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 Bi ∂u ∂xi = 0, A = A∗ > 0, Bi = B∗ i , with the initial condition u|t=0 = φ(x1, . . . , xm). 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 matrices A,B1, . . . , Bm satisfying certain conditions) any initial function φ ∈ C(Q,R) (satisfying certain conditions), p ≥ 2, to the unique solution u ∈ C(H,R), where Q = [0, 1] and H is the nonempty domain of correctness of the system.