Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Diffusion approximation and computation of the critical size

Authors: C. Bardos, R. Santos and R. Sentis
Journal: Trans. Amer. Math. Soc. 284 (1984), 617-649
MSC: Primary 45K05; Secondary 45M05, 82A70
MathSciNet review: 743736
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem.

References [Enhancements On Off] (What's this?)

  • [1] G. Albertoni and B. Montagnin, On the spectrum of neutron transport equations in finite bodies, J. Math. Anal. Appl. 13 (1966), 19-48. MR 0189741 (32:7163)
  • [2] H. Amann, Nonlinear operator in ordered Banach spaces and some applications to nonlinear boundary value problems in nonlinear operator and the calculus of variations (Proc, Bruxelles, 1975), Lecture Notes in Math., vol. 543, Springer-Verlag, Berlin, 1976, pp. 1-55. MR 0513051 (58:23793)
  • [3] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier order, Ann. Sci. École Norm. Sup. 4 (1969), 185-233. MR 0274925 (43:683)
  • [4] A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, J. Publ. RIMS Kyoto Univ. 15 (1979), 53-157. MR 533346 (80i:60122)
  • [5] G. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbance. I, SIAM J. Appl. Math. 34 (1978), 437-476. MR 0476129 (57:15707)
  • [6] K. M. Case and P. F. Zweifel, Linear transport theory, Addison-Wesley, Reading, Mass. MR 0225547 (37:1140)
  • [7] S. Chandrasekhar, Radiative transfer, Dover, New York, 1950. MR 0042603 (13:136a)
  • [8] P. G. Ciarlet and S. Kesavan, Two dimensional approximation of three dimensional eigenvalue problems in plate theory, Comput. Methods Appl. Mech. Engrg. 26 (1981), 145-172. MR 626720 (82h:73052)
  • [9] F. Harris, The theory of branching process, Springer-Verlag, Berlin, 1963. MR 0163361 (29:664)
  • [10] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin and New York, 1976. MR 0407617 (53:11389)
  • [11] S. Kesavan, Homogenisation of eigenvalue problems, Appl. Math. Optim. 5 (1979), 153-168. MR 533617 (80f:65111)
  • [12] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone of positive functions in a Banach space, Trans. Amer. Math. Soc. 26 (1950). MR 0038008 (12:341b)
  • [13] E. Larsen and J. B. Keller, Asymptotic solutions of neutron transport problems, J. Math. Phys. 15 (1974), 75-81. MR 0339741 (49:4498)
  • [14] J. Lehner and G. Wing, On the spectra of an asymmetric operator arising in the transport theory of neutrons, Comm. Pure Appl. Math. 8 (1955), 217-234. MR 0070038 (16:1120f)
  • [15] M. Mokhtar Kharroubi, Propriétés spectrales de l'opérateur de transport dans les cas anisotrope, Thèse 3ème cycle, Université de Paris VI, 1982.
  • [16] R. Sentis, Analyse asymptotique d'équation de transport, Thèse, Université de Paris-Dauphine, Décembre 1981.
  • [17] -, Study of the corrector of the eigenvalue of a transport operator, SIAM J. Math. Anal. 15 (1984).
  • [18] Y. Shmulyan, Completely continuous perturbation of operator, Dokl. Akad. Nauk SSSR 101 (1955), 33-38.
  • [19] I. Vidav, Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator, J. Math. Anal. Appl. 22 (1968), 144-155. MR 0230531 (37:6093)
  • [20] -, Spectrum of perturbed semi-groups with application to transport theory, J. Math. Anal. Appl. 30 (1970), 244-279.
  • [21] M. Williams, Mathematical methods in particle transport theory, Butterworth, London, 1971.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 45K05, 45M05, 82A70

Retrieve articles in all journals with MSC: 45K05, 45M05, 82A70

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society