Product integration for the linear transport equation in slab geometry

Summary In this paper we present a product quadrature rule for the discretization of the well-known linear transport equation in slab geometry. In particular we give an algorithm for constructing the weights of the rule and prove that the order of convergence isO(n ^–3+ ), >0 small as we like. Numerical examples are given, and our formula is also compared with product Simpson rules. Finally, we examine a Nyström method based on our quadrature.Work sponsored by the Ministero della Pubblica Instruzione of Italy     

Numerical methods for the linear Boltzmann transport equation in slab geometry

An iterative method for computing numerical solutions of a finite-difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite-difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well-posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time-increment the discrete system is numerically stable, independent of the mesh-size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time-dependent and the steady-state solutions.     

A numerically efficient variational algorithm to solve a fractional nonlinear elastic string equation

Numerical Algorithms - In this work, we propose a fractional extension of the one-dimensional nonlinear vibration problem on an elastic string. The fractional problem is governed by a hyperbolic...     

A particle method to solve the Navier-Stokes system

Summary We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.     

Multilevel methods to solve the neutron diffusion equation

In this paper, we develop multilevel strategies to solve the time dependent neutron diffusion equation. These methodologies are based on the discretization of the spatial part of the neutron diffusion equation using a nodal collocation method which makes use of an expansion of the neutron flux in terms of Legendre polynomials. The different levels or grids of the multilevel strategies have been defined in terms of the different number of polynomials used for the nodal collocation method expansions instead of using different nodalization sizes. Results for two transients associated with the bidimensional TWIGL reactor are reported.     

On the analytic solutions of the monoenergetic neutron transport equation for slab geometry

The classical initial value problem for the monoenergetic neutron transport equation in slab geometry is solved, using the power series method. A general formula for analytic solutions of this equation is presented. It is shown that a polynomial solution exists only forc=1 and is linear inx and . Other analytic solutions are given in a closed form. The Taylor series expansion method is compared with the spherical harmonic approach.

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Zusammenfassung Das klassische Eigenwertproblem für die monoenergetische Neutronentransportgleichung in ebener Geometrie wird mit Hilfe der Taylorreihe gelöst und eine allgemeine Formel für analytische Lösungen dieser Gleichung angegeben. Es wird gezeigt, dass die Polynomiallösung nur fürc=1 exisitiert und linear ist in undx. Obwohl die analytischen Lösungen nur beschränkt praktische Anwendung haben zeigt sich, dass die abgebrochene Taylorreihe für praktische Probleme verwendbar ist. Rechnungen mit der abgebrochenen Taylorreihe wirden mitP _N -Rechnungen verglichen.

     

Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry

In this paper, we describe a new scattering kernel and general theoretical scheme for the evolution of the discrete and continuum eigenvalue spectrum in one-dimensional slab geometry neutron transport equation. Firstly, some useful properties of the Legendre polynomials which revealed during the definition of the new scattering kernel are discussed. By using the scattering kernel in one-dimensional neutron transport equation we obtained an integral equation for angular part of the angular flux. For the solution of this integral equation and eigenvalue equations, some comments are given.     

Modified splitting method for solving the nonstationary kinetic particle transport equation

A modified splitting method for solving the nonstationary kinetic equation of particle (neutron) transport without iteration with respect to the collision integral is proposed. According to the modification, the solutions of the first-stage integrodifferential equations and the collision integrals are found using analytical rather than finite-difference methods. The solution method is naturally extended to multidimensional problems and is well suited for massive parallelism.     

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

Noncommutative differential geometry related to the Young-Baxter equation

An analogue of the differential calculus associated with a unitary solution of the quantum Young-Baxter equation is constructed. An example of a ring sheaf is considered in which local solutions of the Young-Baxter quantum equation are defined but there is no global section. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 51–70. Translated by I. A. Izergina.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

解一般TSP问题的并行算法

任何连通的带权图均有ＴＳＰ的解．本文用图的最短路径矩阵代替加权邻接矩阵,使所有带权图,ＨＰ模型都能够接受．由于采用并行算法,可以较快获得问题的最优解．     

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.     

How to use RSVD to solve the matrix equation A=BXC

Through the restricted singular value decomposition (RSVD) of the matrix triplet (C, A, B), we show in this note how to choose a variable matrix X such that the matrix pencil A ? BXC attains its maximal and minimal ranks. As applications, we show how to use the RSVD to solve the matrix equation A = BXC.     

Application of the generalized finite difference method to solve the advection-diffusion equation

The study of the advection-diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering.This paper shows the application of the generalized finite difference method to solve the advection-diffusion equation by the explicit method. The convergence of the method has been studied and the truncation error over irregular grids is given. An example has been solved using the explicit finite difference formulae and the criterion of stability.     

A parallel algorithm to solve the stable marriage problem

In this paper a parallel algorithm to solve the stable marriage problem is given. The worst case performance of this algorithm is stated. A theoretical analysis shows that the probability of the occurrence of this worst case is extremely small. For instance, if there are sixteen men and sixteen women involved, then the probability that the worst case occurs is only 10^–45. Possible future research is also discussed in this paper.     

Using group theoretic method to solve multi-dimensional diffusion equation

The nonlinear diffusion equation arises in many important areas of science and technology such as modeling of dopant diffusion in semiconductors. We give analytical solution to N-dimensional radially symmetric nonlinear diffusion equation of the form$\text{∂}\text{∂r}\text{D(C)}\text{∂C}\text{∂r}\text{+}\text{N−1}\text{r}\text{D(C)}\text{∂C}\text{∂r}\text{=}\text{∂C}\text{∂t}\text{,}$where C(r,t) is the concentration and D(C) is diffusion coefficient.The transformation group theoretic approach is applied to present an analysis of the nonlinear diffusion equation. The one-parameter group transformation reduces the number of independent variables by one and the governing partial differential equation with the boundary conditions reduce to an ordinary differential equation with the appropriate boundary conditions. Effect of the time “t” and the number of dimension “N” on the concentration diffusion function C(r,t) has been studied and the results are plotted.