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Meccanica

, Volume 5, Issue 4, pp 253–261 | Cite as

The initial-value problem for neutron transport in a finite body with generalized boundary conditions

  • Aldo Belleni Morante
Article

Summary

We prove the existence and the uniqueness of the solution of the initial-value problem for neutron transport in a finite convex body with generalized boundary conditions which include both the perfect reflection and the vacuum boundary condition as particular cases.

Moreover, we show that the transport operator has at least one real eigenvalue provided a perfect reflection boundary condition is valid over a finite portion of the boundary surface.

Finally, we indicate the asymptotic behavior of the neutron density as t→ + ∞.

Keywords

Boundary Condition Civil Engineer Asymptotic Behavior Convex Body Boundary Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Sommario

Si prova l'esistenza e l'unicità della soluzione di un problema di trasporto di neutroni in un mezzo finito con condizioni al contorno generalizzate.

Si mostra che l'operatore del trasporto ammette almeno un autovalore reale purché la condizione di perfetta riflessione sia soddisfatta su almeno una porzione finita della superficie.

Infine si studia il comportamento asintotico della densità neutronica per t→+∞.

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References

  1. [1]
    J. Lehner andG. M. Wing,On the spectrum of an unsymmetric operator arising in the theory of neutrons, Comm. Pure Appl. Math., 8, pp. 217–234, 1955.Google Scholar
  2. [2]
    J. Lehner andG. M. Wing,Solution of the linearized Boltzmann equation for slab geometry, Duke Mat. J., 23, pp. 125–142, 1955.Google Scholar
  3. [3]
    K. Jörgens,An asymptotic expansion in the theory of neutron transport, Comm. Pure Appl. Math., 11, pp. 219–242, 1958.Google Scholar
  4. [4]
    G. H. Pimbley Jr.,Solution of an initial-value problem for multivelocity neutron transport equation with slab geometry, J. Math. Mech., 8, pp. 837–866, 1959.Google Scholar
  5. [5]
    R. Van Norton,On the real spectrum of a monoenergetic neutron transport operator, Comm. Pure Appl. Math., 15, pp. 149–157, 1962.Google Scholar
  6. [6]
    N. Corngold,Some transient phenomena in thermalization, Nucl. Sci. Eng. 19, pp. 80–94, 1964.Google Scholar
  7. [7]
    J. T. Marti,Existence d'une solution unique de l'équation linéarizée de Boltzmann et développement de cette solution suivent des fonctions propres, C. R. Acad. Sc. Paris, 260, pp. 2692–2694, 1965.Google Scholar
  8. [8]
    B. Montagnini andM. L. Demuru,Complete continuity of the free gas scattering operator in neutron thermalization theory, J. Math. Anal. Appl., 12, pp. 49–57, 1965.Google Scholar
  9. [9]
    S. Albertoni andB. Montagnini,Some spectral properties of the transport equation and their relevance to the theory of pulsed neutron experiments, Pulsed Neut. Res., 1, IAEA, pp. 239–258, Vienna, 1965.Google Scholar
  10. [10]
    R. Bednarz,Spectrum of the Boltzmann operator with an isotropic thermalization kernel, Pulsed Neut. Res., 1, IAEA, pp. 259–270, Vienna, 1965.Google Scholar
  11. [11]
    S. Albertoni andB. Montagnini,On the spectrum of neutron transport operator in finite bodies, J. Math. Anal. Appl., 13, pp. 19–48, 1966.Google Scholar
  12. [12]
    J. T. Marti,Mathematical foundations of kinetics in neutron transport theory, Nukleonik, 8, pp. 159–163, 1966.Google Scholar
  13. [13]
    S. Ukai,Real eigenvalues of the monoenergetic transport operator for a homogeneous medium, J. Nucl. Sci. Technol., 3, pp. 263–266, 1966.Google Scholar
  14. [14]
    J. Mika,The initial-value problem for neutron thermalization, Nukleonik, 9, pp. 303–304, 1967.Google Scholar
  15. [15]
    S. Ukai,Eigenvalues of the neutron transport operator for a homogeneous finite moderator, J. Math. Anal. Appl., 18, pp. 297–314, 1967.Google Scholar
  16. [16]
    J. Mika andM. Borysiewicz,Spectrum of the neutron transport operator in non-uniform media, Nukleonik, 10, pp. 203–206, 1967.Google Scholar
  17. [17]
    J. Mika andM. Borysievicz,Time-dependent neutron transport with elastic scattering, Neutron Thermalization and Reactor Spectra, 1, IAEA, pp. 45–52, Vienna, 1968.Google Scholar
  18. [18]
    J. Mika andM. Borysievicz,Time behaviour of thermal neutrons in moderating media, CNEN Tech. Report Doc. CEC (68) 1.Google Scholar
  19. [19]
    J. Dorning,Thermal. neutron decay in small systems, Nucl. Sci. Eng., 33, pp. 65–80, 1968.Google Scholar
  20. [20]
    E. E. Lewis andF. T. Adler,A Boltzmann integral treatment of neutron resonance absorption in reactor lattices, Nucl. Sci. Eng., 31, pp. 117–126, 1968.Google Scholar
  21. [21]
    A. Belleni-Morante,The initial-value problem for neutron transport in a slab with perfect reflection boundary conditions, In course of publication in the J. Math. Anal. Appl.Google Scholar
  22. [22]
    B. Davison,Neutron Transport Theory, Oxford University Press, London, 1957.Google Scholar
  23. [23]
    E. Hille andR. S. Phillips,Functional Analysis and Semigroups, Am. Math. Soc. Coll. Publ., Vol. 31, Providence, 1957.Google Scholar
  24. [24]
    T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag New York Inc., New York, 1966.Google Scholar
  25. [25]
    F. Riesz andB. SZ.-Nagy,Leçons d'analyse fonctionelle, Gauthier-Villar, Paris, 1955.Google Scholar

Copyright information

© Tamburini Editore s.p.a. Milano 1966

Authors and Affiliations

  • Aldo Belleni Morante
    • 1
  1. 1.Istituto Matematico “U. Dini”Università di FirenzeItaly

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