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Let $\mathbb{F}_{q}$ be a finite field of cardinality $q$,$R=\mathbb{F}_{q}[u]/\langleu^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 integersatisfying ${\rm gcd}(q,n)=1$. 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, andthe 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 $R$ ofodd length $n$ are provided.

On a class of $(\delta+\alpha u^2)$-constacyclic codes over $\mathbb{F}_{q}[u]/\langle u^4\rangle$