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  • 1
    Electronic Resource
    Electronic Resource
    A higher order method for determining nonisolated solutions of a system of nonlinear equations (1984)
    Springer
    Computing 32 (1984), S. 163-166 
    Details
    ISSN: 1436-5057
    Keywords: 65H10 ; 5.1 ; Nonlinear equations ; systems ; order of convergence ; Newton's method ; iteration ; nonisolated solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung In dieser kurzen Mitteilung gewinnen wir eine Methode mindestens vierter Ordnung zur Lösung eines singulären Systems nichtlinearer algebraischer Gleichungen. Dies wird durch eine Vergrößerung des Gleichungssystems zu einem höherdimensionalen Gleichungssystem, dessen entsprechende Lösung isoliert ist, erreicht. Dessen Lösung kann durch eine von B. Neta entwickelte Methode bestimmt werden.
    Notes: Abstract In this note, we obtain a method of order at least four to solve a singular system of nonlinear algebraic equations. This is achieved by enlarging the system to a higher dimensional one whose solution is isolated. For the larger system we use a method developed by B. Neta.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02253690
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  • 2
    Electronic Resource
    Electronic Resource
    On Vlasov–Manev Equations, II: Local Existence and Uniqueness (1998)
    Springer
    Journal of statistical physics 91 (1998), S. 625-654 
    Details
    ISSN: 1572-9613
    Keywords: Vlasov equation with Manev correction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We prove that the initial value problem associated with the Vlasov–Manev system (a Vlasov equation in which a correction of type ε/r 2 is added to the Newtonian or Coulomb potential) has a local in time classical and unique solution for sufficiently regular initial data.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1023029711405
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  • 3
    Electronic Resource
    Electronic Resource
    Convergence of two-dimensional Nyström discrete-ordinates in solving the linear transport equation (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 353-370 
    Details
    ISSN: 0945-3245
    Keywords: 65R05 (Primary) ; 82A75
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the discrete-ordinates approximation to the linear transport equation, the integration over the directional variable is replaced by a numerical quadrature rule involving a weighted sum over functional values at selected directions. The purpose of this paper is to show that the Nyström technique of defining the angular flux in directions other than the quadrature points, as outlined by P.M. Anselone and A. Gibbs and utilized by P. Nelson for anisotropically scattering slabs, produces an approximation scheme which is stable, consistent with, and convergent to the transport equation in two-dimensional geometry.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403674
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  • 4
    Electronic Resource
    Electronic Resource
    On the convergence of the multigroup, discrete-ordinates solutions for subcritical transport media (1988)
    Springer
    Annali di matematica pura ed applicata 153 (1988), S. 229-273 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In the multigroup, discrete-ordinates approximations to the linear transport equation, the integration over the directional variable is replaced by a numerical quadrature rule, involving a weighted sum over functional values at selected directions, with the energy dependence discretized by replacing the cross section data by weighted averages over each energy interval. The stability, consistency, and convergence rely fundamentally on the conditions that the maximum fluctuations in the total cross section — and in the expected number of secondary particles arising from each energy level — tend to zero as the energy mesh becomes finer, and as the number of angular nodes becomes infinite. Our analysis is based on using a natural Nyström method of extending the discrete-ordinates, multigroup approximates to all values of the angular and energy variables. Such an extension enables us to employ generalizations of the collectively compact operator approximation theory of P. M. Anselone to deduce stability and convergence of the approximates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01762394
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  • 5
    Electronic Resource
    Electronic Resource
    On the convergence of the multigroup approximations for multidimensional media (1985)
    Springer
    Annali di matematica pura ed applicata 140 (1985), S. 179-207 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper, we investigate the stability, convergence, and consistency properties of steady-state multigroup models for submultiplying media with spatial dimensions greater than one. We define these concepts in a Banach space whose norm measures the collision density integrated over phase space. Stability and consistency occur under the conditions that the maximum fluctuations in the total cross section, and in the expected number of secondary particles arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding example and discussion deal with pathologies of the multigroup model in situations where these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of both Belleni-Morante and Busoni for isotropic slabs and of Yang Mingzhu and Zhu Guangtian for bounded, three-dimensional media. This work directly extends the steady-state results of Paul Nelson, Jr. and H. D. Victory, Jr. for slab media to steady-state transport in multidimensional media.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01776849
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  • 6
    Electronic Resource
    Electronic Resource
    On Vlasov–Manev Equations. I: Foundations, Properties, and Nonglobal Existence (1997)
    Springer
    Journal of statistical physics 88 (1997), S. 885-911 
    Details
    ISSN: 1572-9613
    Keywords: Vlasov–Poisson equations ; Manev correction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider the classical stellar dynamic (Vlasov) equation with a so-called Manev correction (based on a pair potential γ/r + ε/r 2). For the pure Manev potential γ = 0 we discuss both the continuous case and the N-body problem and show that global solutions will not exist if the initial energy is negative. Certain global solutions can be constructed from local ones by a transformation which is peculiar for the ε/r 2 law. Moreover, scaling arguments are used to show that Boltzmann collision terms are meaningful in conjunction with Manev force terms. In an appendix, a formal justification of the Manev correction based on the quasirelativistic Lagrangian formalism for the motion of a particle in a central force field is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/B:JOSS.0000015177.60491.3c
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  • 7
    Electronic Resource
    Electronic Resource
    Additive Invariant Functionals for Dynamical Systems (1998)
    Springer
    Journal of statistical physics 92 (1998), S. 269-299 
    Details
    ISSN: 1572-9613
    Keywords: Kinetic equations ; dynamical systems ; Liouville equation ; conservation laws ; nonlinear functionals
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider the problem of defining completely a class of additive conservation laws for the generalized Liouville equation whose characteristics are given by an arbitrary system of first-order ordinary differential equations. We first show that if the conservation law, a time-invariant functional, is additive on functions having disjoint compact support in phase space, then it is represented by an integral over phase space of a kernel which is a function of the solution to the Liouville equation. Then we use the fact that in classical mechanics phase space is usually a direct product of physical space and velocity space (Newtonian systems). We prove that for such systems the aforementioned representation of the invariant functionals will hold for conservation laws which are additive only in physical space; i.e., additivity in physical space automatically implies additivity in the whole phase space. We extend the results to include non-degenerate Hamiltonian systems, and, more generally, to include both conservative and dissipative dynamical systems. Some applications of the results are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1023003921053
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  • 8
    Electronic Resource
    Electronic Resource
    On the convergence of the multigroup approximations for submultiplying slab media (1982)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 4 (1982), S. 206-229 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The stability, convergence, and consistency properties of the steady-state multigroup model are investigated for submultiplying slab media. These concepts are defined in a Banach space setting in which the norm of the angular flux is the collision density integrated over phase space. It is shown that the multigroup approximations are stable and are both consistent with, and convergent to, the transport equation under the conditions that the maximum fluctuations in the total cross section and in the expected number of secondary particles, arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding discussion deals with pathologies of the multigroup approximation for situations in which these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of Belleni-Morante and Busoni for isotropic slabs and the results of Nelson for steady-state rod media.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670040114
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