  We numerically solve a discretized model of Levy random walks on a finite
one-dimensional domain in the presence of sources and with a reflection
coefficient $r$. At the domain boundaries, the steady-state density profile is
non-analytic. The meniscus exponent $\mu$, introduced to characterize this
singular behavior, uniquely identifies the whole profile. Numerical data
suggest that $\mu =\alpha/2 + r(\alpha/2-1)$, where $\alpha$ is the Levy
exponent of the step-length distribution. As an application, we show that this
model reproduces the temperature profiles obtained for a chain of oscillators
displaying anomalous heat conduction. Remarkably, the case of free-boundary
conditions in the chain correspond to a Levy walk with negative reflection
coefficient.
