Computing the Solution Operators of Symmetric Hyperbolic Systems of PDE

We study the computability properties of symmetric hyperbolic systems of PDE A ∂u ∂t + m � i=1 B i ∂u ∂xi =0 ,A = A ∗ > 0, Bi = B ∗ i , with the initial condition u|t=0 = ϕ(x1 ,...,x m). Such systems first considered by K.O. Friedrichs can be used to describe a wide variety of physical processes. Using the difference equations approach, we prove computability of the operator that sends (for any fixed computable matri- ces A, B1 ,...,B m satisfying certain conditions) any initial function ϕ ∈ C p+1 (Q, R n ) (satisfying certain conditions), p ≥ 2, to the unique solution u ∈ C p (H, R n ), where Q =( 0, 1) m and H is the nonempty domain of correctness of the system.