Computability
Complexity theory of real functions
Complexity theory of real functions
Selected papers of the workshop on Topology and completion in semantics
COCOON '97 Proceedings of the Third Annual International Conference on Computing and Combinatorics
Effective metric spaces and representations of the reals
Theoretical Computer Science
Turing Computability of a Nonlinear Schrödinger Propagator
COCOON '01 Proceedings of the 7th Annual International Conference on Computing and Combinatorics
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Pour-El/Richards [PER89] and Pour-El/Zhong [PEZ97] have 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 ∈ R, x ∈ R3, such that the unique solution is not computable. This very remarkable result might indicate that the physical process of wave propagation 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 computability concepts on the spaces under consideration. It is shown that the solution operator of the Cauchy problem is computable on continuously differentiable initial conditions, where one order of differentiability is lost. The solution operator is also computable on Sobolev spaces. Finally the results are interpreted in a simple physical model.