Elena Viktorovna Karachanskaya (Far–Eastern State Transport University/Pacific National University) :


517.925:519.2:519.6 Modeling of a differential equations system with dynamical invariants

Karachanskaya E. V. (Far–Eastern State Transport University/Pacific National University)

doi: 10.18698/2309-3684-2019-1-98117

This article is devoted to the construction method of differential equations system with smooth functions as invariants. This method allows us to construct both deterministic differential equations system and Ito’s stochastic differential equations system. These stochastic differential equations are diffusion equations or jump-diffusion ones. The algorithm under consideration is based on the author’s previous research and has no parallel. We study the realization of the algo¬rithm in R2 and in R3 in detail. We represent a MathCad-program for construction of every type of differential equations system de-scribed above. We also show some examples of differential equa¬tions system with the given invariant constructed automatically be the computer program. Rele¬vance of our research is guaranteed by numerical solution of the created system of differential equa-tions. These systems are solved using well-known numerical techniques: Euler method, Euler-Murayama method and Monte-Carlo procedure. The numerical solution of a sys-tem of differential equations and the corresponding values of the invariant function are represented in diagram form. Programmed control with probability 1 for a stochastic SIR-model is an application of the presented theory and the MathCad-program. Results of this research can be used to construct models of dynamical systems with invariant and to further explore these systems.

Карачанская Е.В. Моделирование систем дифференциальных уравнений с ди-намическими инвариантами. Математическое моделирование и численные мето-ды, 2019, № 1, с. 98–117.