The fractional‐order governing equation of Lévy Motion

@article{Benson2000TheFG,  title={The fractional‐order governing equation of L{\'e}vy Motion},  author={David A. Benson and Stephen W. Wheatcraft and Mark M. Meerschaert},  journal={Water Resources Research},  year={2000},  volume={36},  pages={1413 - 1423},  url={https://api.semanticscholar.org/CorpusID:16579630}}
A governing equation of stable random walks is developed in one dimension. This Fokker‐Planck equation is similar to, and contains as a subset, the second‐order advection dispersion equation (ADE) except that the order (α) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Lévy's α‐stable densities that resemble the Gaussian except that they spread proportional to time1/α, have heavier tails, and incorporate any degree of skewness. The measured… 

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83 References

The Fractional Advection-Dispersion Equation: Development and Application A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Hydrogeology by

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