Generalization of the Trotter–Daletsky formula for systems of the "reaction–diffusion" type
Keywords:parabolic equation, semigroup of operators, perturbation theory
An iterative method for constructing a solution to the Cauchy problem for a system of parabolic equations with a nonlinear potential has been proposed and substantiated. The method is based on the Trotter–Daletsky formula, generalized for a nonlinear perturbation of an elliptic operator. The idea of generalization is the construction of a composition of the semigroup generated by the Laplacian and the phase flow corresponding to a system of ordinary differential equations. A computational experiment performed for a two-dimensional system of semilinear parabolic equations of the “reaction–diffusion” type confirms estimates for the convergence of iterations established in the proof of this formula. Obtained results suggest the feasibility of an unconventional approach to modeling dynamic systems with distributed parameters.
A. Yagi, Abstract parabolic evolution equations and their applications. Springer, Berlin, 2010.
J.A. Goldstein, Semigroups of linear operators and applications. Kyiv: Vyshcha shkola, 1989, 347 p.
V.G. Bondarenko, “Trotter–Daletskii Formula for Nonlinear Disturbances”, Ukrainian Mathematical Journal, 70, no.12, pp. 1978–1984, 2019.
V.G. Bondarenko and I.S. Markevych, “On the convergence of iterations in the Trotter–Daletsky formula for nonlinear perturbation”, System Research and Information Technologies, no. 3, pp.118–125, 2019.
J.D. Murray, Mathematical Biology; vol. 1, 2. New York: Springer-Verlag, 2002.
Y.V. Tyutyunov, A.D. Zagrebneva, V.N. Govorukhin, and L.I. Titova, “Numerical study of bifurcations occurring at fast timescale in a predator-prey model with inertial prey-taxis”, in Advanced Mathematical Methods in Bioscience and Applications (eds. F. Berezovskaya and B. Toni), Springer, Cham, 2019, pp. 221–239.