A Suggested Approach to Finite-Difference Representation of Differential Equations,

In a recent paper by Allen and Southwell (1) incidental use was made of a novel method of formulating a finite-difference approximation to a partial differential equation. This method replaces the usually accepted basis of polynomial representation by a more general approach which does however allow polynomial representation as a special case. In the problems treated by Allen and Southwell the new approach succeeded where the old had failed. The method is described in more detail in Part I of this paper and its usefulness as an alternative to polynomial representation is discussed. In Part II the method is applied to the solution of a class of problems of importance in itself, that of determining temperature distributions produced by a moving source of heat. The particular problems considered are of the type where the heat-source is provided by friction at a sliding contact; the temperatures are determined in two solids sliding across one another with contact along a band of finite width. Five solutions are presented for different cases, one of which is shown to be equivalent to that of a single solid sliding across a non-conducting band.