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

The initial attached cell layer in multispecies biolm 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 biolm thickness is equal to zero. No assumptions on initial conditions for biomass concentrations and biolm thickness are required. The data that the problem needs are the concentration of biomass in the bulk liquid and biomass ux 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.