基于正规战的兰彻斯特方程的解析研究

Analytical Research on Lancaster Equation Based on Regular Warfare

摘要: 研究基于正规战的兰彻斯特方程的解析解的算法问题，包括红蓝双方兵力关于时间的函数解析式和红蓝双方兵力在相平面上的函数关系式。首先利用变换得到红蓝各自兵力满足的二阶常系数齐次线性微分方程，由特征方程再结合初值条件得到了红蓝兵力与时间的函数解析式，然后通过变形得到一阶的齐次方程，利用变量代换求解得到红蓝双方兵力的函数关系式，最后进行案例仿真并通过MATLAB编程描绘红蓝双方兵力的曲线示意图，旨在对正规战兰彻斯特方程的解有更深的理解和掌握。

Abstract: This paper studied the calculation method of the analytical solution to the Lanchester equation based on regular warfare, including the analytical expression of the functions of the red and blue forces with respect to time and the functional relationship between the red and blue forces on the phase plane. Firstly, the second-order constant coefficient homogeneous linear differential equations satisfied by the respective forces of the red and blue are obtained through transformation. The analytical expression of the function between the red and blue forces and time is obtained by combining the characteristic equation with the initial conditions. Then, the first-order homogeneous equation is obtained through deformation, and the functional relationship between the red and blue forces is solved through variable substitution. Finally, a case simulation is conducted and the curve diagram of the red and blue forces is drawn through MATLAB programming, aiming to have a deeper understanding and mastery of the solution of the Lanchester equation in regular warfare.