### Approximate solving of differential equations with nonlinear delay and approximate calculation of functionality of quality at known operating influences

#### T. K. Yuldashev^{1}

Annotation | It is considered the questions of approximate solving of differential equations with nonlinear delay and of approximate calculation of functionality of quality at known operating influences. This problem is involved the control bounded by a constant and is contained it as nonlinear function into equation and into functionality of quality. It is considered the case when the variables are integer values. The problem is changed to its discrete analog. For each set of given coordinate and controls the initial value problem is reduced to a summary equation with nonlinear delay. It is proved the existence and uniqueness of solution of the summary equation. It is used the method of successive approximations, combined it with the method of compressing maps. It is estimated the permissible error with respect to state of approximation solution of initial value difference problem. Further it is proved that discrete control sequence is minimizing for the considering problem. As an example it is constructed a simple dynamical model of the economy in the form of differential equations with delay time, which is considered in this paper. This model takes into account the relationship of volume of production and income in certain conditions of market pricing. |
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Keywords | Differential equation, nonlinear delay, initial value condition, optimal control, approximate solution, dynamical model of economics. |

^{1}Associate professor of Higher Mathematics Chair, M. F. Reshetnev Siberian State Aerospace University, Krasnoyarsk, tursunbay@rambler.ru

**Citation**: T. K. Yuldashev, "[Approximate solving of differential equations with nonlinear delay and approximate calculation of functionality of quality at known operating influences]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,16:4 (2014) 75–84 (In Russian)