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 Free Access

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] S. Albertoni and B. Montagnini, On the spectrum of neutron transport equation in finite bodies, J. Math. Anal. Appl. 13 (1966), 19–48. MR 0189741
  • [2] Herbert Amann, Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Springer, Berlin, 1976, pp. 1–55. Lecture Notes in Math., Vol. 543. MR 0513051
  • [3] Claude Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d’approximation; application à l’équation de transport, Ann. Sci. École Norm. Sup. (4) 3 (1970), 185–233 (French). MR 0274925
  • [4] Alain Bensoussan, Jacques-L. Lions, and George C. Papanicolaou, Boundary layers and homogenization of transport processes, Publ. Res. Inst. Math. Sci. 15 (1979), no. 1, 53–157. MR 533346, 10.2977/prims/1195188427
  • [5] G. Blankenship and G. C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbances. I, SIAM J. Appl. Math. 34 (1978), no. 3, 437–476. MR 0476129
  • [6] Kenneth M. Case and Paul F. Zweifel, Linear transport theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 0225547
  • [7] S. Chandrasekhar, Radiative Transfer, Oxford University Press, 1950. MR 0042603
  • [8] P. G. Ciarlet and S. Kesavan, Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory, Comput. Methods Appl. Mech. Engrg. 26 (1981), no. 2, 145–172. MR 626720, 10.1016/0045-7825(81)90091-8
  • [9] Theodore E. Harris, The theory of branching processes, Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin; Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0163361
  • [10] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
  • [11] Srinivasan Kesavan, Homogenization of elliptic eigenvalue problems. I, Appl. Math. Optim. 5 (1979), no. 2, 153–167 (English, with French summary). MR 533617, 10.1007/BF01442551
  • [12] M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation 1950 (1950), no. 26, 128. MR 0038008
  • [13] Edward W. Larsen and Joseph B. Keller, Asymptotic solution of neutron transport problems for small mean free paths, J. Mathematical Phys. 15 (1974), 75–81. MR 0339741
  • [14] Joseph Lehner and G. Milton Wing, On the spectrum of an unsymmetric operator arising in the transport theory of neutrons, Comm. Pure Appl. Math. 8 (1955), 217–234. MR 0070038
  • [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] Ivan Vidav, Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator, J. Math. Anal. Appl. 22 (1968), 144–155. MR 0230531
  • [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