The wave propagator is turing computable

Abstract

Pour-El/Richards [PER89] and Pour-El/Zhong [PEZ97] ha- ve shown that there is a computable initial condition f for the three dimensional wave equation utt = Δu, u(0, x) = f(x), ut(0, x) = 0, t ∈ IR; x ∈ IR3, such that the unique solution is not computable. This very remarkable result might indicate that the physical process of wave pro- pagation is not computable and possibly disprove Turing's thesis. In this paper computability of wave propagation is studied in detail. Concepts from TTE, Type-2 theory of effectivity, are used to define adequate com- putability concepts on the spaces under consideration. It is shown that the solution operator of the Cauchy problem is computable on conti- nuously differentiable initial conditions, where one order of differentiabi- lity is lost. The solution operator is also computable on Sobolev spaces. Finally the results are interpreted in a simple physical model. © Springer-Verlag Berlin Heidelberg 1999.