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


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.


Rothe’s method Modified shooting Integrodifferential equations 

Mathematics Subject Classification

65M20 65R20 



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


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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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