  First hitting times (FHTs) describe the time it takes a random "searcher" to
find a "target" and are used to study timescales in many applications. FHTs
have been well-studied for diffusive search, especially for small targets,
which is called the narrow capture or narrow escape problem. In this paper, we
study the first hitting time to small targets for a one-dimensional
superdiffusive search described by a Levy flight. By applying the method of
matched asymptotic expansions to a fractional differential equation we obtain
an explicit asymptotic expansion for the mean FHT (MFHT). For fractional order
$s\in(0,1)$ (describing a $(2s)$-stable Levy flight whose squared displacement
scales as $t^{1/s}$ in time $t$) and targets of radius $\varepsilon\ll1$, we
show that the MFHT is order one for $s\in(1/2,1)$ and diverges as
$\log(1/\varepsilon)$ for $s=1/2$ and $\varepsilon^{2s-1}$ for $s\in(0,1/2)$.
We then use our asymptotic results to identify the value of $s\in(0,1]$ which
minimizes the average MFHT and find that (a) this optimal value of $s$ vanishes
for sparse targets and (b) the value $s=1/2$ (corresponding to an inverse
square Levy search) is optimal in only very specific circumstances. We confirm
our results by comparison to both deterministic numerical solutions of the
associated fractional differential equation and stochastic simulations.
