  In this paper, we give conditions for the existence of Hermitian self-dual
$\Theta-$cyclic and $\Theta-$negacyclic codes over the finite chain ring
$\mathbb{F}_q+u\mathbb{F}_q$. By defining a Gray map from
$R=\mathbb{F}_q+u\mathbb{F}_q$ to $\mathbb{F}_{q}^{2}$, we prove that the Gray
images of skew cyclic codes of odd length $n$ over $R$ with even characteristic
are equivalent to skew quasi-twisted codes of length $2n$ over $\mathbb{F}_q$
of index $2$. We also extend an algorithm of Boucher and Ulmer \cite{BF3} to
construct self-dual skew cyclic codes based on the least common left multiples
of non-commutative polynomials over $\mathbb{F}_q+u\mathbb{F}_q$.
