An algebraic foundation for factoring linear boundary problems

Motivated by boundary problems for linear ordinary and partial differential equations, we define an abstract boundary problem as a pair consisting of a surjective linear map (representing the differential operator) and a subspace of the dual space (specifying the boundary conditions). This subspace is finite dimensional in the ordinary case, but infinite dimensional for partial differential equations. For so-called regular boundary problems, the given operator has a unique right inverse (called the Green’s operator) satisfying the boundary conditions. The main idea of our approach consists in the passage from a single problem to a compositional structure on boundary problems. We define the composition of boundary problems such that it corresponds to the composition of their Green’s operators in reverse order. If the defining operators are endomorphisms, we can interpret the composition as the multiplication in a semidirect product of certain monoids. Given a factorization of the linear operator defining the problem, we characterize and construct all factorizations of a boundary problem into two factors. In the setting of differential equations, the factor problems have lower order and are often easier to solve. For the case of ordinary differential equations, all the main results can be made algorithmic (in particular the determination of the factor problems). As a first example for partial differential equations, we conclude with a factorization of a boundary problem for the wave equation.