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Domain--Theoretic Solution of Differential Equations (Scalar Fields)

Abbas Edalat, Marko Krznaric, Lieutier

Electronic Journal Article
Electronic Notes in Theoretical Computer Science
Volume 83

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.

London e-Science Centre
Theory of Computational Systems
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