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PreprintCC BY 4.0arXiv

On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations

  • Jemal Rogava
  • Zurab Vashakidze
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Abstract

The solution of the Cauchy problem for homogeneous abstract hyperbolic equations, together with its derivative, admits a vector representation in terms of a unitary operator group associated with a rotation matrix. A rational approximation of this unitary group is constructed and shown to possess optimal fourth-order convergence. The order of convergence is determined in accordance with the smoothness scale. Based on this rational approximation, a two-layer semi-discrete scheme is constructed for the approximate solution of Cauchy problems for nonhomogeneous abstract hyperbolic equations in both the linear and semilinear settings. The convergence properties of the scheme are examined in relation to the regularity of the solution.

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Cite

BibTeXRIS
BibTeX
@misc{rogava2026approximation,
  title = {{On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations}},
  author = {Jemal Rogava and Zurab Vashakidze},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.21293},
  eprint = {2607.21293},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.21293},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Approximation of the Unitary Operator Group Associated with a Rotation Matrix and Its Applications to Abstract Hyperbolic Equations
AU  - Jemal Rogava
AU  - Zurab Vashakidze
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2607.21293
UR  - https://arxiv.org/abs/2607.21293
AB  - The solution of the Cauchy problem for homogeneous abstract hyperbolic equations, together with its derivative, admits a vector representation in terms of a unitary operator group associated with a rotation matrix. A rational approximation of this unitary group is constructed and shown to possess optimal fourth-order convergence. The order of convergence is determined in accordance with the smoothness scale. Based on this rational approximation, a two-layer semi-discrete scheme is constructed for the approximate solution of Cauchy problems for nonhomogeneous abstract hyperbolic equations in both the linear and semilinear settings. The convergence properties of the scheme are examined in relation to the regularity of the solution.
ER  -