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A Rothe-modified shooting method for solving a nonlinear boundary value problem arising in particulate processes

  • Khaldoun El Khaldi
  • Majdi Khasawneh
  • Elias G. SaleebyEmail author
Article
  • 26 Downloads

Abstract

The framework of population balance equations has been employed widely in the modeling of particulate processes. When considering particle growth dispersion and agglomeration, one obtains a second-order nonlinear partial integrodifferential equation. In this article, we develop a rather simple numerical scheme using a Rothe method coupled with a modified simple shooting method (MSSM) to solve initial and boundary value problems for such equations on rectangular regions in the plane. We examine the implementation of this scheme and its performance on some examples. The use of the MSSM is effective in removing the non-physical oscillations that arise using the Rothe method. Moreover, a diffusion–convection equation arises as a special case of our model. We show that the Rothe-MSSM method effectively removes the oscillations that are typically reported with the Rothe method or the method of lines for such equations.

Keywords

Rothe’s method Modified shooting Integrodifferential equations 

Mathematics Subject Classification

65M20 65R20 

Notes

Acknowledgements

The authors would like to thank the referee for his helpful suggestions.

References

  1. Baker CTH (1977) The numerical treatment of integral equations, monographs on numerical analysis. Clarendon Press, OxfordGoogle Scholar
  2. El Khaldi K, Mourany N, Saleeby EG (2013) On the identification of parameters for a nonlinear integrodifferential population balance equation. Int J Comput Math 90:2019–2035MathSciNetCrossRefGoogle Scholar
  3. Gherras N, Fevotte G (2012) On the use of process analytical tectnologies (PAT) and population balance equations for the estimation of crystallization kinetics. A case study. AIChE J 58:2650–2664CrossRefGoogle Scholar
  4. Holsapple R, Venkataraman R, Doman D (2003) A modified simple shooting method for solving two point boundary value problems. In: Proceedings of the IEEE aerospace conference, Big Sky, MT, IEEE, New York, vol 6, pp 2783–2790Google Scholar
  5. Hulburt HM, Katz S (1964) Some problems in particle technology: a statistical mechanical formulation. Chem Eng Sci 19:555–574CrossRefGoogle Scholar
  6. Khanh BD (1994) Hermite predictor–corrector scheme for regular Volterra integral equations and for some integro-differential equations for turbulent diffusion. J Comput Appl Math 51:305–316MathSciNetCrossRefGoogle Scholar
  7. Kohler T, Voss D (1999) Second-order methods for diffusion–convection equations. Commun Numer Methods Eng 15:689–699MathSciNetCrossRefGoogle Scholar
  8. Randolph AD, Larson MA (1988) Theory of particulate processes, 2nd edn. Academic press, New YorkGoogle Scholar
  9. Randolph AD, White ET (1977) Modeling size dispersion in the prediction of crystal-size distribution. Chem Eng Sci 32:1067–1076CrossRefGoogle Scholar
  10. Raphael M, Rohani S (1999) Sunflower protein preciptation in a tubular preciptator. Can J Chem Eng 77:540–554CrossRefGoogle Scholar
  11. Rektorys K (1982) The method of discretization in time. D Reidel Publishing Co, DordrechtzbMATHGoogle Scholar
  12. Saleeby EG, Lee HW (1995) On the solution of the PBE with agglomeration and random growth dispersion. Chem Eng Sci 50:1971–1981CrossRefGoogle Scholar
  13. Tavare NS (1995) Industrial crystallization. Plenum Press, New YorkCrossRefGoogle Scholar
  14. Yang XJ, Baleanu D, Srivastava HM (2016) Local fractional integral transforms and their applications. Academic Press, New YorkzbMATHGoogle Scholar
  15. Yang XJ, Gao F (2017) A new technology for solving diffusion and heat equations. Therm Sci 21:133–140CrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  • Khaldoun El Khaldi
    • 1
  • Majdi Khasawneh
    • 2
  • Elias G. Saleeby
    • 3
    Email author
  1. 1.Notre Dame University-LouaizeZouk MosbehLebanon
  2. 2.Alfaisal UniversityRiyadhSaudi Arabia
  3. 3.Mount LebanonLebanon

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