Canonical Methods in the Solution of Variable-Coefficient Lanchester-Type Equations of Modern Warfare

This paper develops a mathematical theory for solving determin-istic, Lanchester-type, 'square-law' attrition equations for combat between two homogeneous forces with temporal variations in fire effec-tivenesses (as expressed b y the Lanchester attrition-rate coefficients). It gives a general form for expressing the solution of such variable-coefficient combat attrition equations in terms of Lanchester functions , which are introduced here and can be readily tabulated. Different Lanchester functions arise from different mathematical forms for the attrition-rate coefficients. W e give results for two such forms: (1) effectiveness of each side's fire proportional to a power of time, and (2) effectiveness of each side's fire linear with time but with a nonconstant ratio of attrition-rate coefficients. Previous results in the literature for a nonconstant ratio of these attrition-rate coefficients only took a convenient form under rather restrictive conditions. DETERXIINISTIC Lanchester-type equations of warfare (see references 38,44) are of value for identifying trends in weapon system analysis or force structuring studies because of their computational convenience, even though combat between two opposing military forces is indeed a complex random process (see Xote 1). In this paper we present a mathematical theory (including a new standard form) for the solution of variable-coefficient Lanchester-type equations of modern warfare for combat between two homogeneous forces. After some preliminaries, we give a more precise statement of this paper's purpose at the end of Section 3. First, we review Lanchester's classic mathematical model of combat between two homogeneous forces and its extension to cases of time-varying fire effectiveness. Then we discuss previous work on developing analytic solutions to variable-coefficient Lanchester-type equations and explain how we extend these results. Next, we develop a mathematical theory for solving Lanchester-type equations for a 'square-law' attrition process for combat between two homogeneous forces. These general results are then ap