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Solution methods for discrete-state Markovian initial value problems

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Abstract

Solution methods, both numerical and analytical, are considered for solving the Liouville master equation associated with discrete-state Markovian initial value problems. The numerical method, basically a moment (Galerkin) method, is very general and is validated and shown to converge rapidly by comparison with an earlier reported analytical result for the ensemble-averaged transmission of photons through a purely scattering statistical rod. An application of the numerical method to a simple problem in the extended kinetic theory of gases is given. It is also shown that for a certain restricted class of problems, the master equation can be solved analytically using standard Laplace transform techniques. This solution generalizes the analytical solution for the photon transmission problem to a wider class of statistical problems.

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References

  1. D. Vanderhaegen and C. Deutsch, Linear radiation transport in randomly distributed binary mixtures: A one-dimensional and exact treatment for the scattering case,J. Stat. Phys. 54:331 (1989).

    Google Scholar 

  2. G. Milton Wing,An Introduction to Transport Theory (Wiley, New York, 1962).

    Google Scholar 

  3. R. Kubo, Stochastic Liouville equation,J. Math. Phys. 4:174 (1963).

    Google Scholar 

  4. G. C. Pomraning, Radiative transfer in random media with scattering,J. Quant. Spectrosc. Rad. Transfer 40:479 (1988).

    Google Scholar 

  5. V. C. Boffi, V. Franceschini, and G. Spiga, Dynamics of a gas mixture in an extended kinetic theory,Phys. Fluids 28:3232 (1985).

    Google Scholar 

  6. V. C. Boffi and G. Spiga, Extended kinetic theory for gas mixtures in the presence of removal and regeneration effects,Z. Angew. Math. Phys. 37:27 (1986).

    Google Scholar 

  7. N. G. Van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).

    Google Scholar 

  8. D. Vanderhaegen, Radiative transfer in statistically heterogeneous mixtures,J. Quant. Spectrosc. Rad. Transfer 36:557 (1986).

    Google Scholar 

  9. C. D. Levermore, G. C. Pomraning, D. L. Sanzo, and J. Wong, Linear transport theory in a random medium,J. Math. Phys. 27:2526 (1986).

    Google Scholar 

  10. G. E. Roberts and H. Kaufmann,Table of Laplace Transforms (Saunders, Philadelphia, 1966).

    Google Scholar 

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Author notes

  1. V. C. Boffi

    Present address: Faculty of Engineering, University of Rome “La Sapienza,”, 00182, Rome, Italy

Authors and Affiliations

  1. School of Engineering and Applied Science, University of California, Los Angeles, 90024-1597, Los Angeles, California

    V. C. Boffi, F. Malvagi & G. C. Pomraning

Authors
  1. V. C. Boffi
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  2. F. Malvagi
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  3. G. C. Pomraning
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Boffi, V.C., Malvagi, F. & Pomraning, G.C. Solution methods for discrete-state Markovian initial value problems. J Stat Phys 60, 445–472 (1990). https://doi.org/10.1007/BF01314930

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  • DOI: https://doi.org/10.1007/BF01314930

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