We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we...

In this paper, we introduce peridynamic theory and its application to Richards’ equation with a piecewise smooth initial condition. Peridynamic is non-local continuum that models the deformation failure of materials. describes unsaturated flow water through porous media, it plays an essential role in many applications, such as groundwater management, soil science, environmental engineering. We ...