Semi-analytical, physics-informed and numerical solution of a nonlinear reaction–diffusion biofilm model for the biofiltration of ethyl acetate and toluene as single pollutants and in mixture

The biodegradation of volatile organic compounds inside a microbial biofilm is governed by the competition between diffusive transport and inhibited microbial consumption, yet the coupled ethyl-acetate–toluene–oxygen subsystem that controls this process has rarely been examined as a self-contained boundary-value problem. This study isolates the biofilm equations of a dynamic peat-biofilter model and analyses the steady-state, dimensionless reaction–diffusion balances of ethyl acetate, toluene and oxygen, incorporating Haldane self-inhibition of ethyl acetate, Monod kinetics for toluene, competitive cross-inhibition of toluene by ethyl acetate, and dual oxygen limitation of both substrates. The resulting coupled nonlinear system is solved by three independent routes: the semi-analytical Akbari–Ganji method (AGM), a physics-informed neural network (PINN), and a pdepe-based method-of-lines scheme adopted as the numerical benchmark. Mixture, ethyl-acetate-only and toluene-only configurations are each studied across five Thiele-modulus cases spanning reaction-controlled to strongly diffusion-limited regimes. A quantitative point-by-point error analysis shows that the PINN retains sub-0.15% relative error throughout, whereas the AGM is essentially exact at low modulus and deviates by up to about 2% in the film interior at high modulus. A normalized sensitivity analysis identifies oxygen diffusivity as the dominant control on biofilm effectiveness, followed by ethyl-acetate self-inhibition and competitive cross-inhibition. The results also expose a pronounced kinetic asymmetry in which cross-inhibition and shared oxygen demand shape the two substrate profiles in opposite directions.