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

This paper develops a mathematical theory for solving deterministic, Lanchester-type, “square-law” attrition equations for combat between two homogeneous forces with temporal variations in fire effectivenesses (as expressed by 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. We 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.