SOME COSINE OPERATOR FUNCTIONS IN R^3

Abstract

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 \(C(t), \, t\in \mathbb{R}\).