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On eigendistributions in linear transport theory

Published online by Cambridge University Press:  14 November 2011

C. G. Lekkerkerker
C. G. Lekkerkerker
Affiliation:
Mathematisch Instituut, Roetersstraat 15, Amsterdam
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Synopsis

An attempt is made to provide a sound basis for the method of singular eigenfunction expansions which has been in vogue in linear transport theory for some decades. The procedure is exemplified by a treatment of the one-dimensional neutron transport equation with a degenerate scattering function. Full-range as well as half-range results are derived. At the end of the paper the implications for a certain matrix factorization problem are given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 3-4 , 1979 , pp. 303 - 326
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Berezanskiῐ, Ju. M.. Expansions in eigenfunctions of selfadjoint operators. Transl. Math. Monogr. 17 (Providence R.I.: Amer. Math. Soc, 1968).CrossRefGoogle Scholar
2Bukman, C.. The theory of generalized eigenfunctions applied to linear transport theory with anisotropie scattering. Amsterdam Univ., Dept. of Math. Report 76–10 (1976).Google Scholar
3Case, K. M.. Elementary solutions of the transport equation and their applications. Ann. Physics 9 (1960), 1–23.CrossRefGoogle Scholar
4Fel'dman, I. A.. Operator Wiener-Hopf equations and their applications to the transport equation. Mat. Issled. 6 (1971), uyp. 3 (21) 115–132.Google Scholar
5Gohberg, I. C. and Kreῐn, M. G.. Systems of integral equations on a half-line with kernels depending on the difference of arguments. Amer. Math. Soc. Transl. 14 (1960), 217–287.Google Scholar
6Hangelbroek, R. J.. A functional analytic approach to the linear transport equation (Groningen Univ. Thesis, 1973).Google Scholar
7Hangelbroek, R. J.. Derivation of formulas relevant to neutron transport in media with anisotropie scattering Nijmegen Univ. Report (1977).CrossRefGoogle Scholar
8Hangelbroek, R. J.. On the derivation of some formulas in linear transport theory in media with anisotropie scattering. Nijmegen Univ. Report 7720.Google Scholar
9Hangelbroek, R. J. and Lekkerkerker, C. G.. Decompositions of a Hilbert space and factorization of a W-A determinant. SIAM J. Math. Anal. 8 (1977), 459–172.CrossRefGoogle Scholar
10Kato, T.. Perturbation theory for linear operators (Berlin: Springer, 1967).Google Scholar
11Larsen, E. W. and Habetler, G. J.. A functional-analytic derivation of Case's full- and half-range formulas. Comm. Pure Appi Math. 26 (1973), 525–537.CrossRefGoogle Scholar
12Larsen, E. W.. A functional-analytic approach to the steady, one-speed neutron transport equation with anisotropie scattering. Comm. Pure Appl. Math. 27 (1974), 523–545.CrossRefGoogle Scholar
13Lekkerkerker, C. G.. On generalized eigenfunctions and linear transport theory. In New developments in differential equations, ed. Eckhaus, W., 189–197 (Amsterdam: North-Holland, 1976).Google Scholar
14Lekkerkerker, C. G.. The linear transport equation. The degenerate case c = 1. I. Full-range theory; II. Half-range theory. Proc. Roy. Soc. Edinburgh Sect. A 75 (1976), 259–295.CrossRefGoogle Scholar
15Lekkerkerker, C. G.. The albedo problem, in preparation.Google Scholar
16Maslennikov, M. V.. The Milne problem with anisotropie scattering. Proc. Steklov Inst. Math. 97 (1968). (Providence: Amer. Math. Soc, 1969.)Google Scholar
17McCormick, N. J. and Kuščer, J.. Singular eigenfunction expansions in neutron transport theory. Adv. Nucl. Sci. Technol. 7 (1973), 181–282.CrossRefGoogle Scholar
18McCormick, N. J. and Kuščer, J.. Bi-orthogonality relations for solving half-space transport problems. J. Mathematical Phys. 7 (1966), 2036–2045.CrossRefGoogle Scholar
19Mika, J. R.. Neutron transport with anisotropie scattering. Nucl. Sci. Engng 11 (1961), 415–427.CrossRefGoogle Scholar
20Sebastião e Silva, J.. Sur l'intervention du calcul symbolique et des distributions dans l'étude de l'équation de Boltzmann. Atti. Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. la. 12 (1974), 314–363.Google Scholar
21Vekua, N. P.. Systems of Singular integral equations (Groningen: Noordhoflf, 1967).Google Scholar
22Williams, M. M. R.. The Wiener-Hopf technique: an alternative to the singular eigenfunction method. Adv. Nucl. Sci. Technol. 7 (1973), 283–327.CrossRefGoogle Scholar