Asymptotic Behavior of the Solution of the Integral Transport Equation in the Vicinity of a Curved Material Interface

1979 ◽  
Vol 36 (2) ◽  
pp. 200-218 ◽  
Author(s):  
Juhani Pitkäranta
Keyword(s):  
Asymptotic Behavior ◽  
Transport Equation ◽  
Material Interface ◽  
Integral Transport

Approximate solutions to the half-space integral transport equation near a plane boundary

1980 ◽  
Vol 58 (9) ◽  
pp. 1291-1310 ◽  
Author(s):  
Michael S. Milgram
Keyword(s):  
Transport Equation ◽  
Half Space ◽  
Matrix Multiplication ◽  
Solution Space ◽  
Plane Geometry ◽  
Plane Boundary ◽  
Infinite Plane ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Integral Transport

A set of functions spanning the solution space of the integral transport equation near a boundary in semi-infinite plane geometry is obtained and used to reduce the problem to that of a system of linear algebraic equations. Expressions for the boundary angular flux are obtained by matrix multiplication, and the theory is extended to adjacent half-space problems by matching the angular flux at the boundary. Thus a unified theory is obtained for well-behaved arbitrary sources in semi-infinite plane geometry. Numerical results are given for both Milne's problem and the problem of constant production in adjacent half-spaces, and albedo problems in semi-infinite geometry. The solutions for the flux density are best near the boundary, and for the angular flux are best for angles near the plane of the boundary; it is conjectured that the theory will prove most useful when extended to arrays of finite slabs.

Analytic method for the solution of the one-group, integral transport equation for a homogeneous sphere

1978 ◽  
Vol 56 (6) ◽  
pp. 648-655
Author(s):  
Michael S. Milgram
Keyword(s):  
Transport Equation ◽  
Diffusion Theory ◽  
Analytic Method ◽  
Matrix Eigenvalue Problem ◽  
Homogeneous Sphere ◽  
Analytic Approximations ◽  
The Matrix ◽  
The One ◽  
Integral Transport ◽  
Taylor’S Series

The one-group integral transport equation for the spatial distribution of the flux density in a homogeneous critical sphere is reduced to a standard matrix eigenvalue problem, and the matrices are small. The elements of the matrix are simple functions, the eigenvalues correspond to critical values of the number of secondaries/collision, and the eigenvectors represent the coefficients of the Taylor's series of the flux expanded about the centre of the sphere. Analytic approximations and numerical results are cited and the connection with classical diffusion theory is developed.

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