A mathematical theory for variable-coefficient Lanchester-type equations of 'modern warfare'

Abstract

A mathematical theory is developed for the analytic solution to deterministic Lanchester-type "square-law" attrition equations for combat between two homogeneous forces with temporal variations in system effectiveness (as expressed by the Lanchester attrition-rate coefficient). Particular attention is given to solution in terms of tabulated functions. For this purpose Lanchester functions are Introduced and their mathematical properties that facilitate solution given. The above theory is applied to the following cases: (1) lethality of each side's fire proportional to a power of time, and (2) lethality of each side's fire linear with time but a nonconstant ratio of these. By considering the force-ratio equation , the classical Lanchester square law is generalized to variable-coefficient cases in which it provides a "local" condition of "winning."