Based on the idea of separation of variables, a spectral theory for the three-dimensional, stationary, isotropic transport operator in a vector space of complex-valued Borel functions results in continuous sets of regular and generalized eigenfunctions

Abu-Shumays, I. K.

Bareiss, E. H.

English

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September 1969	 Brief 69-10432 
O	
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the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Structure of the Isotropic Transport Operators

in Three Independent Space Variables 
Based on the idea of separation of variables, a 
spectral theory for the three-dimensional, station-
ary, isotropic transport operator is presented (1) in 
a vector space of complex-valued Borel functions 
resulting in continuous sets of regular and generalized 
eige n functions. 
Because of the non-self-adjoint nature of this 
operator, the results could not be anticipated intu-
itively from the known decomposition of this operator 
in the special case of plane geometry. The results 
obtained indicate a promising new approach to ana-
lytic solution of the linear transport equation in 
higher space dimensions. Examples are given for slab 
geometry with and without axial symmetry, spherical 
symmetry, and cylindrical symmetries. 
The use of elementary functions for construction of 
solutions for more than one space dimension, if only 
for special geometries, had been previously attempted. 
In this work (1) a decomposition of the transport 
operator in three space dimensions is carried out. 
This method is suggested by the principle of separa-
tion of variables, which is widely used for the decom-
position of linear partial-differential equations of 
mathematical physics. The solution space for the trans-
port equation is mathematically defined as a set of 
complex-valued Borel functions. For an attempt at 
completeness it was necessary to add to the set of 
regular elgenfunctions, and the corresponding regular 
expansion modes over a complex continuum, a set 
of generalized eigenfunctions and the corresponding 
generalized expansion modes over a complex con-
tinuum of even higher dimension. The regular eigen-
functions are characterized by a complex vector 
called a regular (or proper) eigenvector, A. These
eigenvectors, however, must satisfy the condition 
A . A = A , where A 0 assumes exactly one value 
for any given equation; therefore ± Ao can be called 
eigenvalues. For yield of the generalized eigenfunc-
tions the improper integral must be evaluated. 
A basic result of this research is the general solu-
tion to the transport equation in a sense analogous to 
that of the theory of differential equations. Special 
representations are obtained by imposition of geo-
metric restrictions on the general solution. These 
enable the numerical analyst to construct exact test 
problems for checking of the performance of general 
multidimensional transport codes, and therefore can 
be most helpful in error-analysis. The calculation of 
singular integrals is not difficult if properly executed. 
There is little doubt that the results of the work can 
be generalized to anisotropic scattering, multiregion, 
and multigroup problems. A similar theory can be con-
ceived for the discrete ordinate approximations also; 
their spectra, modes, and general solutions can then 
be compared with this continuous case (I) for a 
measure of fitness for the approximate solutions. 
Many papers on the spectral representation of the 
time-dependent equation give a general expansion in 
exponentials, although little is said regarding the 
structure of the time-independent coefficient func-
tions. This report fills this gap. 
Reference: 
I. E. H. Bareiss and I. K. Abu-Shumays, ANL-7328 
(Argonne National Laboratory, May 1967). 
Notes: 
1. Researchers in mathematical or applied physics 
may be interested.
(continued overleal) 
This document was prepared under the sponsorship of the Atomic 	 liability resulting from the use of the information contained in this 
Energy Commission and/or the National Aeronautics and Space document, or warrants that the use of any information, apparatus, 
Administration. Neither the United States Government nor any per- 	 method, or process disclosed in this document may not infringe pri- 
son acting on behalf of the United States Government assumes any vately owned rights.
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2. Inquiries concerning this innovation may be di-
rected to:
Office of Industrial Cooperation 
Argonne National Laboratory 
9700 South Cass Avenue 
Argonne, Illinois 60439 
Reference: B69-10432 
Source: E. H. Bareiss and I. K. Abu-Shumays

Applied Mathematics Division

(ARG- 10448)
Patent status: 
Inquiries concerning rights for commercial use of 
this innovation may be made to: 
Mr. George H. Lee, Chief 
Chicago Patent Group 
U.S. Atomic Energy Commission 
Chicago Operations Office 
9800 South Cass Avenue 
Argonne, Illinois 60439 
Brief 69-10432	 Category 06


Structure of the isotropic transport operators in three independent space variables

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