Composition Method for Semi-linear Parabolic Systems

An iteration method for constructing a solution to the Cauchy problem for parabolic systems with non-linear potential (reaction–diffusion) is proposed and justified. The method is based on the Trotter formula generalized to the case of a nonlinear perturbation of an elliptic operator. The essence of the generalization is a composition of a semigroup with an elliptic generator and a phase flow generated by an ordinary differential equation. Estimates of the rate of convergence of iterations, established in the proof of this formula, are confirmed with a computational experiment performed for the Kolmogorov–Petrovsky–Piskunov–Fisher equation. The obtained results suggest the feasibility of an unconventional approach to modeling dynamical systems with distributed parameters.