  Let $\mathbb{F}_{q}$ be a finite field of cardinality $q$,
$R=\mathbb{F}_{q}[u]/\langle
u^4\rangle=\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}+u^3\mathbb{F}_{q}$
$(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer
satisfying ${\rm gcd}(q,n)=1$. For any $\delta,\alpha\in
\mathbb{F}_{q}^{\times}$, an explicit representation for all distinct
$(\delta+\alpha u^2)$-constacyclic codes over $R$ of length $n$ is given, and
the dual code for each of these codes is determined. For the case of $q=2^m$
and $\delta=1$, all self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of
odd length $n$ are provided.
