Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting

Abstract

In this short note we consider a nonlinear and spatially nonlocal PDE modelling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We provide a deterministic derivation which allows us to naturally incorporate the nonlinear effects. This reasoning shows that in our setting the so-called nonlocal pressure form of the porous medium equation is preferred as a description of the evolution. In that case the governing nonlocal operator is the fractional gradient rather than the fractional Laplacian.

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Cited by in corpus (3)

• Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology
• Numerical approximations for fractional elliptic equations via the method of semigroups
• Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line