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1. 7. 2024.
SOME COSINE OPERATOR FUNCTIONS IN R^3
In this paper, we find out the cosine operator functions \begin{equation*}C(t)=\begin{bmatrix}a(t) & b(t) & c(t) \\0 & d(t) & e(t) \\0 & 0 & f(t)\end{bmatrix}\end{equation*}in a real vector space \(\mathbb{R}^3 \, ( t\in \mathbb{R})\), as the solutions of the second order Cauchy problem\begin{equation*}C''(t)=\mathcal{A}\cdot C(t),\, C(0)=I, \, C'(0)=0.\end{equation*} We find the solutions for the various cases of a given real matrix \begin{equation*}\mathcal{A}=\begin{bmatrix}A & B & C \\0 & D & E \\0 & 0 & F\end{bmatrix},\end{equation*} which is a generator of these cosine operator functions \(C(t), \, t\in \mathbb{R}\).