Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs

TL;DR

This paper demonstrates the computability of solution operators for certain symmetric hyperbolic PDE systems with boundary conditions, broadening the scope of numerical analysis methods in proving solutions' computability without explicit formulas.

Contribution

It proves the computability of solution operators for symmetric hyperbolic systems with boundary conditions, extending applicability beyond existing methods that rely on explicit solutions.

Findings

Proved computability of solution operators for symmetric hyperbolic systems.

Established computable dependence on system coefficients.

Extended applicability to a broad class of PDE systems.

Abstract

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.