The paper concerns with the approximation of solutions to the nonlinear reaction-diffusion equation, endowed with homogeneous Cauchy-Neumann boundary conditions. It extends the studied types of boundary conditions, already proven by other authors, which makes the problem to be more able to describe many important phenomena of two-phase systems, in particular, the interactions with the walls in confined systems. The convergence and error estimate results for a new iterative scheme of fractional steps type, associated to the nonlinear parabolic problem, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual numerical algorithm is formulated in the end.