Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
For researchers in computable analysis and PDE theory, this provides a general method to prove computability of solution operators for a wide class of hyperbolic systems, partially answering an open question from Weihrauch and Zhong (2002).
The authors prove computability of solution operators for symmetric hyperbolic systems with dissipative boundary conditions and for the Cauchy problem, using numerical analysis methods without requiring explicit solution formulas. This extends computability results to a broader class of PDEs, including elasticity, acoustics, and Maxwell equations.
We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution operator for a symmetric hyperbolic system with computable real coefficients and dissipative boundary conditions, and of the Cauchy problem for the same system (we also prove computable dependence on the coefficients) in a cube $Q\subseteq\mathbb R^m$. Such systems describe a wide variety of physical processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many boundary-value problems for the wave equation also can be reduced to this case, thus we partially answer a question raised in Weihrauch and Zhong (2002). Compared with most of other existing methods of proving computability for PDEs, this method does not require existence of explicit solution formulas and is thus applicable to a broader class of (systems of) equations.