  In this paper I study a model for a growing surface in the presence of
anomalous diffusion, also known as the Fractal Kardar-Parisi-Zhang equation
(FKPZ). This equation includes a fractional Laplacian that accounts for the
possibility that surface transport is caused by a hopping mechanism of a Levy
flight. I show that for a specific choice of parameters of the FKPZ equation,
the equation can be solved exactly in one dimension, so that all the critical
exponents, which describe the surface that grows under FKPZ, can be derived for
that case. Afterwards, I use a Self-Consistent Expansion (SCE) to predict the
critical exponents for the FKPZ model for any choice of the parameters and any
spatial dimension. It is then verified that the results obtained using SCE
recover the exact result in one dimension. At the end a simple picture for the
behavior of the Fractal KPZ equation is suggested, and the upper critical
dimension of this model is discussed.
