Computing solution operators of boundary-value problems for some linear hyperbolic systems of pdes

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Abstract

We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundaryvalue 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 ⊆ ℝm. Such systems describe a wide variety of physical processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many boundaryvalue problems for the wave equation also can be reduced to this case, thus we partially answer a question raised in [WZ02]. 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.

Original languageEnglish
Article number13
JournalLogical Methods in Computer Science
Volume13
Issue number4
DOIs
Publication statusPublished - 2017

Keywords

  • Algebraic real
  • Boundary-value problem
  • Cauchy problem
  • Computability
  • Constructive field
  • Difference scheme
  • Finite-dimensional approximation
  • Solution operator
  • Stability
  • Symmetric hyperbolic system
  • Systems of PDEs
  • Wave equation

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