Blurred derivatives and meshless methods

In this work we rst introduce and describe the concept of blurred derivatives. It is shown how they can be used both to approximate di erential equations and to re-express them in alternative ways. In particular, formulations in terms of functional integrals can be obtained using blurred derivatives and extended to non-linear problems. Blurred derivatives are shown to provide higher exibility for selection of approximation functions than strong and weak formulations. Some computational implementations of one-dimensional problems are discussed and the relationship between several well-known numerical methods is analysed. Finally a meshless numerical scheme for the Poisson equation is described in detail. Its performance is compared with linear nite elements and generalized nite di erences on unstructured meshes of points