一类非线性兰彻斯特方程的摄动解  

Perturbation Solution for a Class of Nonlinear Lanchester Equation

作　　者：谢英超 程燕 李伟兵 XIE Ying-chao;CHENG Yan;LI Wei-bing(Department of Mathematics,Army Antiaircraft and Antiaircraft College of PLA,Hefei 230031,China)

机构地区：[1]陆军炮兵防空兵学院教研室

出　　处：《数学的实践与认识》2018年第13期270-275,共6页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11202106);安徽省自然科学基金(1408085MA06)

摘　　要：研究了一类非线性兰彻斯特方程，描述了现代化战争条件下的战斗模型．在分析实际交战过程中的损耗系数之间的关系的基础上，引入了摄动参数．利用摄动方法，得到了相应非线性方程组的渐近解，再利用微分不等式理论，证明了渐近解的一致有效性，并将得到的渐近解与数值解进行了精度比较．结果表明该摄动方法简单有效，而且它得到的解是近似解析解，能继续进行各种解析运算，这是数值解所无法媲美的优点．从而，所求的渐近解能够更准确地揭示出现代战争的特点和规律，还能为作战决策者提供更多有价值的信息．In this paper, a class of nonlinear Lancaster equation is considered and the combat model for modernized warfare is described. Under the analysis of the relationship between the attrition coefficients in the process of the actual war, the perturbation parameter is introduced. Using the perturbation method, the corresponding asymptotic solution is obtained. Further, the uniform validity for asymptotic solution is proved by using the theory of differential inequalities. To demonstrate the effectiveness of the perturbation method, the asymptotic solution is compared to the numerical solution on the precision. The results show that the perturbation method is simple and effective, and the corresponding solution is approximately analytical, so it can be done all kinds of analytical calculation subsequently, that is the advantage which the numerical solution cannot rival. Consequently, not only can the corresponding asymptotic solution reveal the characteristics and laws of modernized warfare more accurately, but also provide more valuable information for combat decision makers.

关 键 词：兰彻斯特方程 非线性 摄动参数 渐近解 一致有效性 

分 类 号：O175[理学—数学]