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

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.

arXiv commentThis is the third updated version, which incorporates changes to the Introduction and Literature Review, as well as further substantiation of certain facts. The current manuscript consists of 23 pages, including the cover page, and contains 31 references