2 edition of Mathematical solutions of the one-dimensional neutron transport equation found in the catalog.
by Naval Postgraduate School
Written in English
Superposition of solutions When the diffusion equation is linear, sums of solutions are also solutions. Here is an example that uses superposition of error-function solutions: Two step functions, properly positioned, can be summed to give a solution for finite layer placed between two semi-infinite bodies. L3 11/2/06 8. One-Dimensional Neutron Transport Suspended Cable Chapter The Wronskian and Linear Independence Converting Systems of Ordinary Differential Equations Solutions of Ordinary Differential Equation Systems Matrix Mathematics particularly in the medical fields. In using this book, students may review and study the illustrated problems at 5/5(1).
We prove a regularity result for a Fredholm integral equation with weakly singular kernel, arising in connection with the neutron transport equation in an infinite cylindrical domain. The theorem states that the solution has almost two derivatives in L . The work is devoted to direct and inverse problems of the transport equation in the context of a nuclear geophysical technology based on pulsed neutron-gamma logging of inelastic scattering (PNGL-IS). In the first part of the paper we analyze the distribution of fast neutrons from a pulsed source of MeV and study distributions of gamma-quanta of .
forms of the neutron transport equation are reviewed. The solution methods are shown to evolve from only a few basic numerical approximations, such as expansion techniques or the use of quadrature formulas. The emphasis is on the derivation of the approximate equations from the transport equation, and not on the solution of the resulting system. To solve one-dimensional cases, analytic methods have been developed based on expanding the solution in terms of generalized eigenfunctions. The Monte-Carlo method is used to find functionals in the solutions to complex multi-dimensional problems. Finite-difference approximation methods are widely used for transport equations.
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Item Preview remove-circle Mathematical solutions of the one-dimensional neutron transport equation. by. 7. Conclusions. In this work we have proposed a fully meshless method for the numerical solution of the neutron transport equation.
The multiquadric basis function is used as the radial basis function for the spatial approximation and the angular variable is treated by the P N method. The use of Legendre polynomials instead of discrete ordinates has provided a Author: T.
Tanbay, B. Ozgener. tingtheneutron transportequation to an integralequation and then to a singularintegralequation, a solution is found in a method. The subject of this work is computational modeling of neutron trans-port relevant to economical and safe operation of nuclear facilities.
The general mathematical model of neutron transport is provided by the linear Boltzmann’s transport equation and the thesis begins with its precise mathematical formulation and presentation of known con. exact solutions to model problems of elliptic, hyperbolic, and parabolic type.
Next, we review the basic steps involved in the design of numerical approximations and the main criteria that a reliable algorithm should satisfy. The chapter concludes with an outline of the rationale behind the scope and structure of the present book. We consider the combined Walsh function for the three-dimensional case.
A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions.
Miller, W. Jr., E. Lewis, and E. Ris sow, The application of phase-space finite elements to the two-dimensional transport equation in x-y geometry, to appear in Nucl.
Sci. Eng. Ohnishi, T., Application of finite element solution technique to neutron diffusion and transport equations, Proc. Conf. on New Developments in Reactor. By treating one of the space dimensions exactly and approximating the other two by the exp (−iBr) assumption, which is suggested by asymptotic transport theory, it is possible to reduce the three‐dimensional transport equation to an equation that is of one‐dimensional form and that still contains details of the complete three‐dimensional angular distribution.
The spectral analysis of a dissipative linear transport operator with a polynomial collision integral by the Szőkefalvi-Nagy - Foiaş functional model is given. An exact estimate for the remainder in the asymptotic of the corresponding evolution semigroup is proved in the isotropic case.
In the general case, it is shown that the operator has at most finitely many eigenvalues and spectral. The mathematical theory of diffusion is founded on that of heat conduction and correspondingly the early part of this book has developed from 'Con-duction of heat in solids' by Carslaw and Jaeger.
These authors present many solutions of the equation of heat conduction and some of them can be applied. Abstract. This book presents a numerical analysis of neutron transport theory.
Topics considered include the kinetic reactor equation, adjoint equations, the multigroup kinetic reactor equations, the one-group kinetic equation, solution of one-group problems in the transport theory, the method of spherical harmonics, Galerkin's method, the finite-difference equations.
This book presents some recent mathematical developments about neutron transport equations. Several different topics are dealt with including regularity of velocity averages, spectral analysis of transport operators, inverse problems, nonlinear problems arising in the stochastic theory of neutron chain fissions, compactness properties of perturbed of c0.
For the case of one-dimensional, mono-energetic neutron transport, a fractional-order telegraph equation is developed using the continuous-time random walk technique. Request PDF | Exact solution of the neutron transport equation in spherical geometry | Solution of the neutron transport equation in one dimensional slab geometry construct a.
With all the term expresses above, the partial neutron flux J± given by the system equations (35a) and (35b) at the boundaries of the medium are computed in the presence of four term binomial scattering law with internal numerical results are represented graphically in figures 1, 2, 3 and figure 1, evolution of the fluxes J+ and J−versus the.
The code APOLLO, written in Saclay at the Service de Physique Mathematique, makes it possible to calculate the space and energy dependent direct or adjoint flux for a one dimensional medium, by the solution of the integral form of the transport equation, in the. Description; Chapters; Supplementary; This book presents some recent mathematical developments about neutron transport equations.
Several different topics are dealt with including regularity of velocity averages, spectral analysis of transport operators, inverse problems, nonlinear problems arising in the stochastic theory of neutron chain fissions.
The neutron diffusion equation is often used to perform core-level neutronic calculations. It consists of a set of second-order partial differential equations over the spatial coordinates that are, both in the academia and in the industry, usually solved by discretizing the neutron leakage term using a structured grid.
This work introduces the alternatives that unstructured grids can. The density is the solution of an integral-differential equation named the neutron transport equation. Many authors paid attention to this problem and its applications [, ].
In this paper we provide a periodical analytical solution for the one-dimensional stationary problem (1), where u(x,y) is the neutron density and g(x,y) is the source. A solution of the neutron transport equation is obtained by expanding the flux Phi (r Omega) at position r in direction Omega as a series of the form: Phi (r, Omega)= Sigma l=0 N (2l+1) Sigma m=0 l P l m (cos theta)(psi lm (r)cos(m phi)+ gamma lm (r)sin(m phi)) where P l m (cos theta) is the associated Legendre polynomial of order l, m with theta and phi the .Summary.
The linear integral transport operator for slab geometry is formulated and studied as a mapping on the set of measures on the phase space of the underlying system, with the expected number of neutrons emergent from a collision represented by .proximate solutions to the non-scattering one-dimensional neutron transport equation in spherically symmetric geometry.
It is shown that the resulting numerical approximations avoid ux dip and oscillations. The least-squares discretization yields a .