State

KAUST Repository

Peng, Qiujin

2017-09-18

In this paper, we present two second-order numerical schemes to solve the fourth order parabolic equation derived from a diffuse interface model with Peng-Robinson Equation of state (EOS) for pure substance. The mass conservation, energy decay property, unique solvability and L-infinity convergence of these two schemes are proved. Numerical results demonstrate the good approximation of the fourth order equation and confirm reliability of these two schemes.

Elaboration of a nodal method to solve the steady state multigroup diffusion equation. Study and use of the multigroup diffusion code DAHRA

International Nuclear Information System (INIS)

Halilou, A.; Lounici, A.

1981-01-01

The subject is divided in two parts: In the first part a nodal method has been worked out to solve the steady state multigroup diffusion equation. This method belongs to the same set of nodal methods currently used to calculate the exact fission powers and neutron fluxes in a very short computing time. It has been tested on a two dimensional idealized reactors. The effective multiplication factor and the fission powers for each fuel element have been calculated. The second part consists in studying and mastering the multigroup diffusion code DAHRA - a reduced version of DIANE - a two dimensional code using finite difference method

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

Science.gov (United States)

Zhou, L.-Q.; Meleshko, S. V.

2017-07-01

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.

A self-consistent nodal method in response matrix formalism for the multigroup diffusion equations

International Nuclear Information System (INIS)

Malambu, E.M.; Mund, E.H.

1996-01-01

We develop a nodal method for the multigroup diffusion equations, based on the transverse integration procedure (TIP). The efficiency of the method rests upon the convergence properties of a high-order multidimensional nodal expansion and upon numerical implementation aspects. The discrete 1D equations are cast in response matrix formalism. The derivation of the transverse leakage moments is self-consistent i.e. does not require additional assumptions. An outstanding feature of the method lies in the linear spatial shape of the local transverse leakage for the first-order scheme. The method is described in the two-dimensional case. The method is validated on some classical benchmark problems. (author)

Automatic simplification of systems of reaction-diffusion equations by a posteriori analysis.

Science.gov (United States)

Maybank, Philip J; Whiteley, Jonathan P

2014-02-01

Many mathematical models in biology and physiology are represented by systems of nonlinear differential equations. In recent years these models have become increasingly complex in order to explain the enormous volume of data now available. A key role of modellers is to determine which components of the model have the greatest effect on a given observed behaviour. An approach for automatically fulfilling this role, based on a posteriori analysis, has recently been developed for nonlinear initial value ordinary differential equations [J.P. Whiteley, Model reduction using a posteriori analysis, Math. Biosci. 225 (2010) 44-52]. In this paper we extend this model reduction technique for application to both steady-state and time-dependent nonlinear reaction-diffusion systems. Exemplar problems drawn from biology are used to demonstrate the applicability of the technique. Copyright © 2014 Elsevier Inc. All rights reserved.

Image recovery using diffusion equation embedded neural network

International Nuclear Information System (INIS)

Torkamani-Azar, F.

2001-01-01

Artificial neural networks with their inherent parallelism have been shown to perform well in many processing applications. In this paper, a new self-organizing approach for the recovery of gray level images degraded by additive noise based on embedding the diffusion equation in a neural network (without using a priori knowledge about the image point spread function, noise or original image) is described which enhances and restores gray levels of degraded images and is for application in low-level processing. Two learning features have been proposed which would be effective in the practical implementation of such a network. The recovery procedure needs some parameter estimation such as different error goals. While the required computation is not excessive, the procedure dose not require too many iterations and convergence is very fast. In addition, through the simulation the new network showed that it has superior ability to give a better quality result with a minimum of the sum of the squared error

Group-theoretical aspects of the discrete sine-Gordon equation

International Nuclear Information System (INIS)

Orfanidis, S.J.

1980-01-01

The group-theoretical interpretation of the sine-Gordon equation in terms of connection forms on fiber bundles is extended to the discrete case. Solutions of the discrete sine-Gordon equation induce surfaces on a lattice in the SU(2) group space. The inverse scattering representation, expressing the parallel transport of fibers, is implemented by means of finite rotations. Discrete Baecklund transformations are realized as gauge transformations. The three-dimensional inverse scattering representation is used to derive a discrete nonlinear sigma model, and the corresponding Baecklund transformation and Pohlmeyer's R transformation are constructed

The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

OpenAIRE

Jin, Bangti; Lazarov, Raytcho; Liu, Yikan; Zhou, Zhi

2014-01-01

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite...

Measurement and account method for the available standing time of a smoke screen applying the diffusion equation

Energy Technology Data Exchange (ETDEWEB)

Chen-Guang, Zhu; Gong-Pei, Pan; Xiao-Yun, Wu; Hua, Guan [308, School of Chemical Engineering, Nanjing University of Science and Technology, 210094 (China)

2006-06-15

The available shielding area of a smoke screen is an important parameter to evaluate its protection capability. It varies with time according to a parabola equation, which could be obtained by the diffusion equation and a few measured data. The available standing time which could satisfy the request of rating area can be calculated. The process of deriving the equation, the experimental procedure and the method of data collecting and processing is presented in detail in this paper. The calculated result is compared with engineering experience to verify the method. (Abstract Copyright [2006], Wiley Periodicals, Inc.)

Solution of linear and nonlinear matrix systems. Application to a nonlinear diffusion equation

International Nuclear Information System (INIS)

Bonnet, M.; Meurant, G.

1978-01-01

Different methods of solution of linear and nonlinear algebraic systems are applied to the nonlinear system obtained by discretizing a nonlinear diffusion equation. For linear systems, methods in general use of alternating directions type or Gauss Seidel's methods are compared to more recent ones of the type of generalized conjugate gradient; the superiority of the latter is shown by numerical examples. For nonlinear systems, a method on nonlinear conjugate gradient is studied as also Newton's method and some of its variants. It should be noted, however that Newton's method is found to be more efficient when coupled with a good method for solution of the linear system. To conclude, such methods are used to solve a nonlinear diffusion problem and the numerical results obtained are to be compared [fr

Solution of linear and nonlinear matrix systems. Application to a nonlinear diffusion equation

International Nuclear Information System (INIS)

Bonnet, M.; Meurant, G.

1978-01-01

The object of this study is to compare different methods of solving linear and nonlinear algebraic systems and to apply them to the nonlinear system obtained by discretizing a nonlinear diffusion equation. For linear systems the conventional methods of alternating direction type or Gauss Seidel's methods are compared to more recent ones of the type of generalized conjugate gradient; the superiority of the latter is shown by numerical examples. For nonlinear systems, a method of nonlinear conjugate gradient is studied together with Newton's method and some of its variants. It should be noted, however, that Newton's method is found to be more efficient when coupled with a good method for solving the linear system. As a conclusion, these methods are used to solve a nonlinear diffusion problem and the numerical results obtained are compared [fr

Analytical solution of the multigroup neutron diffusion kinetic equation in one-dimensional cartesian geometry by the integral transform technique

International Nuclear Information System (INIS)

Ceolin, Celina

2010-01-01

The objective of this work is to obtain an analytical solution of the neutron diffusion kinetic equation in one-dimensional cartesian geometry, to monoenergetic and multigroup problems. These equations are of the type stiff, due to large differences in the orders of magnitude of the time scales of the physical phenomena involved, which make them difficult to solve. The basic idea of the proposed method is applying the spectral expansion in the scalar flux and in the precursor concentration, taking moments and solving the resulting matrix problem by the Laplace transform technique. Bearing in mind that the equation for the precursor concentration is a first order linear differential equation in the time variable, to enable the application of the spectral method we introduce a fictitious diffusion term multiplied by a positive value which tends to zero. This procedure opened the possibility to find an analytical solution to the problem studied. We report numerical simulations and analysis of the results obtained with the precision controlled by the truncation order of the series. (author)

Numerical solution of the unsteady diffusion-convection-reaction equation based on improved spectral Galerkin method

Science.gov (United States)

Zhong, Jiaqi; Zeng, Cheng; Yuan, Yupeng; Zhang, Yuzhe; Zhang, Ye

2018-04-01

The aim of this paper is to present an explicit numerical algorithm based on improved spectral Galerkin method for solving the unsteady diffusion-convection-reaction equation. The principal characteristics of this approach give the explicit eigenvalues and eigenvectors based on the time-space separation method and boundary condition analysis. With the help of Fourier series and Galerkin truncation, we can obtain the finite-dimensional ordinary differential equations which facilitate the system analysis and controller design. By comparing with the finite element method, the numerical solutions are demonstrated via two examples. It is shown that the proposed method is effective.

Diffusion Influenced Adsorption Kinetics.

Science.gov (United States)

Miura, Toshiaki; Seki, Kazuhiko

2015-08-27

When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir-Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir-Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integro-differential equation using the Grünwald-Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage.

Analytical solutions of linear diffusion and wave equations in semi-infinite domains by using a new integral transform

Directory of Open Access Journals (Sweden)

Gao Lin

2017-01-01

Full Text Available Recently, a new integral transform similar to Sumudu transform has been proposed by Yang [1]. Some of the properties of the integral transform are expanded in the present article. Meanwhile, new applications to the linear wave and diffusion equations in semi-infinite domains are discussed in detail. The proposed method provides an alternative approach to solve the partial differential equations in mathematical physics.

Modelling uncertainties in the diffusion-advection equation for radon transport in soil using interval arithmetic.

Science.gov (United States)

Chakraverty, S; Sahoo, B K; Rao, T D; Karunakar, P; Sapra, B K

2018-02-01

Modelling radon transport in the earth crust is a useful tool to investigate the changes in the geo-physical processes prior to earthquake event. Radon transport is modeled generally through the deterministic advection-diffusion equation. However, in order to determine the magnitudes of parameters governing these processes from experimental measurements, it is necessary to investigate the role of uncertainties in these parameters. Present paper investigates this aspect by combining the concept of interval uncertainties in transport parameters such as soil diffusivity, advection velocity etc, occurring in the radon transport equation as applied to soil matrix. The predictions made with interval arithmetic have been compared and discussed with the results of classical deterministic model. The practical applicability of the model is demonstrated through a case study involving radon flux measurements at the soil surface with an accumulator deployed in steady-state mode. It is possible to detect the presence of very low levels of advection processes by applying uncertainty bounds on the variations in the observed concentration data in the accumulator. The results are further discussed. Copyright © 2017 Elsevier Ltd. All rights reserved.

Mixed-hybrid finite element method for the transport equation and diffusion approximation of transport problems; Resolution de l'equation du transport par une methode d'elements finis mixtes-hybrides et approximation par la diffusion de problemes de transport

Energy Technology Data Exchange (ETDEWEB)

Cartier, J

2006-04-15

This thesis focuses on mathematical analysis, numerical resolution and modelling of the transport equations. First of all, we deal with numerical approximation of the solution of the transport equations by using a mixed-hybrid scheme. We derive and study a mixed formulation of the transport equation, then we analyse the related variational problem and present the discretization and the main properties of the scheme. We particularly pay attention to the behavior of the scheme and we show its efficiency in the diffusion limit (when the mean free path is small in comparison with the characteristic length of the physical domain). We present academical benchmarks in order to compare our scheme with other methods in many physical configurations and validate our method on analytical test cases. Unstructured and very distorted meshes are used to validate our scheme. The second part of this thesis deals with two transport problems. The first one is devoted to the study of diffusion due to boundary conditions in a transport problem between two plane plates. The second one consists in modelling and simulating radiative transfer phenomenon in case of the industrial context of inertial confinement fusion. (author)

PyR@TE. Renormalization group equations for general gauge theories

Science.gov (United States)

Lyonnet, F.; Schienbein, I.; Staub, F.; Wingerter, A.

2014-03-01

Although the two-loop renormalization group equations for a general gauge field theory have been known for quite some time, deriving them for specific models has often been difficult in practice. This is mainly due to the fact that, albeit straightforward, the involved calculations are quite long, tedious and prone to error. The present work is an attempt to facilitate the practical use of the renormalization group equations in model building. To that end, we have developed two completely independent sets of programs written in Python and Mathematica, respectively. The Mathematica scripts will be part of an upcoming release of SARAH 4. The present article describes the collection of Python routines that we dubbed PyR@TE which is an acronym for “Python Renormalization group equations At Two-loop for Everyone”. In PyR@TE, once the user specifies the gauge group and the particle content of the model, the routines automatically generate the full two-loop renormalization group equations for all (dimensionless and dimensionful) parameters. The results can optionally be exported to LaTeX and Mathematica, or stored in a Python data structure for further processing by other programs. For ease of use, we have implemented an interactive mode for PyR@TE in form of an IPython Notebook. As a first application, we have generated with PyR@TE the renormalization group equations for several non-supersymmetric extensions of the Standard Model and found some discrepancies with the existing literature. Catalogue identifier: AERV_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AERV_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.: 924959 No. of bytes in distributed program, including test data, etc.: 495197 Distribution format: tar.gz Programming language: Python. Computer

Direct method of solving finite difference nonlinear equations for multicomponent diffusion in a gas centrifuge

International Nuclear Information System (INIS)

Potemki, Valeri G.; Borisevich, Valentine D.; Yupatov, Sergei V.

1996-01-01

This paper describes the the next evolution step in development of the direct method for solving systems of Nonlinear Algebraic Equations (SNAE). These equations arise from the finite difference approximation of original nonlinear partial differential equations (PDE). This method has been extended on the SNAE with three variables. The solving SNAE bases on Reiterating General Singular Value Decomposition of rectangular matrix pencils (RGSVD-algorithm). In contrast to the computer algebra algorithm in integer arithmetic based on the reduction to the Groebner's basis that algorithm is working in floating point arithmetic and realizes the reduction to the Kronecker's form. The possibilities of the method are illustrated on the example of solving the one-dimensional diffusion equation for 3-component model isotope mixture in a ga centrifuge. The implicit scheme for the finite difference equations without simplifying the nonlinear properties of the original equations is realized. The technique offered provides convergence to the solution for the single run. The Toolbox SNAE is developed in the framework of the high performance numeric computation and visualization software MATLAB. It includes more than 30 modules in MATLAB language for solving SNAE with two and three variables. (author)

A numerical solution for a class of time fractional diffusion equations with delay

Directory of Open Access Journals (Sweden)

Pimenov Vladimir G.

2017-09-01

Full Text Available This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ2−α+ h4 in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

Scalable implicit methods for reaction-diffusion equations in two and three space dimensions

Energy Technology Data Exchange (ETDEWEB)

Veronese, S.V.; Othmer, H.G. [Univ. of Utah, Salt Lake City, UT (United States)

1996-12-31

This paper describes the implementation of a solver for systems of semi-linear parabolic partial differential equations in two and three space dimensions. The solver is based on a parallel implementation of a non-linear Alternating Direction Implicit (ADI) scheme which uses a Cartesian grid in space and an implicit time-stepping algorithm. Various reordering strategies for the linearized equations are used to reduce the stride and improve the overall effectiveness of the parallel implementation. We have successfully used this solver for large-scale reaction-diffusion problems in computational biology and medicine in which the desired solution is a traveling wave that may contain rapid transitions. A number of examples that illustrate the efficiency and accuracy of the method are given here; the theoretical analysis will be presented.

BASHAN: A few-group three-dimensional diffusion code with burnup and fuel management features

International Nuclear Information System (INIS)

Pearce, D.F.

1970-12-01

The diffusion equation for a two or three-dimensional, two-group or multi-group downscatter problem is solved by conventional finite difference techniques. An x-y-z geometry is assumed with an 'in-channel' mesh point representation. Options are available which allow representation of a soluble poison dispersed throughout the reactor, and also absorber rods in specified channels. The power distribution and multiplication factor k eff are calculated and a point rating map is used to advance the irradiation at each mesh point by a specified time-step so that burnup is followed. Fuel changes may be made so that radial shuffling and axial shuffling fuel management schemes can be studies. The code has been written in FORTRAN S2 for an IBM 7030 (STRETCH) computer which, with a fast store of 80,000 locations, allows problems of up to 15,000 mesh points to be dealt with. Conversion to FORTRAN IV for IBM 360 has now been completed. (author)

Diffusion and the self-measurability

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Holeček M.

2009-06-01

Full Text Available The familiar diffusion equation, ∂g/∂t = DΔg, is studied by using the spatially averaged quantities. A non-local relation, so-called the self-measurability condition, fulfilled by this equation is obtained. We define a broad class of diffusion equations defined by some "diffusion inequality", ∂g/∂t · Δg ≥ 0, and show that it is equivalent to the self-measurability condition. It allows formulating the diffusion inequality in a non-local form. That represents an essential generalization of the diffusion problem in the case when the field g(x, t is not smooth. We derive a general differential equation for averaged quantities coming from the self-measurability condition.

Calculation of similarity solutions of partial differential equations

International Nuclear Information System (INIS)

Dresner, L.

1980-08-01

When a partial differential equation in two independent variables is invariant to a group G of stretching transformations, it has similarity solutions that can be found by solving an ordinary differential equation. Under broad conditions, this ordinary differential equation is also invariant to another stretching group G', related to G. The invariance of the ordinary differential equation to G' can be used to simplify its solution, particularly if it is of second order. Then a method of Lie's can be used to reduce it to a first-order equation, the study of which is greatly facilitated by analysis of its direction field. The method developed here is applied to three examples: Blasius's equation for boundary layer flow over a flat plate and two nonlinear diffusion equations, cc/sub t/ = c/sub zz/ and c/sub t/ = (cc/sub z/)/sub z/

Evaluating Equating Accuracy and Assumptions for Groups that Differ in Performance

Science.gov (United States)

Powers, Sonya; Kolen, Michael J.

2014-01-01

Accurate equating results are essential when comparing examinee scores across exam forms. Previous research indicates that equating results may not be accurate when group differences are large. This study compared the equating results of frequency estimation, chained equipercentile, item response theory (IRT) true-score, and IRT observed-score…

SNAP - a three dimensional neutron diffusion code

International Nuclear Information System (INIS)

McCallien, C.W.J.

1993-02-01

This report describes a one- two- three-dimensional multi-group diffusion code, SNAP, which is primarily intended for neutron diffusion calculations but can also carry out gamma calculations if the diffusion approximation is accurate enough. It is suitable for fast and thermal reactor core calculations and for shield calculations. SNAP can solve the multi-group neutron diffusion equations using finite difference methods. The one-dimensional slab, cylindrical and spherical geometries and the two-dimensional case are all treated as simple special cases of three-dimensional geometries. Numerous reflective and periodic symmetry options are available and may be used to reduce the number of mesh points necessary to represent the system. Extrapolation lengths can be specified at internal and external boundaries. (Author)

ADI as a preconditioning for solving the convection-diffusion equation

International Nuclear Information System (INIS)

Chin, R.C.Y.; Manteuffel, T.A.; De Pillis, J.

1984-01-01

The authors examine a splitting of the operator obtained from a steady convection-diffusion equation with variable coefficients in which the convection term dominates. The operator is split into a dominant and subdominant parts consistent with the inherent directional property of the partial differential equation. The equations involving the dominant parts should be easily solved. The authors accelerate the convergence of this splitting or the outer iteration by a Chebyshev semi-iterative method. When the dominant part has constant coefficients, it can be easily solved using alternating direction implicit (ADI) methods. This is called the inner iteration. The optimal parameters for a stationary two-parameter ADI method are obtained when the eigenvalues become complex. This corresponds to either the horizontal or the vertical half-grid Reynolds number larger than unity. The Chebyshev semi-iterative method is used to accelerate the convergence of the inner ADI iteration. A two-fold increase in speed is obtained when the ADI iteration matrix has real eigenvalues, and the increase is less significant when the eigenvalues are complex. If either the horizontal or the vertical half-grid Reynolds number is equal to one, the spectral radius of the optimal ADI iterative matrix is zero. However, a high degree of nilpotency impairs rapid convergence. This problem is removed by introducing a more implicit iterative method called ADI/Gauss-Seidel (ADI/GS). ADI/GS resolves the nilpotency and, thus, converges more rapidly for half-grid Reynolds number near 1. Finally, these methods are compared with several well-known schemes on test problems. 23 references, 5 figures

Correction of the calculation of beam loading based in the RF power diffusion equation

International Nuclear Information System (INIS)

Silva, R. da.

1980-01-01

It is described an empirical correction based upon experimental datas of others authors in ORELA, GELINA and SLAC accelerators, to the calculation of the energy loss due to the beam loading effect as stated by the RF power diffusion equation theory an accelerating structure. It is obtained a dependence of this correction with the electron pulse full width half maximum, but independent of the electron energy. (author) [pt

Comparisons of hybrid radiosity-diffusion model and diffusion equation for bioluminescence tomography in cavity cancer detection

Science.gov (United States)

Chen, Xueli; Yang, Defu; Qu, Xiaochao; Hu, Hao; Liang, Jimin; Gao, Xinbo; Tian, Jie

2012-06-01

Bioluminescence tomography (BLT) has been successfully applied to the detection and therapeutic evaluation of solid cancers. However, the existing BLT reconstruction algorithms are not accurate enough for cavity cancer detection because of neglecting the void problem. Motivated by the ability of the hybrid radiosity-diffusion model (HRDM) in describing the light propagation in cavity organs, an HRDM-based BLT reconstruction algorithm was provided for the specific problem of cavity cancer detection. HRDM has been applied to optical tomography but is limited to simple and regular geometries because of the complexity in coupling the boundary between the scattering and void region. In the provided algorithm, HRDM was first applied to three-dimensional complicated and irregular geometries and then employed as the forward light transport model to describe the bioluminescent light propagation in tissues. Combining HRDM with the sparse reconstruction strategy, the cavity cancer cells labeled with bioluminescent probes can be more accurately reconstructed. Compared with the diffusion equation based reconstruction algorithm, the essentiality and superiority of the HRDM-based algorithm were demonstrated with simulation, phantom and animal studies. An in vivo gastric cancer-bearing nude mouse experiment was conducted, whose results revealed the ability and feasibility of the HRDM-based algorithm in the biomedical application of gastric cancer detection.

Predicting in vivo glioma growth with the reaction diffusion equation constrained by quantitative magnetic resonance imaging data

International Nuclear Information System (INIS)

Hormuth II, David A; Weis, Jared A; Barnes, Stephanie L; Miga, Michael I; Yankeelov, Thomas E; Rericha, Erin C; Quaranta, Vito

2015-01-01

Reaction–diffusion models have been widely used to model glioma growth. However, it has not been shown how accurately this model can predict future tumor status using model parameters (i.e., tumor cell diffusion and proliferation) estimated from quantitative in vivo imaging data. To this end, we used in silico studies to develop the methods needed to accurately estimate tumor specific reaction–diffusion model parameters, and then tested the accuracy with which these parameters can predict future growth. The analogous study was then performed in a murine model of glioma growth. The parameter estimation approach was tested using an in silico tumor ‘grown’ for ten days as dictated by the reaction–diffusion equation. Parameters were estimated from early time points and used to predict subsequent growth. Prediction accuracy was assessed at global (total volume and Dice value) and local (concordance correlation coefficient, CCC) levels. Guided by the in silico study, rats (n = 9) with C6 gliomas, imaged with diffusion weighted magnetic resonance imaging, were used to evaluate the model’s accuracy for predicting in vivo tumor growth. The in silico study resulted in low global (tumor volume error 0.92) and local (CCC values >0.80) level errors for predictions up to six days into the future. The in vivo study showed higher global (tumor volume error >11.7%, Dice <0.81) and higher local (CCC <0.33) level errors over the same time period. The in silico study shows that model parameters can be accurately estimated and used to accurately predict future tumor growth at both the global and local scale. However, the poor predictive accuracy in the experimental study suggests the reaction–diffusion equation is an incomplete description of in vivo C6 glioma biology and may require further modeling of intra-tumor interactions including segmentation of (for example) proliferative and necrotic regions. (paper)

A modified two-fluid model for the application of two-group interfacial area transport equation

International Nuclear Information System (INIS)

Sun, X.; Ishii, M.; Kelly, J.

2003-01-01

This paper presents the modified two-fluid model that is ready to be applied in the approach of the two-group interfacial area transport equation. The two-group interfacial area transport equation was developed to provide a mechanistic constitutive relation for the interfacial area concentration in the two-fluid model. In the two-group transport equation, bubbles are categorized into two groups: spherical/distorted bubbles as Group 1 while cap/slug/churn-turbulent bubbles as Group 2. Therefore, this transport equation can be employed in the flow regimes spanning from bubbly, cap bubbly, slug to churn-turbulent flows. However, the introduction of the two groups of bubbles requires two gas velocity fields. Yet it is not desirable to solve two momentum equations for the gas phase alone. In the current modified two-fluid model, a simplified approach is proposed. The momentum equation for the averaged velocity of both Group-1 and Group-2 bubbles is retained. By doing so, the velocity difference between Group-1 and Group-2 bubbles needs to be determined. This may be made either based on simplified momentum equations for both Group-1 and Group-2 bubbles or by a modified drift-flux model

DIFFUSION - WRS system module number 7539 for solving a set of multigroup diffusion equations in one dimension

International Nuclear Information System (INIS)

Grimstone, M.J.

1978-06-01

The WRS Modular Programming System has been developed as a means by which programmes may be more efficiently constructed, maintained and modified. In this system a module is a self-contained unit typically composed of one or more Fortran routines, and a programme is constructed from a number of such modules. This report describes one WRS module, the function of which is to solve a set of multigroup diffusion equations for a system represented in one-dimensional plane, cylindrical or spherical geometry. The information given in this manual is of use both to the programmer wishing to incorporate the module in a programme, and to the user of such a programme. (author)

On the implementation of an accurate and efficient solver for convection-diffusion equations

Science.gov (United States)

Wu, Chin-Tien

In this dissertation, we examine several different aspects of computing the numerical solution of the convection-diffusion equation. The solution of this equation often exhibits sharp gradients due to Dirichlet outflow boundaries or discontinuities in boundary conditions. Because of the singular-perturbed nature of the equation, numerical solutions often have severe oscillations when grid sizes are not small enough to resolve sharp gradients. To overcome such difficulties, the streamline diffusion discretization method can be used to obtain an accurate approximate solution in regions where the solution is smooth. To increase accuracy of the solution in the regions containing layers, adaptive mesh refinement and mesh movement based on a posteriori error estimations can be employed. An error-adapted mesh refinement strategy based on a posteriori error estimations is also proposed to resolve layers. For solving the sparse linear systems that arise from discretization, goemetric multigrid (MG) and algebraic multigrid (AMG) are compared. In addition, both methods are also used as preconditioners for Krylov subspace methods. We derive some convergence results for MG with line Gauss-Seidel smoothers and bilinear interpolation. Finally, while considering adaptive mesh refinement as an integral part of the solution process, it is natural to set a stopping tolerance for the iterative linear solvers on each mesh stage so that the difference between the approximate solution obtained from iterative methods and the finite element solution is bounded by an a posteriori error bound. Here, we present two stopping criteria. The first is based on a residual-type a posteriori error estimator developed by Verfurth. The second is based on an a posteriori error estimator, using local solutions, developed by Kay and Silvester. Our numerical results show the refined mesh obtained from the iterative solution which satisfies the second criteria is similar to the refined mesh obtained from

Applications of Lie-group methods to the equations of magnetohydrodynamics

International Nuclear Information System (INIS)

Mandrekas, J.

1987-01-01

The invariance properties of various sets of magnetohydrodynamic (MHD) equations are studied using techniques from the theory of differential forms. Equations considered include the ideal MHD equations in different geometries and with different magnetic field configurations, the MHD equations in the presence of gravitational forces due to self-attraction or external fields, and the MHD equations including finite thermal conductivity and magnetic viscosity. The knowledge of the group structure of these equations is then used to introduce similarity variables to these equations. For each choice of similarity variables, the original set of partial differential equations is transformed into a set of ordinary differential equations and the most general form of the initial conditions is determined. Three cases are studied in detail and the corresponding sets of ordinary differential equations are solved numerically: the problem of a blast wave in an inhomogeneous atmosphere, the problem of a piston moving according to a power law in time, and the problem of a piston moving according to an exponential law in time

Application of COMSOL in the solution of the neutron diffusion equations for fast nuclear reactors in stationary state

International Nuclear Information System (INIS)

Silva A, L.; Del Valle G, E.

2012-10-01

This work shows an application of the program COMSOL Multi physics Ver. 4.2a in the solution of the neutron diffusion equations for several energy groups in nuclear reactors whose core is formed by assemblies of hexagonal transversal cut as is the cas of fast reactors. A reference problem of 4 energy groups is described of which takes the cross sections which are processed by means of a program that prepares the definition of the constants utilized in COMSOL for the generic partial differential equations that this uses. The considered solution domain is the sixth part of the core which is applied frontier conditions of reflection and incoming flux zero. The discretization mesh is elaborated in automatic way by COMSOL and the solution method is one of finite elements of Lagrange grade two. The reference problem is known as the Knk with and without control rod which led to propose the calculation of the effective multiplication factor in function of the control rod fraction from a value 0 (completely inserted control rod) until the value 1 (completely extracted control rod). Besides this the reactivity was determined as well as the change of this in function of control rod fraction. The neutrons scalar flux for each energy group with and without control rod is proportioned. The reported results show a behavior similar to the one reported in other works but using the discreet ordinates S 2 approximation. (Author)

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Directory of Open Access Journals (Sweden)

Ben C. Yee

2017-09-01

Full Text Available In this paper, we present a new multilevel in space and energy diffusion (MSED method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1 a grey (one-group diffusion equation used to efficiently converge the fission source and eigenvalue, (2 a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3 a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

On the fundamental equation of nonequilibrium statistical physics—Nonequilibrium entropy evolution equation and the formula for entropy production rate

Institute of Scientific and Technical Information of China (English)

无

2010-01-01

In this paper the author presents an overview on his own research works. More than ten years ago, we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation. That is the stochastic velocity type’s Langevin equation in 6N dimensional phase space or its equivalent Liouville diffusion equation. This equation is time-reversed asymmetrical. It shows that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality, and the law of motion of statistical thermodynamics is expressed by a superposition of both the law of dynamics and the stochastic velocity and possesses both determinism and probability. Hence it is different from the law of motion of particles in dynamical systems. The stochastic diffusion motion of the particles is the microscopic origin of macroscopic irreversibility. Starting from this fundamental equation the BBGKY diffusion equation hierarchy, the Boltzmann collision diffusion equation, the hydrodynamic equations such as the mass drift-diffusion equation, the Navier-Stokes equation and the thermal conductivity equation have been derived and presented here. What is more important, we first constructed a nonlinear evolution equation of nonequilibrium entropy density in 6N, 6 and 3 dimensional phase space, predicted the existence of entropy diffusion. This entropy evolution equation plays a leading role in nonequilibrium entropy theory, it reveals that the time rate of change of nonequilibrium entropy density originates together from its drift, diffusion and production in space. From this evolution equation, we presented a formula for entropy production rate (i.e. the law of entropy increase) in 6N and 6 dimensional phase space, proved that internal attractive force in nonequilibrium system can result in entropy decrease while internal repulsive force leads to another entropy increase, and derived a common expression for this entropy decrease rate or

A new finite element formulation for CFD:VIII. The Galerkin/least-squares method for advective-diffusive equations

International Nuclear Information System (INIS)

Hughes, T.J.R.; Hulbert, G.M.; Franca, L.P.

1988-10-01

Galerkin/least-squares finite element methods are presented for advective-diffusive equations. Galerkin/least-squares represents a conceptual simplification of SUPG, and is in fact applicable to a wide variety of other problem types. A convergence analysis and error estimates are presented. (author) [pt

On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

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Yuri Luchko

2017-12-01

Full Text Available In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

Lattice Boltzmann scheme for mixture modeling: analysis of the continuum diffusion regimes recovering Maxwell-Stefan model and incompressible Navier-Stokes equations.

Science.gov (United States)

Asinari, Pietro

2009-11-01

A finite difference lattice Boltzmann scheme for homogeneous mixture modeling, which recovers Maxwell-Stefan diffusion model in the continuum limit, without the restriction of the mixture-averaged diffusion approximation, was recently proposed [P. Asinari, Phys. Rev. E 77, 056706 (2008)]. The theoretical basis is the Bhatnagar-Gross-Krook-type kinetic model for gas mixtures [P. Andries, K. Aoki, and B. Perthame, J. Stat. Phys. 106, 993 (2002)]. In the present paper, the recovered macroscopic equations in the continuum limit are systematically investigated by varying the ratio between the characteristic diffusion speed and the characteristic barycentric speed. It comes out that the diffusion speed must be at least one order of magnitude (in terms of Knudsen number) smaller than the barycentric speed, in order to recover the Navier-Stokes equations for mixtures in the incompressible limit. Some further numerical tests are also reported. In particular, (1) the solvent and dilute test cases are considered, because they are limiting cases in which the Maxwell-Stefan model reduces automatically to Fickian cases. Moreover, (2) some tests based on the Stefan diffusion tube are reported for proving the complete capabilities of the proposed scheme in solving Maxwell-Stefan diffusion problems. The proposed scheme agrees well with the expected theoretical results.

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.

STAB: A kinetic, three-dimensional, one-group digital computer program

International Nuclear Information System (INIS)

Curtis, A.R.; Tyror, J.G.; Wrigley, H.E.

1961-10-01

A computer program for solving the one-group, time dependent, three-dimensional diffusion equation together with auxiliary equations representing heat transfer, xenon production and control rod movements, is presented. The equations and the methods of solution are discussed. (author)

Oceanic diffusion in the coastal area

International Nuclear Information System (INIS)

Rukuda, Masaaki

1980-03-01

Described in this paper is the eddy diffusion in the area off Tokai Village investigated by means of dye diffusion experiment and of oceanic observation. In order to assess the oceanic diffusion in coastal areas, improved methods effective in complex field were developed. The oceanic diffusion was separated in two groups, horizontal and vertical diffusion respectively. Both these diffusions are combined and their analysis together is difficult. The oceanic diffusion is thus considered separately. Instantaneous point release is the basis of horizontal diffusion analysis. Continuous release is then the overlap of numerous instantaneous releases. It was shown that the diffusion parameters derived from the results of diffusion experiment or oceanic observation vary widely with time and place and with sea conditions. A simple diffusion equation was developed from the equation of continuity. The results were in good agreement with seasonal mean horizontal distribution of river water in the sea area. The vertical observation in diffusion experiment is difficult and the vertical structure of oceanic condition is complex, so that the research on vertical diffusion generally is not advanced yet. With river water as the tracer, a method of estimating vertical diffusion parameters with a Gaussian model or one-dimensional model was developed. The vertical diffusion near sea bottom was numerically analized with suspended particles in seawater as the tracer. Diffusion was computed for each particle size, and by summing up the vertical distribution of beam attenuation coefficient was estimated. By comparing the results of estimation and those of observation the vertical diffusivity and the particle size distribution at sea bottom could be estimated. (author)

Pseudospectral operational matrix for numerical solution of single and multiterm time fractional diffusion equation

OpenAIRE

GHOLAMI, SAEID; BABOLIAN, ESMAIL; JAVIDI, MOHAMMAD

2016-01-01

This paper presents a new numerical approach to solve single and multiterm time fractional diffusion equations. In this work, the space dimension is discretized to the Gauss$-$Lobatto points. We use the normalized Grunwald approximation for the time dimension and a pseudospectral successive integration matrix for the space dimension. This approach shows that with fewer numbers of points, we can approximate the solution with more accuracy. Some examples with numerical results in tables and fig...

Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation

OpenAIRE

Li, Zhiyuan; Yamamoto, Masahiro

2014-01-01

This article proves the uniqueness for two kinds of inverse problems of identifying fractional orders in diffusion equations with multiple time-fractional derivatives by pointwise observation. By means of eigenfunction expansion and Laplace transform, we reduce the uniqueness for our inverse problems to the uniqueness of expansions of some special function and complete the proof.

Diffusion of Charged Species in Liquids