Method Of Elementary Solutions Research Articles

Elementary solutions for a model Boltzmann equation in one dimension and the connection to grossly determined solutions

The Fourier-transformed version of the time dependent slip-flow model Boltzmann equation associated with the linearized BGK model is solved in order to determine the solution’s asymptotics. The ultimate goal of this paper is to demonstrate that there exists a robust set of solutions to this model Boltzmann equation that possess a special property that was conjectured by Truesdell and Muncaster: that solutions decay to a subclass of the solution set uniquely determined by the initial mass density of the gas called the grossly determined solutions. First we determine the spectrum and eigendistributions of the associated homogeneous equation. Then, using Case’s method of elementary solutions, we find analytic time-dependent solutions to the model Boltzmann equation for initial data with a specialized compact support condition under the Fourier transform. In doing so, we show that the spectrum separates the solutions into two distinct parts: one that behaves as a set of transient solutions and the other limiting to a stable subclass of solutions. Thus, we demonstrate that for gas flows with this specialized initial density condition, in time all gas flows for the one dimensional model Boltzmann equation act as grossly determined solutions.

Extension of the HN method for solving radiation transfer problems in plane geometry for general anisotropic scattering

Abstract The HN method is used to solve radiation transfer problems for plane parallel media for anisotropic scattering. This approach is based on the use of the angular distributions in terms of singular eigenfunctions of the method of elementary solutions. Two analytical expressions are derived for the coefficients of the power series expansions in exit distributions.

The effects of different expansions of the exit distribution on the extrapolation length for linearly anisotropic scattering

Abstract HN and singular eigenfunction methods are used to determine the neutron distribution everywhere in a source-free half space with zero incident flux for a linearly anisotropic scattering kernel. The singular eigenfunction expansion of the method of elementary solutions is used. The orthogonality relations of the discrete and continuous eigenfunctions for linearly anisotropic scattering provides the determination of the expansion coefficients. Different expansions of the exit distribution are used: the expansion in powers of μ, the expansion in terms of Legendre polynomials and the expansion in powers of 1/(1 + μ). The results are compared to each other. In the second part of our work, the transport equation and the infinite medium Green function are used. The numerical results of the extrapolation length obtained for the different expansions is discussed.

The solution of the third form transport equation using singular eigenfunctions: the slab and the sphere criticality problems

A singular eigenfunction approach is used to solve the classical slab and the sphere criticality problems. This approach is based on the use of the infinite medium Green's function which is obtained by the method of elementary solutions in the C N equations, that is in the third form of the transport equation. It is shown that our numerical results, even in the approximations of the lowest order are very accurate.

The singular eigenfunction method: the Milne problem for isotropic and extremely anisotropic scattering

In the C N method of solving the third form of the transport equation, the medium as a result of Placzek lemma is extended to infinity. Infinite medium Green function which is obtained by the Fourier transform technique is used and the method is applied to one velocity problems in plane and cylindrical geometries. As the result of physical applications of the C N method in different geometries, it is seen that the only difficulty lies in writting the expression of the Green function in a form easy to handle. In the new method of solving the third form of the transport equation (that we have generated recently), three methods, namely, C N, F N and the method of elementary solutions are considered, compared and the Green function in terms of the singular eigenfunctions is used. This method yields simple analytical expressions that can be solved numerically more efficiently than the C N method because the expression of the Green function is in the form easy to handle. Here this new method is applied to calculate the extrapolation length for the Milne problem which is a classical problem in astrophysics concerned with the diffusion of radiation through a stellar atmosphere for both isotropic and anisotropic scatterings. It is shown that the numerical results which are tabulated for selected cases are accurate even in the approximations of the lowest order and are in good agreement with the numerical results obtained by the other methods.

Particular solutions for the radiative transfer equation

Particular solutions for both the (formally exact) method of elementary solutions and the P N method are derived for the case of monochromatic radiative transfer in a homogeneous plane-parallel medium which contains a source that varies with position and direction.

The one-speed albedo and the slab albedo problems for the FBIS kernel

The one-speed albedo and the slab albedo problems are studied for the forward–backward–isotropic scattering (FBIS) kernel. The method of elementary solutions is used and the expansion coefficients are obtained by solving the resulting Fredholm integral equations. The zeroth-order results for the FBIS kernel are reduced to BIS, FIS models, and also to the well-known IS model. In the limit as the slab thickness becomes infinite, the special cases for the albedo problem for the same models are also obtained.

The criticality problems with the FN method for the FBIS model

In one-speed, time-independent, neutron transport theory, the F N method is used for the FBIS (forward-backward-isotropic scattering) model to reinvestigate the behaviour of the critical size in plane and spherical geometries. For the FIS (forward-isotropic scattering) model the numerical results are compared with previously obtained variational results and it is shown that they are in agreement. For the BIS (backward-isotropic scattering) model exact results are obtained and compared with the first-order approximate results obtained using the method of elementary solutions.

The critical slab with the backward-forward-isotropic scattering model

The variation of the critical slab thickness from backward to forward scattering is studied for specified isotropic scattering. Numerical results are obtained from the zeroth and first-order approximate analytical expressions, derived using the method of elementary solutions. It is shown that the critical slab thickness increases monotonically with increasing forward anisotropy.

Some analytical results for thermal transpiration and the mechanocaloric effect in a cylindrical tube

Analytical expressions for the phenomenological coefficients for thermal transpiration and the mechanocaloric effect in a cylindrical tube, relating to flow of a rarefied gas, are obtained by considering the BGK model. Neumann iteration and variational techniques are used in the near free molecular regime and the method of elementary solutions is used in the near continuum regime. The previously reported results of Ferziger and Loyalka on Poiseuille and thermal creep flows are extended and augmented.

Particle Transport Theory in a Finite Sphere Containing a Spherical-Shell Source

The method of elementary solutions is used to compute the radiation field due to a spherical-shell source located in a finite sphere. In addition, exact analysis and the FN method are used to establish an approximate solution that can be evaluated with modest computational effort to yield results accurate to five significant figures.

A Multiregion Calculation in the Theory of Neutron Diffusion

The method of elementary solutions and the F /SUB N/ method are used to compute and report especially accurate numerical results relevant to a previously established four-region test problem in the theory of neutron diffusion.

Some exact numerical results for the BGK model: Couette, Poiseuille and thermal creep flow between parallel plates

Numerically “exact” results for the Couette flow, Poiseuille flow and thermal creep flow between parallel plates are given for the BGK model. Equations obtained earlier by Cercignani and Williams, who used the method of elementary solutions, are solved by convergent Liouville-Series and results are compared with works of other authors.

Three Basic Neutron-Transport Problems in Spherical Geometry

The method of elementary solutions and the FN method are used to solve three basic problems concerning neutron diffusion in spheres.

On Several Good Methods for Computing the Distortion Factor Relevant to a Foil Placed in an Exponentially Varying Flux

The method of elementary solutions, the CN method, and the integral transform method are used to compute the flux-distortion factor for a foil placed in an exponentially varying flux.

Twenty years of transport theory

This review of transport theory covers the history, the Case method of elementary solutions, the contour integration approach, and the present situation. 43 references. (JFP)

On an exceptional case concerning plasma oscillations

The method of elementary solutions is used to analyze the situation in plasma oscillations when there exists a real discrete eigenvalue.

Flux-Depression Factors for Scattering and Absorbing Media

The method of elementary solutions, along with Chandrasekhar's invariance principles, is used to solve a two-region problem relevant to flux-depression calculations. An improved P-L-type approximation is also discussed.

The effect of anisotropic scattering on the critical slab problem in neutron transport theory using a synthetic kernel

The effect of widely differing scattering laws on the critical thickness of a multiplying slab in one-speed neutron transport theory is studied using a scattering kernel which consists of a linear combination of backward, forward and isotropic scattering. An extensive numerical survey is carried out for the critical thickness in order to provide benchmark results for future studies. 'Exact' solutions are obtained by two methods: the Wiener-Hopf technique and the method of elementary solutions. Although both methods lead to the same results, it is shown that the method of elementary solutions is superior, in terms of operational simplicity, for this type of problem.

Elementary solutions of coupled model equations in the kinetic theory of gases

The method of elementary solutions is employed to solve two coupled integrodifferential equations sufficient for determining temperature-density effects in a linearized BGK model in the kinetic theory of gases. Full-range completeness and orthogonality theorems are proved for the developed normal modes and the infinite-medium Green's function is constructed as an illustration of the full-range formalism. The appropriate homogeneous matrix Riemann problem is discussed, and half-range completeness and orthogonality theorems are proved for a certain subset of the normal modes. The required existence and uniqueness theorems relevant to the H matrix, basic to the half-range analysis, are proved, and an accurate and efficient computational method is discussed. The half-space temperature-slip problem is solved analytically. and a highly accurate value of the temperature-slip coefficient is reported.