The modelling of advection-dispersion by fractional di erential equations

In this article, we introduce a fractional di erential equation model for advection-dispersions, that describe the Brownian motion of particles. In the following, wee concerns the existence of solutions to a fractional boundary-value problem. First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the rst eigenvalue is also given. Then under di erent assumptions on the nonlinearity we show the existence of weak solutions of the problem. Our main tools are variational methods and critical point theorems.