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Fundamental solutions for the anisotropic neutron transport equation

Published online by Cambridge University Press:  14 November 2011

Joseph G. Conlon
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri, U.S.A.

Synopsis

We construct a fundamental solution for the n dimensional time independent anisotropic neutron transport equation. This is an operator valued distribution G(x) with a singularity at the origin. By estimating G(x) we are able to construct smooth solutions to the transport equation. We are also able to derive in a straightforward fashion results of Birkhoff and Abu-Shumays on the existence of harmonic solutions to the isotropic transport equation. When n = 1, G(x) is a function which is continuous except at x = 0. We show that the classical formula for the jump of G(x) at the origin is equivalent to the completeness of Case's full range eigenfunction expansion.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1978

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References

1Case, K.Elementary solutions of the transport equation and their applications. Ann. Physics 9 (1960), 123.CrossRefGoogle Scholar
2Sattinger, D. H.The eigenvalues of an integral equation in anisotropic neutron transport theory. J. Math, and Phys. 45 (1966), 188196.CrossRefGoogle Scholar
3Sattinger, D. H.A singular eigenfunction expansion in anisotropic transport theory. J. Math. Anal. Appl. 15 (1966), 497511.CrossRefGoogle Scholar
4Birkhoff, G. and Abu-Shumays, I.Harmonic solutions of transport equations. J. Math. Anal. Appl. 28 (1969), 211221.CrossRefGoogle Scholar
5Birkhoff, G. and Abu-Shumays, I.Exact analytic solutions of transport equations. J. Math. Anal. Appl. 32 (1970), 468481.CrossRefGoogle Scholar
6Larsen, E. and Habetler, G.A functional analytic derivation of Case's full and half range formulas. Comm. Pure Appl. Math. 26 (1973), 525537.CrossRefGoogle Scholar
7Maslennikov, M.The Milne problem with anisotropic scattering. Proc. Steklov Inst. Math. 97 (1968). (Transl. Providence, R.I: Amer. Math. Soc, 1969).Google Scholar
8Fel'dman, I. A.Operator Wiener-Hopf equations and their applications to the transport equation. Mat. Issled. 6 (1971), 3 (21), 115132.Google Scholar
9Fel'dman, I. A.The finiteness of the discrete spectrum of the characteristic equation in the theory of radiative transfer. Soviet Math. Dokl. 15 (1974), 379383.Google Scholar
10Silva, J. Sebastião eSus l'intervention du calcul symbolique et des distributions dans l'étude de l'equation de Boltzmann, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. 112 (1974), 314363.Google Scholar
11Case, K.Scattering theory, orthogonal polynomials, and the transport equation. J. Mathematical Phys. 15 (1974), 974983.CrossRefGoogle Scholar
12Norton, R. VanOn the real spectrum of a mono-energetic neutron transport operator. Comm. Pure Appl. Math. 15 (1962), 149158.CrossRefGoogle Scholar