On the convergence of iterations in the Trotter–Daletsky formula for nonlinear perturbation
DOI:
https://doi.org/10.20535/SRIT.2308-8893.2019.3.11Keywords:
parabolic equation, Semigroup of operators, perturbation theoryAbstract
An iterative method for constructing a solution to the Cauchy problem for a parabolic equation with a nonlinear potential ("reaction–diffusion" type equation) is proposed and substantiated. The method is based on the Trotter–Daletsky formula that is generalized for a nonlinear perturbation of an elliptic operator. The essence of the generalization is the composition of the semigroup of an elliptic generator and the phase flow generated by an ordinary differential equation. The estimates of the rate of convergence of iterations established in the proof of this formula were confirmed by the computational experiment performed for the Kolmogorov–Petrovsky–Piskunov–Fisher equation. The obtained results suggest the feasibility of an unconventional approach to the modeling of dynamic systems with distributed parameters. A model of the space-time dynamics of the water community in terms of the two-species "predator–prey" system was shown as an example.References
Aronson D.G. Multidimensional Nonlinear Diffusion Arising in Population Genetics // D.G. Aronson, H.F.Weinberger // Advances Mathematics. — 1978. — V. 30. — P. 33–76.
Amann H. Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems / H. Amann // Differential Integral Equations. — 1990. — V.3, N 1. — P. 13–75.
Yagi A. Abstract parabolic evolution equations and their applications / A. Yagi. — Berlin: Springer, 2010.
Bondarenko V.G. Formula Trottera–Daletskogo dlja nelinejnogo vozmuschenija / V.G. Bondarenko // Ukrainskij matematicheskij zhurnal. — 2018. — 70, № 12. — S. 1717–1722.
Bondarenko V.G. Metod kompozitsii dlja sistem s raspredelennymi parametrami / V.G. Bondarenko // Problemy upravlenija i informatiki. — 2018. — № 4. — S. 112–120.
Goldstejn Dzh. Polugruppy linejnyh operatorov i ih prilozhenija / Dzh. Holdstejn. — K.: Vyshcha shk., 1989. — 347 s.
Trotter T.F. Of the product of semi-groups of operators / T.F. Trotter // Pros. Am. Math. Soc. — V. 959, N 10. — P. 545–551.
Daletskij Ju.L. Kontinual'nye integraly, svjazannye s operatornymi evoljutsionnymi uravnenijami / Ju.L. Daletskij // UMN. — 1962.— T. 17, № 5. — S. 3–115.
Taylor M.E. Partial Differential Equations III / M.E. Taylor. — New York: Springer Verlag, 1997.
Svirezhev Ju.M. Nelinejnye volny, dissipativnye struktury i katastrofy v ekologii / Ju.M. Svirezhev. — M.: Nauka, 1987. — 368s.
Murray J.D. Mathematical Biology / J.D. Murray. — New York: Springer-Verlag, 2002. — Vol. 1, 2.
Medvinskij A.B. Formirovanie prostranstvenno-vremennyh struktur, fraktaly i haos v kontseptual'nyh ekologicheskih modeljah na primere dinamiki vzaimodejstvujuschih populjatsij planktona i ryby / A.B. Medvinskij, S.V. Petrovskij, I.A. Tihonova i dr. // Uspehi fizicheskih nauk. — 2002. — T. 172, № 1. — S. 31–66.
Medvinsky A.B. Spatiotemporal complexity of plankton and fish dynamics / A.B. Medvinsky, S.V. Petrovskii, I.A. Tikhonova, H. Malchow // SIAM Review. — 2002. — V. 44, N 3. — P. 311–370.
