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Abstract

In this chapter, the closure problem for non-integer moments as discussed in Sect. 5.3 will be considered.

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References

  1. Rigopoulos S (2010) Population balance modelling of polydispersed particles in reactive flows. Prog Energy Combust Sci 36:412–443

    Article  Google Scholar 

  2. Frenklach M, Harris SJ (1987) Aerosol dynamics modeling using the method of moments. J Colloid Interface Sci 118:252–261

    Article  Google Scholar 

  3. Diemer RB, Olson JH (2002c) A moment methodology for coagulation and breakage problems: part 3-generalized daughter distribution functions. Chem Eng Sci 57:4187–4198

    Article  Google Scholar 

  4. Diemer RB, Olson JH (2002a) A moment methodology for coagulation and breakage problems: part 1-analytical solution of the steady-state population balance. Chem Eng Sci 57:2193–2209

    Article  Google Scholar 

  5. Diemer RB, Olson JH (2002b) A moment methodology for coagulation and breakage problems: part 2 moment models and distribution reconstruction. Chem Eng Sci 57:2211–2228

    Article  Google Scholar 

  6. Diemer R, Olson J (2006) Bivariate moment methods for simultaneous coagulation, coalescence and breakup. J Aerosol Sci 37:363–385

    Article  Google Scholar 

  7. Beck J, Watkins A (2002) On the development of spray submodels based on droplet size moments. J Comput Phys 182:586–621

    Article  MATH  Google Scholar 

  8. Beck JC, Watkins AP (2003c) The droplet number moments approach to spray modelling: the development of heat and mass transfer sub-models. Int J Heat Fluid Flow 24:242–259

    Article  Google Scholar 

  9. John V, Angelov I, Oncul AA, Thevenin D (2007) Techniques for the reconstruction of a distribution from a finite number of its moments. Chem Eng Sci 62:2890–2904

    Article  Google Scholar 

  10. Tagliani A (1999) Hausdorff moment problem and maximum entropy: a unified approach. Appl Math Comput 105:291–305

    Article  MATH  MathSciNet  Google Scholar 

  11. Inverardi P, Pontuale G, Petri A, Tagliani A (2003) Hausdorff moment problem via fractional moments. Appl Math Comput 144:61–74

    Article  MATH  MathSciNet  Google Scholar 

  12. Inverardi P, Pontuale G, Petri A, Tagliani A (2005) Stieltjes moment problem via fractional moments. Appl Math Comput 166:664–677

    Article  MATH  MathSciNet  Google Scholar 

  13. Pintarellia M, Vericat F (2003) Generalized Hausdorff inverse moment problem. Phys A 324:568–588

    Article  MathSciNet  Google Scholar 

  14. Talenti G (1987) Recovering a function from a finite number of moments. Inverse Prob 3: 501–517

    Google Scholar 

  15. Volpe EV, Baganoff D (2003) Maximum entropy PDFs and the moment problem under near-Gaussian conditions. Prob Eng Mech 18:17–29

    Article  Google Scholar 

  16. Koopman BO (1969) Relaxed motion in irreversible molecular statistics. Stoch Process Chem Phys 15:37–63

    Google Scholar 

  17. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–623

    Article  MATH  MathSciNet  Google Scholar 

  18. Paris J, Vencovska A (1997) In defense of the maximum entropy inference process. Int J Approx Reasoning 17:77–103

    Article  MATH  MathSciNet  Google Scholar 

  19. Archambault MR, Edwards CF, McCormack RW (2003d) Computation of spray dynamics by moment transport equations I: theory and development. Atomization Sprays 13:63–87

    Google Scholar 

  20. Blinnikov S, Moessner R (1998) Expansions for nearly Gaussian distributions. Astron Astrophys Suppl Ser 130:193–205

    Article  Google Scholar 

  21. Abramowitz M, Stegun I (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications, New York

    Google Scholar 

  22. Majumdar AK, Luna C, Idell P (2007) Reconstruction of probability density function of intensity fluctuations relevant to free-space laser communications through atmospheric turbulence. In: Proceedings of SPIE

    Google Scholar 

  23. Kendall M, Stuart A, Ord J (1991) Kendall’s advanced theory of statistics: distribution theory, vol 1. Wiley, New York

    Google Scholar 

  24. Gaztanaga E, Fosalba P, Elizalde E (2000) Gravitational evolution of the large-scale probability density. Astrophys J 539:522–531

    Article  Google Scholar 

  25. Haeri S, Shrimpton J (2012) Closure of non-integer moments arising in multiphase flow phenomena. Chem Eng Sci 75(0):424–434. http://dx.doi.org/10.1016/j.ces.2012.03.052

  26. Mood A, Graybill FA, Boes D (1974) Introduction to the theory of statistics. McGraw-Hill, New York

    Google Scholar 

  27. Scott SJ (2006) A PDF based method formodelling polysized particle laden turbulent flows without size class discretisation. Ph.D. thesis, Imperial College, London

    Google Scholar 

  28. Jaynes E (1957a) Information theory and statistical mechanics. Phys Rev 106:620–630

    Article  MATH  MathSciNet  Google Scholar 

  29. Sellens RW, Brzustowski TA (1985) A prediction of the drop size distribution in a spray from first principles. Atomisation Spray Technol 1:89–102

    Google Scholar 

  30. Sellens RW (1989) Prediction of the drop size and velocity distribution in a spray, based on the maximum entropy formalism. Part Part Syst Charact 6:17–27

    Article  Google Scholar 

  31. Ahmadi M, Sellens RW (1993) A simplified maximum-entropy-based drop size distribution. Atomization Sprays 3:291–310

    Google Scholar 

  32. Boyaval S, Dumouchel C (2001) Investigation on the drop size distribution of sprays produced by a high-pressure swirl injector. Measurements and application of the maximum entropy formalism. Part Part Syst Charact 18:33–49

    Article  Google Scholar 

  33. Dumouchel C, Boyaval S (1999) Use of the maximum entropy formalism to determine drop size distribution characteristics. Part Part Syst Charact 16:177–184

    Article  Google Scholar 

  34. Archambault MR, Edwards CF, McCormack RW (2003a) Computation of spray dynamics by moment transport equations II: application to quasi-one dimensional spray. Atomization Sprays 13:89–115

    Article  Google Scholar 

  35. Zwillinger D (2003) CRC standard mathematical tables and formulae. CRC Press, Boca Raton

    Google Scholar 

  36. Kreyszig E (1999) Advanced engineering mathematics. Wiley, New York

    Google Scholar 

  37. Alhassid Y, Agmon N, Levine RD (1978) An upper bound for the entropy and its applications to the maximal entropy problem. Chem Phys Lett 53:22–26

    Article  MathSciNet  Google Scholar 

  38. Alhassid Y, Agmon N, Levine RD (1979) An algorithm for finding the distribution of maximum entropy. J Comput Phys 30:250–258

    Article  MATH  Google Scholar 

  39. Mustapha H, Dimitrakopoulos R (2010) Generalized Laguerre expansions of multivariate probability densities with moments. Comput Math Appl 60:2178–2189

    Article  MATH  MathSciNet  Google Scholar 

  40. Cody W (1976) An overview of software development for special functions. Lect Notes Math 506:38–48

    Article  Google Scholar 

  41. Lebedev N (1972) Special functions and their applications. Dover Publications, New York

    Google Scholar 

  42. Ross B (1977) Fractional calculus: an historical apologia for the development of a calculus using differentiation and antidifferentiation of noninteger orders. Math Mag 50:115–122

    Google Scholar 

  43. Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    Google Scholar 

  44. Hilfer R (2000) Fractional calculus in physics. World Scientific, Singapore

    Google Scholar 

  45. Schiavone S, Lamb W (1990) A fractional power approach to fractional calculus. J Math Anal Appl 149:337–401

    Article  MathSciNet  Google Scholar 

  46. Cottone G, Di Paola M (2009) On the use of fractional calculus for the probabilistic characterization of random variables. Probab Eng Mech 24:321–334

    Article  Google Scholar 

  47. Caputo M (1967) Linear models of dissipation whose q is almost frequency independent-II. Geophys J R Astr Soc 13:529–539

    Article  Google Scholar 

  48. Cressie N, Borkent M (1986) The moment generating function has its moments. J Stat Plan Infer 13:337–344

    Article  MATH  MathSciNet  Google Scholar 

  49. Deng W (2007) Short memory principle and a predictor-corrector approach for fractional differential equations. J Comput Appl Math 206:174–188

    Article  MATH  MathSciNet  Google Scholar 

  50. Gzyl H, Tagliani A (2010) Hausdorff moment problem and fractional moments. Appl Math Comput 216:3319–3328

    Article  MATH  MathSciNet  Google Scholar 

  51. Alexiadisa A, Vanni M, Gardin P (2004) Extension of the method of moments for population balances involving fractional moments and application to a typical agglomeration problem. J Colloid Interface Sci 276:106–112

    Article  Google Scholar 

  52. Lubich C (1986) Discretized fractional calculus. SIAM J Math Anal 17:704–719

    Google Scholar 

  53. Behboodian J (1970) On the modes of a mixture of two normal distributions on the modes of a mixture of two normal distributions. Technometrics 12:131–139

    Google Scholar 

  54. Schilling M, Watkins A, Watkins W (2002) Is human height bimodal? Am Stat 56:223–229

    Article  MathSciNet  Google Scholar 

  55. Shampine L (2008) Vectorized adaptive quadrature in Matlab. J Comput Appl Math 211: 131–140

    Google Scholar 

  56. Majumdar AK, Gamo H (1982) Statistical measurements of irradiance fluctuations of a multipass laser beam propagated through laboratory-simulated atmospheric turbulence. Appl Opt 21:2229–2235

    Article  Google Scholar 

  57. Majumdar A (1984) Uniqueness of statistics derived from moments of irradiance fluctuations in atmospheric optical propagation. Opt Commun 50:1–7

    Article  Google Scholar 

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Shrimpton, J.S., Haeri, S., Scott, S.J. (2014). PDF Reconstruction Methods. In: Statistical Treatment of Turbulent Polydisperse Particle Systems. Green Energy and Technology. Springer, London. https://doi.org/10.1007/978-1-4471-6344-2_6

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  • DOI: https://doi.org/10.1007/978-1-4471-6344-2_6

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