AREAL INTERPRETATION OF THE CONFLICT SITUATION MODEL WITH TWO OPPOSING SIDES’ STRATEGIC POSITIONS AREAS INCREASE

Abstract

This study is devoted to the application of differential equation systems for modeling combat operations under certain conditions. The article considers a mathematical model of confrontation between two hostile parties, focusing on an area model of combat operations, taking into account the relative increase in the areas of the parties compared to the initial sizes of the bridgeheads per unit of time, based on a system of nonlinear differential equations. The authors study the dynamics of the development of a conflict unfolding in a limited but unstable territory, where the gradual expansion of the controlled area by one or both sides at the same time plays a decisive role. The area model takes into account the probability of hitting targets, the combat characteristics of the sides, and the dynamics of the transformation of bridgehead areas over time. A system of equations was obtained that describes the change in the relative areas of the intact parts of the parties’ bridgeheads at a given moment in time during combat operations. An analytical transformation of the system of equations was performed for the case of expansion of the bridgehead area by one side, in particular by reduction to the Riccati and Bernoulli equations. The following method was used to solve the Bernoulli equation: the initial value was presented as a definite integral with a variable upper limit of integration. In the course of further calculations, the integral with a variable upper limit of integration was transformed into a Laplace function, which made it possible to obtain an explicit representation of the solutions to the Cauchy problem. A numerical solution using the fourth-order Runge-Kutta method is proposed for the general case when both sides of the conflict increase the area of their strongholds by a certain amount relative to the initial area per unit of time, which is implemented by programming in Python. An example of numerical calculations using the created software for the general case is given. The example demonstrates that even if one side’s relative damage per unit of time is greater than that of the other side, an advantage in battle can be gained by expanding the area to a greater extent. The results of the study are suitable for constructing models of combat situations, evaluating the effectiveness of selected strategies, and optimizing the use of resources in conflict zones.