Domain-theoretic Solution of Differential Equations (Scalar Fields)

Abstract

We provide an algorithmic formalization of ordinary differential equations in the framework of domain theory. Given a Scott continuous, interval-valued and time-dependent scalar field and a Scott continuous initial function consistent with the scalar field, the domain-theoretic analogue of the classical Picard operator, whose fix-points give the solutions of the differential equation, acts on the domain of continuously differentiable functions by successively updating the information about the solution and the information about its derivative. We present a linear and a quadratic algorithm respectively for updating the function information and the derivative information on the basis elements of the domain. In the generic case of a classical initial value problem with a continuous scalar field, which is Lipschitz in the space component, this provides a novel technique for computing the unique solution of the differential equation up to any desired accuracy, such that at each stage of computation one obtains two continuous piecewise linear maps which bound the solution from below and above, thus giving the precise error. When the scalar field is continuous and computable but not Lipschitz, it is known that no computable classical solution may exist. We show that in this case the interval-valued domain-theoretic solution is computable and contains all classical solutions. This framework also allows us to compute an interval-valued solution to a differential equation when the initial value and/or the scalar field are interval-valued, i.e. imprecise.