The initial-value problem for neutron transport in a finite body with generalized boundary conditions

Summary We prove the existence and the uniqueness of the solution of the initial-value problem for neutron transport in a finite convex body with generalized boundary conditions which include both the perfect reflection and the vacuum boundary condition as particular cases.Moreover, we show that the transport operator has at least one real eigenvalue provided a perfect reflection boundary condition is valid over a finite portion of the boundary surface.Finally, we indicate the asymptotic behavior of the neutron density as t + .

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Sommario Si prova l'esistenza e l'unicità della soluzione di un problema di trasporto di neutroni in un mezzo finito con condizioni al contorno generalizzate.Si mostra che l'operatore del trasporto ammette almeno un autovalore reale purché la condizione di perfetta riflessione sia soddisfatta su almeno una porzione finita della superficie.Infine si studia il comportamento asintotico della densità neutronica per t+.

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This paper was partly written during the author's stay at the Department of Physics of the University of Illinois (Urbana, Illinois) with a NATO fellowship.     

The initial-value problem for elastodynamics of incompressible bodies

We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials. Our method consists in:

1. Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.
2. Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:

1. Using energy estimates to bound the growth of various Sobolev norms of solutions.
2. Finding the solution as the limit of a sequence of solutions of linearized problems.

Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.     

A Riemann-Hilbert boundary value problem in neutron transport theory

Summary A Riemann-Hilbert boundary value problem, occurring in neutron transport theory, is presented and solved in this note for the case of an infinite straight line and zero index.

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Sommario Viene presentato e risolto in questa nota un problema al contorno di Riemann-Hilbert di indice zero e relativo ad un intervallo infinito. Esso ha origine da un problema di diffusione di neutroni, studiato nell'ambito della teoria del trasporto.

     

An initial-value problem for a heavy viscous liquid flowing down an inclined plane

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 90–98, September–October, 1991.     

On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off

We consider a Boltzmann gas which fills all of space and is under the influence of a field of conservative external force whose potential is bounded from below.Assuming the intermolecular force has a cut-off, we prove existence and uniqueness for the general solution of the nonlinear Maxwell-Boltzmann equation at least in a finite interval of time. The solution can be constructed by the method of successive approximations in corresponding complete spaces. Definitions of these spaces are connected with an exponential function of the total energy of the molecule.Some indications of future generalizations and investigations are given.     

Solution of a nonlinear Boltzmann equation for neutron transport in L1 space

The aim of this paper is to give a constructive method for the solution of the Boltzmann equation for neutron transport in a bounded space domain subject to typical boundary and initial conditions. Sufficient conditions are given to insure the existence of a unique solution. The method entails the use of a semiinner product in a Banach space together with successive approximations, and leads to a recursion formula for the calculation of approximate solutions and error estimates. The linearized Boltzmann equation for neutron transport is included as a special case.     

Conditions for the unique solvability of the initial-value problem for linear second-order differential equations with argument deviations

We establish new conditions under which the initial-value problem for a system of linear second-order differential equations with argument deviations has a unique solution, which depends monotonically on additive perturbations of the problem. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 535–547, October–December, 2006.     

The problem of the neutron diffusion: A survey

Summary This paper expounds researches done by the author and others on the theory of the diffusion of neutrons in a moderating medium from the point of view of phenomenological theory and of transport integro-differential equations.In particular the transformational methods with a rigorous analysis for the case of slab-geometry are considered.

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Sommario In questo lavoro si espongono ricerche dell'autore e di altri concernenti il problema della diffusione dei neutroni in un mezzo moderatore, sia dal punto di vista fenomenologico, sia dal punto di vista della teoria del trasporto.Si rivolge particolare attenzione ai metodi trasformazionali e ad una analisi rigorosa nel caso della lastra infinitamente estesa.

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Here we do not consider the kinetics of the nuclear reactor or related problems.     

Functional methods in three-dimensional neutron transport theory

Summary In this work, we are concerned with the stationary neutron transport Boltzmann equation (in its integral form) in a parallelepiped. Functional methods allow us to prove that the integral transport operator, which is defined in L² space, has eigenvalues depending continuously and monotonically on geometrical and physical parameters. We show that the eigenfunctions are continuous with respect to set of the spatial variables and the optical parameters. Finally, we remark that the same results are valid if the study is carried out in the Banach space C.

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Sommario In questo lavoro consideriamo l'equazione stazionaria di Boltzmann (nella forma integrale) per neutroni monoenergetici nel caso di un sistema tridimensionale a forma di parallelepipedo. L'uso di alcuni metodi dell'analisi funzionale ci permette di provare che l'operatore integrale del trasporto, definito nello spazio L², ha autovalori che dipendono continuamente dai parametri geometrici e fisici. Si prova che le autofunzioni sono continue rispetto all'insieme delle variabili spaziali e dei parametri ottici. Infine, si osserva che gli stessi risultati sono validi se l'operatore del trasporto agisce nello spazio di Banach C.

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Work performed under contract C.N.R. (Gruppo Nazionale per la Fisica Matematica).     

Parallel solution of a coupled flow and transport model for shallow water

We present an integrated approach for the concurrent solution of a 3D hydrodynamical model coupled with a 3D transport model. Since both models are quite similar in nature, the same numerical method has been employed. This leads to a code that is more efficient than when two existing codes would have been combined. Discretization of the spatial differential operators, and the boundary conditions, results in a stiff initial value problem. To cope with the stiffness, we select an implicit time‐integration formula, viz. the second‐order, L‐stable BDF method because of its excellent stability properties. To reduce the huge amount of linear algebra involved in solving the implicit relations, an Approximate Factorization technique has been used. Essentially, this technique replaces a ‘multi‐dimensional’ system by a series of ‘one‐dimensional’ systems. Since the output of the hydrodynamical model (i.e., the flow field) serves as input for the transport model, we solve the hydrodynamical model one time step ahead in time. This allows us to solve the models in parallel, using two different groups of processors. By a little tuning of the parameters in the algorithm, a load‐balancing has been obtained that is close to optimal. As a result, both models require roughly the same amount of CPU time, so that one of them, effectively, can be considered as obtained ‘for free’. Copyright © 2002 John Wiley & Sons, Ltd.     

Initial-value problem for coupled Boussinesq equations and a hierarchy of Ostrovsky equations

We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg-de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes.