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**

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July 2021

Pages 899-907

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