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Meshfree methods based on radial basis function (RBF) approximation are ofinterest for numerical solution of partial differential equations (PDEs)because they are flexible with respect to the geometry of the computationaldomain, they can provide high order convergence, they are not more complicatedfor 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, asan example of an initial-boundary value problem with multiple boundaryconditions, can be implemented using RBF approximation methods. We extend thefictitious point method and the resampling method to work in combination withan RBF collocation method. Both approaches are implemented in one and two spacedimensions. The accuracy of the RBF fictitious point method is analysed partlytheoretically and partly numerically. The error estimates indicate that a highorder of convergence can be achieved for the Rosenau equation. The numericalexperiments show that both methods perform well. In the one-dimensional case,the accuracy of the RBF approaches is compared with that of a pseudospectralresampling method, showing similar or slightly better accuracy for the RBFmethods. In the two-dimensional case, the Rosenau problem is solved both in asquare domain and in a starfish-shaped domain, to illustrate the capability ofthe RBF-based methods to handle irregular geometries.

Radial basis function methods for the Rosenau equation and other higher order PDEs