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In this paper I study a model for a growing surface in the presence ofanomalous diffusion, also known as the Fractal Kardar-Parisi-Zhang equation(FKPZ). This equation includes a fractional Laplacian that accounts for thepossibility that surface transport is caused by a hopping mechanism of a Levyflight. 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 criticalexponents, which describe the surface that grows under FKPZ, can be derived forthat case. Afterwards, I use a Self-Consistent Expansion (SCE) to predict thecritical exponents for the FKPZ model for any choice of the parameters and anyspatial dimension. It is then verified that the results obtained using SCErecover the exact result in one dimension. At the end a simple picture for thebehavior of the Fractal KPZ equation is suggested, and the upper criticaldimension of this model is discussed.Comment: 11 page

Growing surfaces with anomalous diffusion: Results for the fractal Kardar-Parisi-Zhang equation