University of Tabriz Computational Methods for Differential Equations 2345-3982 9 3 2021 07 01 Solving free boundary problem for an initial cell layer in multispecies biofilm formation by Newton-Raphson method 899 907 10908 10.22034/cmde.2020.39940.1746 EN Karim Ivaz Faculty of Mathematical sciences, University of Tabriz, Tabriz, Iran. Mohammad Asadpour Fazlallahi Faculty of Mathematical sciences, University of Tabriz, Tabriz, Iran. Journal Article 2020 05 23 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. https://cmde.tabrizu.ac.ir/article_10908_082747e2679c9a440406ec4a05720457.pdf