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 $δ,α\in \mathbb{F}_{q}^{\times}$, an explicit representation for all distinct $(δ+αu^2)$-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 $δ=1$, all self-dual $(1+αu^2)$-constacyclic codes over $R$ of odd length $n$ are provided.