ABSTRACT
In this paper we will introduce a direct discretization method of collocation type for ordinary and partial differential equations, with possible extension to integral equations or equations with time lags. The resulting discretization is a general Hermite-Birkhoff interpolation problem in a certain space of functions. If the latter consists of radial basis functions, the solvability of the discrete system can be proven under very weak assumptions. By taking radial basis functions of sufficient smoothness and problems with sufficiently smooth solutions, the approximation order of the true solution by the discrete solution can be made arbitrarily large. For the special case of initial–value problems for ordinary differential equations, a proof of this fact is provided. Generalizations are to appear in a forthcoming paper.
