Solving free boundary problem for an initial cell layer in multispecies biofilm formation by Newton-Raphson method

Document Type : Research Paper

Authors

Faculty of Mathematical sciences, University of Tabriz, Tabriz, Iran.

Abstract

The initial attached cell layer in multispecies biofilm growth is studied in this paper. The corresponding mathematical model leads to discuss a free boundary problem for a system of nonlinear hyperbolic partial differential equations, where the initial biofilm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biofilm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass flux from the bulk liquid. The differential equations are converted into an equivalent system of Volterra integral equations. We use Newton-Raphson method to solve the nonlinear system of Volterra integral equations (SVIEs) of the second kind. This method converts the nonlinear system of integral equations into a linear integral equation at each step.

Keywords

References

[1] D. Ahmadian, O. Farkhondeh Rouz, K. Ivaz, and A. Safdari-Vaighani, Robust numerical algorithm to the European option with illiquid markets, Applied Mathematics and Computation, 366 (2020), 124693.
[2] E. Alpkvist, C. Picioreanu, M. C. M. Van Loosdrecht, and A. Heyden, Three-dimensional biofilm model with individual cells and continuum EPS matrix, Biotechnol. Bioeng. 94 (2006), 961–979.
[3] K. Atkinson and W. Han, Theoretical Numerical Analysis; A Functional Analysis Framework. Third Edition. 2009.
[4] B. Babayar-Razlighi and K. Ivaz, M. R. Mokhtarzadeh, Newton-product integration for a Stefan problem with kinetics, J. Sci. I.R. Iran 22 (2011) 51–61.
[5] B.Babayar-Razlighi and B.Soltanalizadeh, Numerical solution for system of singular nonlinear Volterra integro differential equations by Newton-Product method, Applied Mathematics and Computation, 219 (2013), 8375–8383.
[6] N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, Multiscale biological tissue models and fluxlimited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci. 20 (2010), 1179–1207.
[7] B. D’Acunto and L. Frunzo, Free boundary problem for an initial cell layer in multispecies biofilm formation, Applied Mathematics Letters. 25 (2012), 20–26.
[8] B. D’Acunto and L. Frunzo, Qualitative analysis and simulations of a free boundary problem for multispecies biofilm models, Math. Comput. Modelling. 53 (2011), 1596–1606.
[9] P. Darania and K. Ivaz, Numerical solution of nonlinear VolterraFredholm integro-differential equations, Computers and Mathematics with Applications. 56 (2008), 2197–2209.
[10] P. Darania, J. Ahmadi Shali, and K. Ivaz, New computational method for solving some 2-dimensional nonlinear Volterra integro-differential equations, Numerical Algorithms, 57 (2011), 125–147.
[11] L. Hall-Stoodley, J.W. Costerton, and P. Stoodley, Bacterial biofilms: from the natural environment to infectious diseases, Nat. Rev. Microbiol. 2 (2004), 95–108.
[12] H. C. Jones, I. L. Roth, and W. M. Sanders, Electron microscopic study of a slime layer, J. Bacteriol. 99 (1969), 316–325.
[13] C. Picioreanu, J. U. Kreft, and M. C. M. van Loosdrecht, Particle-based multidimensional multispecies biofilm model, Appl. Environ. Microbiol. 70 (2004), 3024–3040.
[14] O. Wanner and W. Gujer, A multispecies biofilm model, Biotechnol. Bioeng. 28 (1986), 314–328.
[15] E. Zeidler, Nonlinear Functional Analysis and its Applications. I: Fixedpoint Theorems, Springer-Verlag, New York, 1985.
Computational Methods for Differential Equations
Volume 9, Issue 3
July 2021
Pages 899-907

Files

History

  • Receive Date: 23 May 2020
  • Revise Date: 09 July 2020
  • Accept Date: 14 July 2020

Share

How to cite

Statistics