Syntactic methods for recursive equations and applications to differential equations

Abstract

The objectives of this paper are two‐fold: (1) to show that a proper appreciation of the roles that syntax and semantics play in the solution of equations can not only simplify the solution but also lead to greater generality, and (2) to demonstrate the utility of recursion as a practical tool in analysis.

The use of recursion in analysis is not new; Liouville [8] and Neumann [9] used it to obtain formal series solutions of certain integral equations, and Poincaré [10] used iterated mappings in his study of periodic motions. This paper is based on the more general interpretation of iteration as a transformation of the recursive equation u = P(u). Use is made of the notions of ‘string’, ‘binary tree’ and ‘formal language’, and those unfamiliar with these may consult [12].

The method of solving recursive equations (and inequalities) developed in this article is based on the scheme u = Pn(u) = Pn_1(P(u)),P0(u) = P(u). The generality of the scheme permits the exploitation of both the syntax and the semantics of equations. The method is applied to the solution of various ordinary and partial differential equations (d.e.). The notion of partial solutions of the equation u = P(u,u) is introduced and used in the transformation of equations. Based on syntactic considerations, generalizations of the exponential and related functions are given and the method of variation of parameters is extended to partial differential equations.