Domain-Theoretic Formulation of Linear Boundary Value Problems

We present a domain theoretic framework for obtaining exact solutions of linear boundary value problems. Based on the domain of compact real intervals, we show how to approximate both a fundamental system and a particular solution up to an arbitrary degree of accuracy. The boundary conditions are then satisfied by solving a system of imprecisely given linear equations at every step of the approximation. By restricting the construction to effective bases of the involved domains, we not only obtain results on the computability of boundary value problems, but also directly implementable algorithms, based on proper data types, that approximate solutions up to an arbitrary degree of accuracy. As these data types are based on rational numbers, no numerical errors are incurred in the computation process.