  Meshfree methods based on radial basis function (RBF) approximation are of
interest for numerical solution of partial differential equations (PDEs)
because they are flexible with respect to the geometry of the computational
domain, they can provide high order convergence, they are not more complicated
for problems with many space dimensions and they allow for local refinement.
The aim of this paper is to show that the solution of the Rosenau equation, as
an example of an initial-boundary value problem with multiple boundary
conditions, can be implemented using RBF approximation methods. We extend the
fictitious point method and the resampling method to work in combination with
an RBF collocation method. Both approaches are implemented in one and two space
dimensions. The accuracy of the RBF fictitious point method is analysed partly
theoretically and partly numerically. The error estimates indicate that a high
order of convergence can be achieved for the Rosenau equation. The numerical
experiments show that both methods perform well. In the one-dimensional case,
the accuracy of the RBF approaches is compared with that of a pseudospectral
resampling method, showing similar or slightly better accuracy for the RBF
methods. In the two-dimensional case, the Rosenau problem is solved both in a
square domain and in a starfish-shaped domain, to illustrate the capability of
the RBF-based methods to handle irregular geometries.
