Dominant critical eigenfunction and Neumann series convergence for inhomogeneous slabs and spheres in integral neutron transport
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Summary
In this paper many properties of the solutions of the monoenergetic neutron transport stationary integral equation are investigated for slabs and spheres in which the macroscopic cross sections depend on the position and the scattering is isotropic. Such an equation in spherical geometry is derived from the third form of the Boltzmann equation. It is proved that in the Hilbert space L2 the kernels, though asymmetric and discontinuous, are bounded and continuous in the mean and the associated integral operators are of finite double norm. This implies the boundedness and continuity of the total fluxes and the existence of a dominant positive eigenfunction, which represents the critical total flux. Finally the connection between the existence of the dominant eigenfunction and the convergence both in the mean and point-wise uniform and absolute of the Neumann series in all subcritical cases is shown.
Keywords
Hilbert Space Integral Operator Boltzmann Equation Total Flux Spherical GeometrySommario
In questo lavoro vengono esaminate molte proprietà delle soluzioni dell'equazione integrale del trasporto stazionaria per neutroni monoenergetici nel caso di lastre piane e sfere, in cui le sezioni d'urto macroscopiche dipendono dalla posizione e lo scattering è isotropo. Tale equazione in geometria sferica è ricavata dalla terza forma dell'equazione di Boltzmann. Si prova che nello spazio di Hilbert L2 i kernel, pur essendo asimmetrici e discontinui, sono limitati e continui in media e gli operatori integrali ad essi associati sono di doppia norma finita. Ciò implica la limitatezza e continuità dei flussi totali e l'esistenza di una auto funzione dominante positiva che rappresenta il flusso totale critico. Viene infine mostrata la connessione tra l'esistenza dell'auto funzione dominante e la convergenza sia in media che punto per punto uniforme ed assoluta della serie di Neumann in tutti i casi sottocritici.
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References
- [1]F. Premuda,Boundedness, continuity, positivity and dominance of the solution of the neutron Boltzmann equation, Meccanica, no. 4, Vol. VI, 1971.Google Scholar
- [2]A. Belleni-Morante,Neutron transport in a non-uniform slab with generalized boundary conditions, Jour. Math. Phys., Vol. 11, no. 5, 1553, 1970.Google Scholar
- [3]M. M. R. Williams,Mathematical methods in particle transport theory, Butterworths, London, first published 1971.Google Scholar
- [4]J. Le Caine,A table of integrals involving the functions E n (x), MT-131, NRC-1553.Google Scholar
- [5]F. Premuda,Boundedness, continuity, positivity, convergence and summability for the solution of the neutron Boltzmann equation in a multi-region problem, CNEN Report RT/FI (73) 11, Roma, 1973.Google Scholar
- [6]F. Premuda,Integral transport theory of neutron conservation and flux summability for inhomogeneous slabs and spheres, to be published.Google Scholar
- [7]R. F. Peierls,Critical conditions in neutron multiplication, Proc. Camb. Phil. Soc. Math. Phys. Sci. 35, 610, 1939.Google Scholar
- [8]S. G. Mikhlin,Linear integral equations, Hindustan Publishing Corp., India, Delhi, 1960.Google Scholar
- [9]A. C. Zaanen,Linear Analysis, North Holland publishing Co. Amsterdam P. Noordhoff N. V., Groningen, 1964.Google Scholar
- [10]F. Riesz andB. Sz. Nagy,Functional Analysis, Frederick Ungar Publishing Co. New York, 1965.Google Scholar